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-Earl_R._Babbie-_The_Practice_of_Social_Research_((BookFi)

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174 ■ Chapter 6: Indexes, Scales, and Typologies examine the relationship between the index and the 1’s, we find that a higher percentage of the the individual item. 2’s (91 percent) say “basic mechanisms” than the 1’s (16 percent). Index of Scientific Orientations 01 23 An item analysis of the other two components of the index yields similar results, as shown here. ?? ?? ?? ?? Percent who said they were Index of Scientific Orientations more interested in basic 01 2 3 mechanisms 0 4 14 100 If you take a minute to reflect on the table, you Percent who said they could may see that we already know the numbers that teach best as medical researchers 0 80 97 100 go in two of the cells. To get a score of 3 on the index, respondents had to say “basic mechanisms” Percent who said they in response to this question and give the “scientific” preferred reading about rationales answers to the other two items as well. Thus, 100 percent of the 3’s on the index said “basic Each of the items, then, seems an appropriate mechanisms.” By the same token, all the 0’s had to component in the index. Each seems to reflect the answer this item with “total patient management.” same quality that the index as a whole measures. Thus, 0 percent of those respondents said “basic mechanisms.” Here’s how the table looks with the In a complex index containing many items, this information we already know. step provides a convenient test of the independent contribution of each item to the index. If a given Percent who said they were Index of Scientific Orientations item is found to be poorly related to the index, it more interested in basic 01 2 3 may be assumed that other items in the index can- mechanisms cel out the contribution of that item, and it should 0 ?? ?? 100 be excluded from the index. If the item in question contributes nothing to the index’s power, it should If the individual item is a good reflection of the be excluded. overall index, we should expect the 1’s and 2’s to fill in a progression between 0 percent and 100 per- Although item analysis is an important first cent. More of the 2’s should choose “basic mecha- test of an index’s validity, it is not a sufficient test. nisms” than 1’s. This result is not guaranteed by If the index adequately measures a given variable, the way the index was constructed, however; it is it should successfully predict other indications of an empirical question— one we answer in an item that variable. To test this, we must turn to items not analysis. Here’s how this particular item analysis included in the index. turned out. External Validation Percent who said they were Index of Scientific Orientations more interested in basic 01 2 3 People scored as politically conservative on an in- mechanisms dex should appear conservative by other measures 0 16 91 100 as well, such as their responses to other items in a questionnaire. Of course, we’re talking about As you can see, in accord with our assumption relative conservatism, because we can’t define that the 2’s are more scientifically oriented than conservatism in any absolute way. However, those respondents scored as the most conservative on the index should score as the most conservative in an- swering other questions. Those scored as the least conservative on the index should score as the least conservative on other items. Indeed, the ranking of groups of respondents on the index should predict

Index Construction ■ 175 the ranking of those groups in answering other TABLE 6-1 questions dealing with political orientations. Validation of Scientific Orientation Index In our example of the scientific orientation Index of Scientific Orientation index, several questions in the questionnaire of- fered the possibility of such external validation. Low High Table 6-1 presents some of these items, which 0 12 3 provide several lessons regarding index validation. First, we note that the index strongly predicts the Percent interested in 34 42 46 65 responses to the validating items in the sense that attending scientific lectures the rank order of scientific responses among the at the medical school 43 60 65 89 four groups is the same as the rank order provided 0 8 32 66 by the index itself. That is, the percentages reflect Percent who say faculty 61 76 94 99 greater scientific orientation as you read across members should have the rows of the table. At the same time, each item experience as medical gives a different description of scientific orienta- researchers tion overall. For example, the last validating item indicates that the great majority of all faculty were Percent who would prefer engaged in research during the preceding year. If faculty duties involving this were the only indicator of scientific orienta- research activities only tion, we would conclude that nearly all faculty were scientific. Nevertheless, those scored as more Percent who engaged in scientific on the index are more likely to have research during the engaged in research than were those scored as preceding academic year relatively less scientific. The third validating item provides a different descriptive picture: Only a mi- second conclusion compelling. Typically, you’ll nority of the faculty overall say they would prefer feel you have included the best indicators of the duties limited exclusively to research. Nevertheless, variable in the index; the validating items are, the relative percentages giving this answer corre- therefore, second-rate indicators. Nevertheless, spond to the scores assigned on the index. you should recognize that the index is purportedly a very powerful measure of the variable; thus, it Bad Index versus Bad Validators should be somewhat related to any item that taps the variable even poorly. Nearly every index constructor at some time must face the apparent failure of external items When external validation fails, you should to validate the index. If the internal item analysis reexamine the index before deciding that the vali- shows inconsistent relationships between the items dating items are insufficient. One way to do this is included in the index and the index itself, some- to examine the relationships between the validat- thing is wrong with the index. But if the index fails ing items and the individual items included in the to predict strongly the external validation items, the index. If you discover that some of the index items conclusion to be drawn is more ambiguous. In this relate to the validators and others do not, you’ll situation we must choose between two possibili- have improved your understanding of the index as ties: (1) the index does not adequately measure the it was initially constituted. variable in question, or (2) the validation items do not adequately measure the variable and thereby There’s no cookbook solution to this problem; do not provide a sufficient test of the index. it is an agony serious researchers must learn to Having worked long and conscientiously on external validation The process of testing the the construction of an index, you’ll likely find the validity of a measure, such as an index or scale, by examining its relationship to other, presumed indi- cators of the same variable. If the index really mea- sures prejudice, for example, it should correlate with other indicators of prejudice.

176 ■ Chapter 6: Indexes, Scales, and Typologies survive. Ultimately, the wisdom of your decision to poorly on the GEM. Thus, while women were accept an index will be determined by the useful- doing fairly well in terms of income, education, ness of that index in your later analyses. Perhaps and life expectancy, they were still denied access to you’ll initially decide that the index is a good one power. And whereas the GDI scores were higher and that the validators are defective, but you’ll later in the wealthier nations than in the poorer ones, find that the variable in question (as measured by GEM scores showed that women’s empowerment the index) is not related to other variables in the depended less on national wealth, with many poor, ways you expected. You may then have to compose developing countries outpacing some rich, indus- a new index. trial ones in regard to such empowerment. The Status of Women: An By examining several different dimensions Illustration of Index Construction of the variables involved in their study, the UN researchers also uncovered an aspect of women’s For the most part, our discussion of index construc- earnings that generally goes unnoticed. Population tion has focused on the specific context of survey Communications International (1996: 1) summa- research, but other types of research also lend rizes the finding nicely: themselves to this kind of composite measure. For example, when the United Nations (1995) set out Every year, women make an invisible contribu- to examine the status of women in the world, they tion of eleven trillion U.S. dollars to the global chose to create two indexes, reflecting two different economy, the UNDP [United Nations Devel- dimensions. opment Programme] report says, counting both unpaid work and the underpayment of The Gender-related Development Index (GDI) women’s work at prevailing market prices. This compared women to men in terms of three indica- “underevaluation” of women’s work not only tors: life expectancy, education, and income. These undermines their purchasing power, says the indicators are commonly used in monitoring the 1995 HDR [Human Development Report], but status of women in the world. The Scandinavian also reduces their already low social status and countries of Norway, Sweden, Finland, and Den- affects their ability to own property and use mark ranked highest on this measure. credit. Mahbub ul Haq, the principal author of the report, says that “if women’s work were The second index, the Gender Empowerment accurately reflected in national statistics, it Measure (GEM), aimed more at power issues and would shatter the myth that men are the main comprised three different indicators: breadwinner of the world.” The UNDP report finds that women work longer hours than men • The proportion of parliamentary seats held by in almost every country, including both paid and unpaid duties. In developing countries, women women do approximately 53% of all work and spend two-thirds of their work time on • The proportion of administrative, managerial, unremunerated activities. In industrialized countries, women do an average of 51% of professional, and technical positions held by the total work, and—like their counterparts women in the developing world—perform about two- thirds of their total labor without pay. Men in • A measure of access to jobs and wages industrialized countries are compensated for two-thirds of their work. Once again, the Scandinavian countries ranked high but were joined by Canada, New Zealand, the “Indexing the World” gives some other exam- Netherlands, the United States, and Austria. Having ples of indexes that have been created to monitor two different measures of gender equality rather the state of the world. than one allowed the researchers to make more- sophisticated distinctions. For example, in several countries, most notably Greece, France, and Japan, women fared relatively well on the GDI but quite

Scale Construction ■ 177 “Indexing the World” If you browse the web in search of indexes,you’ll be handsomely The well-being of America’s young people is the focus of the rewarded.Here are just a few examples of the ways in which people Child and Youth Well-Being Index,housed at Duke University. have used the logic of social indexes to monitor the state of the world. Go to http://www.cengage.com/sociology/babbie for links to each of Money Magazine has indexed the 100 best places to live in the following examples. America,using factors such as economics,housing,schools,health, crime,weather,and public facilities. The well-being of nations is commonly measured in economic terms,such as the Gross Domestic Product per capita,average income,or The Heritage Foundation offers the Index of Economic Freedom stock market averages.In 1972,however,the mountainous kingdom of for those planning business ventures around the world. Bhutan drew global attention by proposing an index of“Gross National Happiness,”augmenting economic factors with measures of physical For Christians who believe in prophecies of the end of times,the and mental health,freedom,environment,marital stability,and other Rapture Index uses 45 indicators—including inflation,famine,floods, indicators of noneconomic well-being.The World Data Base of Happiness liberalism,and Satanism—and offers a gauge of how close or far expands this general idea to 24 countries. away the end is. Columbia University’s Environmental Sustainability Index is one of Can you find other,similar indexes online? several measures that seek to monitor the environmental impact of the nations of the planet. As you can see, indexes can be constructed Scales offer more assurance of ordinality from many different kinds of data for a variety of by tapping the intensity structures among the purposes. Now we’ll turn our attention from the indicators. The several items going into a com- construction of indexes to an examination of scal- posite measure may have different intensities in ing techniques. terms of the variable. Many methods of scaling are available. We’ll look at four scaling procedures to Scale Construction illustrate the variety of techniques available, along with a technique called the semantic differential. Good indexes provide an ordinal ranking of cases Although these examples focus on questionnaires, on a given variable. All indexes are based on this the logic of scaling, like that of indexing, applies to kind of assumption: A senator who voted for other research methods as well. seven conservative bills is considered to be more conservative than one who voted for only four of Bogardus Social Distance Scale them. What an index may fail to take into account, however, is that not all indicators of a variable are Let’s suppose you’re interested in the extent to equally important or equally strong. The first sena- which U.S. citizens are willing to associate with, tor might have voted in favor of seven mildly con- say, sex offenders. You might ask the following servative bills, whereas the second senator might questions: have voted in favor of four extremely conservative bills. (The second senator might have considered 1. Are you willing to permit sex offenders to live the other seven bills too liberal and voted against in your country? them.) 2. Are you willing to permit sex offenders to live in your community?

178 ■ Chapter 6: Indexes, Scales, and Typologies 3. Are you willing to permit sex offenders to live The Bogardus social distance scale illustrates in your neighborhood? the important economy of scaling as a data-reduc- tion device. By knowing how many relationships 4. Would you be willing to let a sex offender live with sex offenders a given respondent will accept, next door to you? we know which relationships were accepted. Thus, a single number can accurately summarize five or 5. Would you let your child marry a sex offender? six data items without a loss of information. These questions increase in terms of the close- Motoko Lee, Stephen Sapp, and Melvin Ray ness of contact with sex offenders. Beginning (1996) noticed an implicit element in the Bogardus with the original concern to measure willingness social distance scale: It looks at social distance from to associate with sex offenders, you have thus the point of view of the majority group in a society. developed several questions indicating differing These researchers decided to turn the tables and degrees of intensity on this variable. The kinds create a “reverse social distance” scale: looking at of items presented constitute a Bogardus social social distance from the perspective of the minor- distance scale (created by Emory Bogardus). This ity group. Here’s how they framed their questions scale is a measurement technique for determining (1996: 19): the willingness of people to participate in social relations— of varying degrees of closeness—with Considering typical Caucasian Americans you other kinds of people. have known, not any specific person nor the worst or the best, circle Y or N to express your The clear differences of intensity suggest a opinion. structure among the items. Presumably if a person is willing to accept a given kind of association, he Y N 5. Do they mind your being a citizen in this or she would be willing to accept all those preced- country? ing it in the list—those with lesser intensities. For example, the person who is willing to permit sex Y N 4. Do they mind your living in the same offenders to live in the neighborhood will surely neighborhood? accept them in the community and the nation but may or may not be willing to accept them as next- Y N 3. Would they mind your living next door neighbors or relatives. This, then, is the logical to them? structure of intensity inherent among the items. Y N 2. Would they mind your becoming a close Empirically, one would expect to find the larg- friend to them? est number of people accepting co-citizenship and the fewest accepting intermarriage. In this sense, Y N 1. Would they mind your becoming their we speak of “easy items” (for example, residence in kin by marriage? the United States) and “hard items” (for example, intermarriage). More people agree to the easy items As with the original scale, the researchers found than to the hard ones. With some inevitable excep- that knowing the number of items minority tions, logic demands that once a person has refused respondents agreed with also told the researchers a relationship presented in the scale, he or she will which ones were agreed with, 98.9 percent of the also refuse all the harder ones that follow it. time in this case. Bogardus social distance scale A measurement Thurstone Scales technique for determining the willingness of people to participate in social relations— of varying degrees Often, the inherent structure of the Bogardus social of closeness—with other kinds of people. It is an distance scale is not appropriate to the variable especially efficient technique in that one can sum- being measured. Indeed, such a logical structure marize several discrete answers without losing any among several indicators is seldom apparent. of the original details of the data. A Thurstone scale (created by Louis Thurstone) is an attempt to develop a format for generating groups of indicators of a variable that have at least

Scale Construction ■ 179 an empirical structure among them. A group of meanings conveyed by the several items indicating judges is given perhaps a hundred items that are a given variable tend to change over time. Thus, an thought to be indicators of a given variable. Each item having a given weight at one time might have judge is then asked to estimate how strong an quite a different weight later on. For a Thurstone indicator of a variable each item is—by assign- scale to be effective, it would have to be updated ing scores of perhaps 1 to 13. If the variable were periodically. prejudice, for example, the judges would be asked to assign the score of 1 to the very weakest indica- Likert Scaling tors of prejudice, the score of 13 to the strongest indicators, and intermediate scores to those felt to You may sometimes hear people refer to a ques- be somewhere in between. tionnaire item containing response categories such as “strongly agree,” “agree,” “disagree,” Once the judges have completed this task, the and “strongly disagree” as a Likert scale. This is researcher examines the scores assigned to each technically a misnomer, although Rensis Likert item by all the judges, then determines which (pronounced “LICK-ert”) did create this com- items produced the greatest agreement among the monly used question format. Likert also created a judges. Those items on which the judges disagreed technique for combining the items into a scale, but broadly would be rejected as ambiguous. Among while Likert’s scaling technique is rarely used, his those items producing general agreement in scor- answer format is one of the most frequently used ing, one or more would be selected to represent in survey research. each scale score from 1 to 13. The particular value of this format is the The items selected in this manner might then unambiguous ordinality of response categories. If be included in a survey questionnaire. Respondents respondents were permitted to volunteer or select who appeared prejudiced on those items repre- such answers as “sort of agree,” “pretty much senting a strength of 5 would then be expected to agree,” “really agree,” and so forth, you would find appear prejudiced on those having lesser strengths, it impossible to judge the relative strength of agree- and if some of those respondents did not appear ment intended by the various respondents. The prejudiced on the items with a strength of 6, it Likert format solves this problem. would be expected that they would also not appear prejudiced on those with greater strengths. Though seldom used, Likert’s scaling method is fairly easy to understand, based on the relative If the Thurstone scale items were adequately intensity of different items. As a simple example, developed and scored, the economy and effective- suppose we wish to measure prejudice against ness of data reduction inherent in the Bogardus women. To do this, we create a set of 20 state- social distance scale would appear. A single score ments, each of which reflects that prejudice. One might be assigned to each respondent (the strength of the items might be “Women can’t drive as well of the hardest item accepted), and that score would as men.” Another might be “Women shouldn’t be adequately represent the responses to several allowed to vote.” Likert’s scaling technique would questionnaire items. And as is true of the Bogar- demonstrate the difference in intensity between dus scale, a respondent who scored 6 might be these items as well as pegging the intensity of the regarded as more prejudiced than one who scored other 18 statements. 5 or less. Thurstone scale A type of composite measure, Thurstone scaling is not often used in research constructed in accord with the weights assigned by today, primarily because of the tremendous expen- “judges” to various indicators of some variables. diture of energy and time required to have 10 to 15 judges score the items. Because the quality of their judgments would depend on their experience with the variable under consideration, they might need to be professional researchers. Moreover, the

180 ■ Chapter 6: Indexes, Scales, and Typologies Let’s suppose we ask a sample of people to agree used in the creation of simple indexes. With, say, or disagree with each of the 20 statements. Simply five response categories (including “no opinion” giving one point for each of the indicators of preju- or something similar), scores of 0 to 4 or 1 to 5 dice against women would yield the possibility of might be assigned, taking the direction of the items index scores ranging from 0 to 20. A true Likert into account (for example, assign a score of 5 to scale goes one step beyond that and calculates “strongly agree” for positive items and to “strongly the average index score for those agreeing with disagree” for negative items). Each respondent each of the individual statements. Let’s say that all would then be assigned an overall score represent- those who agreed that women are poorer drivers ing the summation of the scores he or she received than men had an average index score of 1.5 (out for responses to the individual items. of a possible 20). Those who agreed that women should be denied the right to vote might have Semantic Differential an average index score of, say, 19.5—indicating the greater degree of prejudice reflected in that Like the Likert format, the semantic differential response. asks questionnaire respondents to choose between two opposite positions by using qualifiers to bridge As a result of this item analysis, respondents the distance between the two opposites. Here’s how could be rescored to form a scale: 1.5 points for it works. agreeing that women are poorer drivers, 19.5 points for saying women shouldn’t vote, and points Suppose you’re evaluating the effectiveness for other responses reflecting how those items of a new music-appreciation lecture on subjects’ related to the initial, simple index. If those who appreciation of music. As a part of your study, you disagreed with the statement “I might vote for a want to play some musical selections and have the woman for president” had an average index score subjects report their feelings about them. A good of 15, then the scale would give 15 points to people way to tap those feelings would be to use a seman- disagreeing with that statement. tic differential format. As I’ve said earlier, Likert scaling is seldom used To begin, you must determine the dimensions today. The item format devised by Likert, however, along which subjects should judge each selection. is one of the most commonly used formats in con- Then you need to find two opposite terms, repre- temporary questionnaire design. Typically, it is now senting the polar extremes along each dimension. Let’s suppose one dimension that interests you is Likert scale A type of composite measure devel- simply whether subjects enjoyed the piece or not. oped by Rensis Likert, in an attempt to improve the Two opposite terms in this case could be “enjoy- levels of measurement in social research through able” and “unenjoyable.” Similarly, you might want the use of standardized response categories in survey to know whether they regarded the individual questionnaires, to determine the relative intensity of selections as “complex” or “simple,” “harmonic” or different items. Likert items are those using such re- “discordant,” and so forth. sponse categories as strongly agree, agree, disagree, and strongly disagree. Such items may be used in Once you have determined the relevant di- the construction of true Likert scales as well as other mensions and have found terms to represent the types of composite measures. extremes of each, you might prepare a rating sheet each subject would complete for each piece of mu- semantic differential A questionnaire format in sic. Figure 6-5 shows what it might look like. which the respondent is asked to rate something in terms of two, opposite adjectives (e.g., rate textbooks On each line of the rating sheet, the subject as “boring” or “exciting”), using qualifiers such as would indicate how he or she felt about the piece “very,” “somewhat,” “neither,” “somewhat,” and of music: whether it was enjoyable or unenjoyable, “very” to bridge the distance between the two op- for example, and whether it was “somewhat” that posites. way or “very much” so. To avoid creating a biased pattern of responses to such items, it’s a good idea

Scale Construction ■ 181 FIGURE 6-5 Semantic Differential: Feelings about Musical Selections. The semantic differential asks respondents to describe something or someone in terms of opposing adjectives. to vary the placement of terms that are likely to Earlier, when we talked about attitudes re- be related to each other. Notice, for example, that garding a woman’s right to have an abortion, we “discordant” and “traditional” are on the left side of discussed several conditions that can affect people’s the sheet, with “harmonic” and “modern” on the opinions: whether the woman is married, whether right. Most likely, those selections scored as “dis- her health is endangered, and so forth. These dif- cordant” would also be scored as “modern” rather fering conditions provide an excellent illustration of than “traditional.” Guttman scaling. Both the Likert and semantic differential for- Here are the percentages of the people in the mats have a greater rigor and structure than other 2000 GSS sample who supported a woman’s right question formats do. As I indicated earlier, these to an abortion, under three different conditions: formats produce data suitable to both indexing and scaling. Woman’s health is seriously endangered 89% Pregnant as a result of rape 81% Guttman Scaling Woman is not married 39% Researchers today often use the scale developed The different percentages supporting abor- by Louis Guttman. Like Bogardus, Thurstone, and tion under the three conditions suggest something Likert scaling, Guttman scaling is based on the fact about the different levels of support that each item that some items under consideration may prove to indicates. For example, if someone supported abor- be more-extreme indicators of the variable than tion when the mother’s life is seriously endangered, others. Here’s an example to illustrate this pattern. that’s not a very strong indicator of general support for abortion, because almost everyone agreed with In the earlier example of measuring scientific that. Supporting abortion for unmarried women orientation among medical school faculty members, seems a much stronger indicator of support for you’ll recall that a simple index was constructed. abortion in general—fewer than half the sample As it happens, however, the three items included in took that position. the index essentially form a Guttman scale. Guttman scaling is based on the idea that any- The construction of a Guttman scale begins one who gives a strong indicator of some variable with some of the same steps that initiate index con- will also give the weaker indicators. In this case, struction. You begin by examining the face validity we would assume that anyone who supported of items available for analysis. Then, you examine the bivariate and perhaps multivariate relations Guttman scale A type of composite measure used among those items. In scale construction, however, to summarize several discrete observations and to you also look for relatively “hard” and “easy” indi- represent some more-general variable. cators of the variable being examined.

182 ■ Chapter 6: Indexes, Scales, and Typologies TABLE 6-2 (It would be extremely rare for such data to form a Scaling Support for Choice of Abortion Guttman scale perfectly.) Scale types Women’s Result Woman Number Recall at this point that one of the chief func- Health of Rape Unmarried of Cases tions of scaling is efficient data reduction. Scales provide a technique for presenting data in a sum- ϩ ϩ ϩ 677 mary form while maintaining as much of the origi- ϩ ϩ Ϫ 607 nal information as possible. When the scientific ϩ Ϫ Ϫ 165 orientation items were formed into an index in Ϫ Ϫ Ϫ 147 our earlier discussion, respondents were given one point for each scientific response they gave. If these Mixed types Ϫ ϩ Total ϭ 1,596 same three items were scored as a Guttman scale, ϩ Ϫ Ϫ 42 some respondents would be assigned scale scores Ϫ Ϫ ϩ5 that would permit the most accurate reproduction Ϫ ϩ ϩ2 of their original responses to all three items. ϩ4 In the present example of attitudes regarding Total ϭ 53 abortion, respondents fitting into the scale types would receive the same scores as would be assigned abortion for unmarried women would also support in the construction of an index. Persons selecting it in the case of rape or of the woman’s health being all three pro-choice responses (ϩ ϩ ϩ) would threatened. Table 6-2 tests this assumption by pre- still be scored 3, those who selected pro-choice senting the number of respondents who gave each responses to the two easier items and were opposed of the possible response patterns. on the hardest item (ϩ ϩ Ϫ) would be scored 2, and so on. For each of the four scale types we could The first four response patterns in the table predict accurately all the actual responses given by compose what we would call the scale types: those all the respondents based on their scores. patterns that form a scalar structure. Following those respondents who supported abortion under The mixed types in the table present a problem, all three conditions (line 1), we see (line 2) that however. The first mixed type (Ϫ ϩ Ϫ) was those with only two pro-choice responses have scored 1 on the index to indicate only one pro- chosen the two easier ones; those with only one choice response. But, if 1 were assigned as a scale such response (line 3) chose the easiest of the score, we would predict that the 42 respondents three (the woman’s health being endangered). And in this group had chosen only the easiest item finally, there are some respondents who opposed (approving abortion when the woman’s life was abortion in all three circumstances (line 4). endangered), and we would be making two errors for each such respondent: thinking their response The second part of the table presents those pattern was (ϩ Ϫ Ϫ) instead of (Ϫ ϩ Ϫ). Scale response patterns that violate the scalar structure scores are assigned, therefore, with the aim of of the items. The most radical departures from the minimizing the errors that would be made in re- scalar structure are the last two response patterns: constructing the original responses. those who accepted only the hardest item and those who rejected only the easiest one. Table 6-3 illustrates the index and scale scores that would be assigned to each of the response The final column in the table indicates the patterns in our example. Note that one error is number of survey respondents who gave each of made for each respondent in the mixed types. This the response patterns. The great majority (1,596, is the minimum we can hope for in a mixed-type or 99 percent) fit into one of the scale types. The pattern. In the first mixed type, for example, we presence of mixed types, however, indicates that would erroneously predict a pro-choice response to the items do not form a perfect Guttman scale. the easiest item for each of the 42 respondents in this group, making a total of 42 errors.

Typologies ■ 183 TABLE 6-3 structed in fact measures the concept under con- Index and Scale Scores sideration. What it does is increase confidence that all the component items measure the same thing. Response Number Index Scale Total Also, you should realize that a high coefficient of Pattern of Cases Scores Scores Scale Errors reproducibility is most likely when few items are involved. Scale Types ϩ ϩ ϩ 677 33 0 ϩϩϩ 607 One concluding remark with regard to Gutt- ϩϪϪ 165 22 0 man scaling: It’s based on the structure observed ϪϪϪ 147 among the actual data under examination. This is 42 11 0 an important point that is often misunderstood. It Mixed Types Ϫ ϩ Ϫ does not make sense to say that a set of question- ϩϪϩ 5 00 0 naire items (perhaps developed and used by a ϪϪϩ 2 previous researcher) constitutes a Guttman scale. Ϫϩϩ 4 1 2 42 Rather, we can say only that they form a scale within a given body of data being analyzed. Scal- 23 5 ability, then, is a sample-dependent, empirical mat- ter. Although a set of items may form a Guttman 10 2 scale among one sample of survey respondents, for example, there is no guarantee that this set will 23 4 form such a scale among another sample. In this sense, then, a set of questionnaire items in and Total Scale errors ϭ 53 of itself never forms a scale, but a set of empirical observations may. Coefficient of reproducibility ϭ 1 Ϫ number of errors number of guesses This concludes our discussion of indexing and scaling. Like indexes, scales are composite measures This table presents one common method for scoring mixed types,but you should of a variable, typically broadening the meaning of be advised that other methods are also used. the variable beyond what might be captured by a single indicator. Both scales and indexes seek to The extent to which a set of empirical re- measure variables at the ordinal level of measure- sponses form a Guttman scale is determined by the ment. Unlike indexes, however, scales take advan- accuracy with which the original responses can tage of any intensity structure that may be present be reconstructed from the scale scores. For each among the individual indicators. To the extent that of the 1,649 respondents in this example, we’ll such an intensity structure is found and the data predict three questionnaire responses, for a total from the people or other units of analysis comply of 4,947 predictions. Table 6-3 indicates that we’ll with the logic of that intensity structure, we can make 53 errors using the scale scores assigned. have confidence that we have created an ordinal The percentage of correct predictions is called the measure. coefficient of reproducibility: the percentage of original responses that could be reproduced by knowing Typologies the scale scores used to summarize them. In the present example, the coefficient of reproducibility Indexes and scales, then, are constructed to provide is 4,894/4,947, or 98.9 percent. ordinal measures of given variables. We attempt to assign index or scale scores to cases in such a way Except for the case of perfect (100 percent) re- as to indicate a rising degree of prejudice, religios- producibility, there is no way of saying that a set of ity, conservatism, and so forth. In such cases, we’re items does or does not form a Guttman scale in any dealing with single dimensions. absolute sense. Virtually all sets of such items ap- proximate a scale. As a general guideline, however, coefficients of 90 or 95 percent are the commonly used standards. If the observed reproducibility exceeds the level you’ve set, you’ll probably decide to score and use the items as a scale. The decision concerning criteria in this regard is, of course, arbitrary. Moreover, a high degree of reproducibility does not insure that the scale con-

184 ■ Chapter 6: Indexes, Scales, and Typologies TABLE 6-4 extraction of elements derived from specific ex- A Political Typology of Newspapers amples, provide a theoretical model by which and from which we may examine reality” (2006: 87). Foreign Policy Conservative Liberal Frequently, you arrive at a typology in the course of an attempt to construct an index or scale. Domestic Policy Conservative A B The items that you felt represented a single vari- Liberal C D able appear to represent two. We might have been attempting to construct a single index of political Often, however, the researcher wishes to sum- orientations for newspapers but discovered—em- marize the intersection of two or more variables, pirically—that foreign and domestic politics had to thereby creating a set of categories or types—a be kept separate. nominal variable—called a typology. You may, for example, wish to examine the political orienta- In any event, you should be warned against a tions of newspapers separately in terms of domestic difficulty inherent in typological analysis. When- issues and foreign policy. The fourfold presentation ever the typology is used as the independent in Table 6-4 describes such a typology. variable, there will probably be no problem. In the preceding example, you might compute the Newspapers in cell A of the table are conserva- percentages of newspapers in each cell that nor- tive on both foreign policy and domestic policy; mally endorse Democratic candidates; you could those in cell D are liberal on both. Those in cells B then easily examine the effects of both foreign and and C are conservative on one and liberal on the domestic policies on political endorsements. other. It’s extremely difficult, however, to analyze a As another example, Rodney Coates (2006) typology as a dependent variable. If you want to created a typology of “racial hegemony” from two discover why newspapers fall into the different cells dimensions: of typology, you’re in trouble. That becomes ap- parent when we consider the ways you might con- 1. Political Ideology struct and read your tables. Assume, for example, that you want to examine the effects of community a. Democratic size on political policies. With a single dimension, you could easily determine the percentages of rural b. Nondemocratic and urban newspapers that were scored conserva- tive and liberal on your index or scale. 2. Military and Industrial Sophistication With a typology, however, you would have to a. Low present the distribution of the urban newspapers in your sample among types A, B, C, and D. Then you b. High would repeat the procedure for the rural ones in the sample and compare the two distributions. Let’s He then used the typology to examine modern suppose that 80 percent of the rural newspapers examples of colonial rule, with specific reference are scored as type A (conservative on both dimen- to race relations. The specific cases he examined sions), compared with 30 percent of the urban allowed him to illustrate and refine the typology. ones. Moreover, suppose that only 5 percent of the He points out that such a device represents Max rural newspapers are scored as type B (conserva- Weber’s “ideal type”: “As stipulated by Weber, ideal tive only on domestic issues), compared with 40 types represent a type of abstraction from reality. percent of the urban ones. It would be incorrect to These abstractions, constructed from the logical conclude from an examination of type B that urban newspapers are more conservative on domestic typology The classification (typically nominal) of issues than rural ones are, because 85 percent of observations in terms of their attributes on two or the rural newspapers, compared with 70 percent of more variables. The classification of newspapers as liberal-urban, liberal-rural, conservative-urban, or conservative-rural would be an example.

Main Points ■ 185 the urban ones, have this characteristic. The rela- which a dimension is to be measured, and the tive sparsity of rural newspapers in type B is due to amount of variance provided by the items. their concentration in type A. It should be apparent that an interpretation of such data would be very • If different items are indeed indicators of the same difficult for anything other than description. variable, then they should be related empirically In reality, you’d probably examine two such to one another. In constructing an index, the dimensions separately, especially if the dependent researcher needs to examine bivariate and multi- variable has more categories of responses than the variate relationships among the items. given example does. • Index scoring involves deciding the desirable Don’t think that typologies should always be avoided in social research; often they provide the range of scores and determining whether items most appropriate device for understanding the will have equal or different weights. data. To examine the pro-life orientation in depth, for example, you might create a typology involving • There are various techniques that allow items to both abortion and capital punishment. Libertarian- ism could be seen in terms of both economic and be used in an index in spite of missing data. social permissiveness. You’ve now been warned, however, against the special difficulties involved in • Item analysis is a type of internal validation, based using typologies as dependent variables. on the relationship between individual items in MAIN POINTS the composite measure and the measure itself. External validation refers to the relationships be- Introduction tween the composite measure and other indicators of the variable—indicators not included in the • Single indicators of variables seldom (1) cap- measure. ture all the dimensions of a concept, (2) have Scale Construction sufficiently clear validity to warrant their use, or (3) permit the desired range of variation to al- • Four types of scaling techniques are represented low ordinal rankings. Composite measures, such as scales and indexes, solve these problems by by the Bogardus social distance scale, a device including several indicators of a variable in one for measuring the varying degrees to which a summary measure. person would be willing to associate with a given class of people; Thurstone scaling, a technique Indexes versus Scales that uses judges to determine the intensities of different indicators; Likert scaling, a measurement • Although both indexes and scales are intended technique based on the use of standardized re- sponse categories; and Guttman scaling, a method as ordinal measures of variables, scales typically of discovering and using the empirical intensity satisfy this intention better than indexes do. structure among several indicators of a given variable. Guttman scaling is probably the most • Whereas indexes are based on the simple cumula- popular scaling technique in social research today. tion of indicators of a variable, scales take advan- tage of any logical or empirical intensity structures • The semantic differential is a question format that that exist among a variable’s indicators. asks respondents to make ratings that lie between Index Construction two extremes, such as “very positive” and “very negative.” • The principal steps in constructing an index Typologies include selecting possible items, examining their empirical relationships, scoring the index, and • A typology is a nominal composite measure often validating it. used in social research. Typologies may be used • Criteria of item selection include face validity, effectively as independent variables, but interpre- tation is difficult when they are used as dependent unidimensionality, the degree of specificity with variables. KEY TERMS The following terms are defined in context in the chapter and at the bottom of the page where the term

186 ■ Chapter 6: Indexes, Scales, and Typologies is introduced, as well as in the comprehensive glossary 5. Economists often use indexes to measure eco- at the back of the book. nomic variables, such as the cost of living. Go to the Bureau of Labor Statistics link on this book’s Bogardus social distance Likert scale website and find the Consumer Price Index scale scale survey. What are some of the dimensions of living external validation semantic differential costs included in this measure? Guttman scale Thurstone scale index typology SPSS EXERCISES item analysis See the booklet that accompanies your text for ex- PROPOSING SOCIAL RESEARCH: ercises using SPSS (Statistical Package for the Social COMPOSITE MEASURES Sciences). There are exercises offered for each chapter, and you’ll also find a detailed primer on using SPSS. This chapter has extended the issue of measurement to include those in which variables are measured by Online Study Resources more than one indicator. What you have learned here may extend the discussion of measurement in your If your book came with an access code card, visit proposal. As in the case of operationalization, you may www.cengage.com/login to register. To purchase find this easier to formulate in the case of quantitative access, please visit www.ichapters.com. studies, but the logic of multiple indicators may be ap- 1. Before you do your final review of the chapter, plied to all research methods. take the CengageNOW pretest to help identify the If your study will involve the use of composite areas on which you should concentrate. You’ll measures, you should identify the type(s), the indica- find information on this online tool, as well as tors to be used in their construction, and the methods instructions on how to access all of its great re- you’ll use to create and validate them. If the study you sources, in the front of the book. are planning in this series of exercises will not include 2. As you review, take advantage of the CengageNOW composite measures, you can test your understand- personalized study plan, based on your quiz ing of the chapter by exploring ways in which they results. Use this study plan with its interactive ex- could be used, even if you need to temporarily vary ercises and other resources to master the material. the data-collection method and/or variables you have 3. When you’re finished with your review, take the in mind. posttest to confirm that you’re ready to move on to the next chapter. REVIEW QUESTIONS AND EXERCISES WEBSITE FOR THE PRACTICE 1. In your own words, describe the difference OF SOCIAL RESEARCH 12TH EDITION between an index and a scale. Go to your book’s website at www.cengage.com/ 2. Suppose you wanted to create an index for rating sociology/babbie for tools to aid you in studying for the quality of colleges and universities. Name your exams. You’ll find Tutorial Quizzes with feedback, three data items that might be included in such an Internet Exercises, Flash Cards, Glossaries, and Essay Quiz- index. zes, as well as InfoTrac College Edition search terms, sug- gestions for additional reading, Web Links, and primers 3. Make up three questionnaire items that measure for using data-analysis software such as SPSS. attitudes toward nuclear power and that would probably form a Guttman scale. 4. Construct a typology of pro-life attitudes as dis- cussed in the chapter.

CHAPTER SEVEN The Logic of Sampling CHAPTER OVERVIEW Now you’ll see how social scientists can select a few people for study— and discover things that apply to hundreds of millions of people not studied. Introduction Populations and Sampling Frames Review of Populations and A Brief History of Sampling Sampling Frames President Alf Landon President Thomas E. Dewey Types of Sampling Designs Two Types of Sampling Methods Simple Random Sampling Systematic Sampling Nonprobability Sampling Stratified Sampling Reliance on Available Subjects Implicit Stratification in Purposive or Judgmental Systematic Sampling Sampling Illustration: Sampling University Snowball Sampling Students Quota Sampling Selecting Informants Multistage Cluster Sampling Multistage Designs and Sampling The Theory and Logic of Error Probability Sampling Stratification in Multistage Cluster Sampling Conscious and Unconscious Probability Proportionate to Size Sampling Bias (PPS) Sampling Representativeness and Disproportionate Sampling and Probability of Selection Weighting Random Selection Probability Theory, Sampling Probability Sampling in Review Distributions, and Estimates of Sampling Error The Ethics of Sampling CengageNOW for Sociology Use this online tool to help you make the grade on your next exam. After reading this chapter, go to “Online Study Resources” at the end of the chapter for instructions on how to benefit from CengageNOW.

188 ■ Chapter 7: The Logic of Sampling Introduction TABLE 7-1 Election Eve Polls Reporting Percentage of Population One of the most visible uses of survey sampling lies Voting for U.S.Presidential Candidates,2004 in the political polling that is subsequently tested by election results. Whereas some people doubt the Poll Date Begun Bush Kerry accuracy of sample surveys, others complain that political polls take all the suspense out of cam- Fox/OpinDynamics Oct 28 50 50 paigns by foretelling the result. In recent presiden- tial elections, however, the polls have not removed TIPP Oct 28 53 47 the suspense. CBS/NYT Oct 28 52 48 Going into the 2004 presidential elections, pollsters generally agreed that the election was ARG Oct 28 50 50 “too close to call,” a repeat of their experience four years earlier. The Roper Center has compiled a ABC Oct 28 51 49 list of polls conducted throughout the campaign; Table 7-1 reports those conducted during the few Fox/OpinDynamics Oct 29 49 51 days preceding the election. Despite some varia- tions, the overall picture they present is amaz- Gallup/CNN/USA Oct 29 49 51 ingly consistent and was played out in the election results. NBC/WSJ Oct 29 51 49 Now, how many interviews do you suppose it TIPP Oct 29 51 49 took each of these pollsters to come within a cou- ple of percentage points in estimating the behavior Harris Oct 29 52 48 of more than 115 million voters? Often fewer than 2,000! In this chapter, we’re going to find out how Democracy Corps Oct 29 49 51 social researchers can pull off such wizardry. Harris Oct 29 51 49 For another powerful illustration of the potency of sampling, look at this graphic portrayal of Presi- CBS Oct 29 51 49 dent George W. Bush’s approval ratings prior to and following the September 11, 2001, terrorist attack Fox/OpinDynamics Oct 30 49 52 on the U.S. (see Figure 7-1). The data reported by several different polling agencies describe the same TIPP Oct 30 51 49 pattern. Marist Oct 31 50 50 Political polling, like other forms of social re- search, rests on observations. But neither pollsters GWU Battleground 2004 Oct 31 52 48 nor other social researchers can observe everything that might be relevant to their interests. A critical Actual vote Nov 2 52 48 part of social research, then, is deciding what to observe and what not. If you want to study voters, Source:Poll data adapted from the Roper Center,Election 2004 (http:// for example, which voters should you study? www.ropercenter.uconn.edu/elect_2004/pres_trial_heats.html), accessed November 16,2004.I’ve apportioned the undecided and other votes according The process of selecting observations is called to the percentages saying they were voting for Bush or Kerry. sampling. Although sampling can mean any pro- cedure for selecting units of observation—for ex- a larger population is probability sampling, which ample, interviewing every tenth passerby on a busy involves the important idea of random selection. street—the key to generalizing from a sample to Much of this chapter is devoted to the logic and skills of probability sampling. This topic is more rigorous and precise than some of the other topics in this book. Whereas social research as a whole is both art and science, sampling leans toward science. Although this subject is somewhat technical, the basic logic of sampling is not difficult to understand. In fact, the logical neatness of this topic can make it easier to comprehend than, say, conceptualization. Although probability sampling is central to social research today, we’ll take some time to

A Brief History of Sampling ■ 189 FIGURE 7-1 Bush Approval: Raw Poll Data. This graph demonstrates how independent polls produce the same picture of reality. This also shows the impact of a national crisis on the president’s popularity: in this case, the September 11 terrorist attack and President George W. Bush’s popularity. Source:Copyright © 2001,2002 by drlimerick.com.(http://www.pollkatz.homestead.com/files/MyHTML2.gif ).All rights reserved. examine a variety of nonprobability methods as opportunities social researchers have to discover well. These methods have their own logic and can the accuracy of their estimates. On election day, provide useful samples for social inquiry. they find out how well or how poorly they did. Before we discuss the two major types of President Alf Landon sampling, I’ll introduce you to some basic ideas by way of a brief history of sampling. As you’ll see, President Alf Landon? Who’s he? Did you sleep the pollsters who correctly predicted the election through an entire presidency in your U.S. his- cliff-hanger of 2000 did so in part because research- tory class? No—but Alf Landon would have been ers had learned to avoid some pitfalls that earlier president if a famous poll conducted by the Liter- pollsters had fallen into. ary Digest had proved to be accurate. The Literary Digest was a popular newsmagazine published A Brief History of Sampling between 1890 and 1938. In 1920, Digest editors mailed postcards to people in six states, asking Sampling in social research has developed hand them whom they were planning to vote for in the in hand with political polling. This is the case, no presidential campaign between Warren Harding doubt, because political polling is one of the few and James Cox. Names were selected for the poll

190 ■ Chapter 7: The Logic of Sampling from telephone directories and automobile registra- Actually, there was a better explanation—what tion lists. Based on the postcards sent back, the is technically called the sampling frame used by the Digest correctly predicted that Harding would be Digest. In this case the sampling frame consisted of elected. In the elections that followed, the Literary telephone subscribers and automobile owners. In Digest expanded the size of its poll and made correct the context of 1936, this design selected a dispro- predictions in 1924, 1928, and 1932. portionately wealthy sample of the voting popula- tion, especially coming on the tail end of the worst In 1936, the Digest conducted its most ambi- economic depression in the nation’s history. The tious poll: Ten million ballots were sent to people sample effectively excluded poor people, and the listed in telephone directories and on lists of poor voted predominantly for Roosevelt’s New Deal automobile owners. Over two million people recovery program. The Digest’s poll may or may not responded, giving the Republican contender, Alf have correctly represented the voting intentions Landon, a stunning 57 to 43 percent landslide over of telephone subscribers and automobile owners. the incumbent, President Franklin Roosevelt. The Unfortunately for the editors, it decidedly did not editors modestly cautioned, represent the voting intentions of the population as a whole. We make no claim to infallibility. We did not coin the phrase “uncanny accuracy” which has President Thomas E. Dewey been so freely applied to our Polls. We know only too well the limitations of every straw The 1936 election also saw the emergence of a vote, however enormous the sample gathered, young pollster whose name would become syn- however scientific the method. It would be onymous with public opinion. In contrast to the a miracle if every State of the forty-eight Literary Digest, George Gallup correctly predicted behaved on Election Day exactly as forecast by that Roosevelt would beat Landon. Gallup’s suc- the Poll. cess in 1936 hinged on his use of something called quota sampling, which we’ll look at more closely (Literary Digest 1936a: 6) later in the chapter. For now, it’s enough to know that quota sampling is based on a knowledge of the Two weeks later, the Digest editors knew the characteristics of the population being sampled: limitations of straw polls even better: The voters what proportion are men, what proportion are gave Roosevelt a second term in office by the women, what proportions are of various incomes, largest landslide in history, with 61 percent of the ages, and so on. Quota sampling selects people to vote. Landon won only 8 electoral votes to Roo- match a set of these characteristics: the right num- sevelt’s 523. ber of poor, white, rural men; the right number of rich, African American, urban women; and so on. The editors were puzzled by their unfortunate The quotas are based on those variables most rel- turn of luck. A part of the problem surely lay in the evant to the study. In the case of Gallup’s poll, the 22 percent return rate garnered by the poll. The sample selection was based on levels of income; the editors asked, selection procedure ensured the right proportion of respondents at each income level. Why did only one in five voters in Chicago to whom the Digest sent ballots take the trouble Gallup and his American Institute of Public to reply? And why was there a preponder- Opinion used quota sampling to good effect in ance of Republicans in the one-fifth that did 1936, 1940, and 1944—correctly picking the reply? . . . We were getting better cooperation presidential winner each of those years. Then, in in what we have always regarded as a public 1948, Gallup and most political pollsters suffered service from Republicans than we were getting the embarrassment of picking Governor Thomas from Democrats. Do Republicans live nearer to Dewey of New York over the incumbent, President mailboxes? Do Democrats generally disapprove of straw polls? (Literary Digest 1936b: 7)

A Brief History of Sampling ■ 191 W. Eugene Smith/Time & Life Pictures/Getty Images Based on early political polls that showed Dewey leading Truman, the Chicago Daily Tribune sought to scoop the competition with this unfortunate headline. Harry Truman. The pollsters’ embarrassing miscue the total population (of voters in this instance). For continued right up to election night. A famous national political polls, such information came pri- photograph shows a jubilant Truman—whose marily from census data. By 1948, however, World followers’ battle cry was “Give ’em hell, Harry!”— War II had produced a massive movement from the holding aloft a newspaper with the banner head- country to cities, radically changing the character line “Dewey Defeats Truman.” of the U.S. population from what the 1940 census showed, and Gallup relied on 1940 census data. Several factors accounted for the pollsters’ City dwellers, moreover, tended to vote Demo- failure in 1948. First, most pollsters stopped polling cratic; hence, the overrepresentation of rural voters in early October despite a steady trend toward Tru- in his poll had the effect of underestimating the man during the campaign. In addition, many vot- number of Democratic votes. ers were undecided throughout the campaign, and these went disproportionately for Truman when Two Types of Sampling Methods they stepped into the voting booth. By 1948, some academic researchers had already More important, Gallup’s failure rested on the been experimenting with a form of sampling based unrepresentativeness of his samples. Quota sam- on probability theory. This technique involves the pling—which had been effective in earlier years— was Gallup’s undoing in 1948. This technique requires that the researcher know something about

192 ■ Chapter 7: The Logic of Sampling selection of a “random sample” from a list contain- sampling. This is a common method for journalists ing the names of everyone in the population being in their “person-on-the-street” interviews, but it sampled. By and large, the probability-sampling is an extremely risky sampling method for social methods used in 1948 were far more accurate than research. Clearly, this method does not permit any quota-sampling techniques. control over the representativeness of a sample. It’s justified only if the researcher wants to study the Today, probability sampling remains the pri- characteristics of people passing the sampling point mary method of selecting large, representative sam- at specified times or if less-risky sampling methods ples for social research, including national political are not feasible. Even when this method is justified polls. At the same time, probability sampling can be on grounds of feasibility, researchers must exercise impossible or inappropriate in many research situ- great caution in generalizing from their data. Also, ations. Accordingly, before turning to the logic and they should alert readers to the risks associated techniques of probability sampling, we’ll first take a with this method. look at techniques for nonprobability sampling and how they’re used in social research. University researchers frequently conduct sur- veys among the students enrolled in large lecture Nonprobability Sampling classes. The ease and frugality of such a method explains its popularity, but it seldom produces data Social research is often conducted in situations that of any general value. It may be useful for pretest- do not permit the kinds of probability samples used ing a questionnaire, but such a sampling method in large-scale social surveys. Suppose you wanted should not be used for a study purportedly describ- to study homelessness: There is no list of all home- ing students as a whole. less individuals, nor are you likely to create such a list. Moreover, as you’ll see, there are times when Consider this report on the sampling design in probability sampling wouldn’t be appropriate even an examination of knowledge and opinions about if it were possible. Many such situations call for nutrition and cancer among medical students and nonprobability sampling. family physicians: In this section, we’ll examine four types of The fourth-year medical students of the Uni- nonprobability sampling: reliance on available versity of Minnesota Medical School in Minne- subjects, purposive (judgmental) sampling, snow- apolis comprised the student population in this ball sampling, and quota sampling. We’ll conclude study. The physician population consisted of all with a brief discussion of techniques for obtaining physicians attending a “Family Practice Review information about social groups through the use of and Update” course sponsored by the Univer- informants. sity of Minnesota Department of Continuing Medical Education. Reliance on Available Subjects (Cooper-Stephenson and Theologides 1981: 472) Relying on available subjects, such as stopping people at a street corner or some other location, is After all is said and done, what will the results sometimes called “convenience” or “haphazard” of this study represent? They do not provide a meaningful comparison of medical students and nonprobability sampling Any technique in family physicians in the United States or even in which samples are selected in some way not sug- Minnesota. Who were the physicians who attended gested by probability theory. Examples include the course? We can guess that they were probably reliance on available subjects as well as purposive more concerned about their continuing educa- (judgmental), quota, and snowball sampling. tion than other physicians were, but we can’t say for sure. Although such studies can provide useful insights, we must take care not to overgeneralize from them.

Nonprobability Sampling ■ 193 Purposive or Judgmental In qualitative research projects, the sampling of Sampling subjects may evolve as the structure of the situation being studied becomes clearer and certain types of Sometimes it’s appropriate to select a sample on the subjects seem more central to understanding than basis of knowledge of a population, its elements, others do. Let’s say you’re conducting an interview and the purpose of the study. This type of sampling study among the members of a radical politi- is called purposive or judgmental sampling. cal group on campus. You may initially focus on In the initial design of a questionnaire, for ex- friendship networks as a vehicle for the spread of ample, you might wish to select the widest variety group membership and participation. In the course of respondents to test the broad applicability of of your analysis of the earlier interviews, you may questions. Although the study findings would not find several references to interactions with faculty represent any meaningful population, the test run members in one of the social science departments. might effectively uncover any peculiar defects in As a consequence, you may expand your sample to your questionnaire. This situation would be consid- include faculty in that department and other stu- ered a pretest, however, rather than a final study. dents that they interact with. This is called “theo- retical sampling,” since the evolving theoretical In some instances, you may wish to study a understanding of the subject directs the sampling small subset of a larger population in which many in certain directions. members of the subset are easily identified, but the enumeration of them all would be nearly impos- Snowball Sampling sible. For example, you might want to study the leadership of a student protest movement; many Another nonprobability sampling technique, which of the leaders are easily visible, but it would not some consider to be a form of accidental sampling, be feasible to define and sample all the leaders. In is called snowball sampling. This procedure studying all or a sample of the most visible leaders, is appropriate when the members of a special you may collect data sufficient for your purposes. population are difficult to locate, such as homeless individuals, migrant workers, or undocumented Or let’s say you want to compare left-wing and immigrants. In snowball sampling, the researcher right-wing students. Because you may not be able collects data on the few members of the target to enumerate and sample from all such students, population he or she can locate, then asks those you might decide to sample the memberships of individuals to provide the information needed to left- and right-leaning groups, such as the Green locate other members of that population whom Party and the Young Americans for Freedom. they happen to know. “Snowball” refers to the Although such a sample design would not provide process of accumulation as each located subject a good description of either left-wing or right-wing suggests other subjects. Because this procedure also students as a whole, it might suffice for general results in samples with questionable representative- comparative purposes. ness, it’s used primarily for exploratory purposes. Field researchers are often particularly inter- purposive (judgmental) sampling A type of ested in studying deviant cases—cases that don’t fit nonprobability sampling in which the units to be into fairly regular patterns of attitudes and behav- observed are selected on the basis of the researcher’s iors—in order to improve their understanding of judgment about which ones will be the most useful the more-regular pattern. For example, you might or representative. gain important insights into the nature of school spirit, as exhibited at a pep rally, by interviewing snowball sampling A nonprobability sampling people who did not appear to be caught up in the method, often employed in field research, whereby emotions of the crowd or by interviewing students each person interviewed may be asked to suggest who did not attend the rally at all. Selecting devi- additional people for interviewing. ant cases for study is another example of purposive study.

194 ■ Chapter 7: The Logic of Sampling Suppose you wish to learn a community or- tion. Depending on your research purposes, you ganization’s pattern of recruitment over time. You may need to know what proportion of the popula- might begin by interviewing fairly recent recruits, tion is male and what proportion female as well as asking them who introduced them to the group. what proportions of each gender fall into various You might then interview the people named, ask- age categories, educational levels, ethnic groups, ing them who introduced them to the group. You and so forth. In establishing a national quota might then interview those people named, asking, sample, you might need to know what proportion in part, who introduced them. Or, in studying a of the national population is urban, eastern, male, loosely structured political group, you might ask under 25, white, working class, and the like, and all one of the participants who he or she believes to the possible combinations of these attributes. be the most influential members of the group. You might interview those people and, in the course of Once you’ve created such a matrix and as- the interviews, ask who they believe to be the most signed a relative proportion to each cell in the influential. In each of these examples, your sample matrix, you proceed to collect data from people would “snowball” as each of your interviewees sug- having all the characteristics of a given cell. You gested other people to interview. then assign to all the people in a given cell a weight appropriate to their portion of the total population. Examples of this technique in social science re- When all the sample elements are so weighted, the search abound. Karen Farquharson (2005) provides overall data should provide a reasonable represen- a detailed discussion of how she used snowball tation of the total population. sampling to discover a network of tobacco policy makers in Australia: both those at the core of the Although quota sampling resembles probability network and those on the periphery. Kath Browne sampling, it has several inherent problems. First, (2005) used snowballing through social networks the quota frame (the proportions that different cells to develop a sample of nonheterosexual women in represent) must be accurate, and it’s often difficult a small town in the United Kingdom. She reports to get up-to-date information for this purpose. The that her own membership in such networks greatly Gallup failure to predict Truman as the presiden- facilitated this type of sampling, and that potential tial victor in 1948 was due partly to this problem. subjects in the study were more likely to trust her Second, the selection of sample elements within a than to trust heterosexual researchers. given cell may be biased even though its proportion of the population is accurately estimated. In- Quota Sampling structed to interview five people who meet a given, complex set of characteristics, an interviewer may Quota sampling is the method that helped still avoid people living at the top of seven-story George Gallup avoid disaster in 1936—and set up walk-ups, having particularly run-down homes, or the disaster of 1948. Like probability sampling, owning vicious dogs. quota sampling addresses the issue of representa- tiveness, although the two methods approach the In recent years, attempts have been made to issue quite differently. combine probability- and quota-sampling meth- ods, but the effectiveness of this effort remains to Quota sampling begins with a matrix, or table, be seen. At present, you would be advised to treat describing the characteristics of the target popula- quota sampling warily if your purpose is statistical description. quota sampling A type of nonprobability sam- pling in which units are selected into a sample on At the same time, the logic of quota sampling the basis of prespecified characteristics, so that the can sometimes be applied usefully to a field re- total sample will have the same distribution of char- search project. In the study of a formal group, for acteristics assumed to exist in the population being example, you might wish to interview both leaders studied. and nonleaders. In studying a student political orga- nization, you might want to interview radical, mod- erate, and conservative members of that group. You may be able to achieve sufficient representativeness

Nonprobability Sampling ■ 195 in such cases by using quota sampling to ensure societies nor even many subgroups within English- that you interview both men and women, both speaking countries. younger and older people, and so forth. Simply because they’re the ones willing to Selecting Informants work with outside investigators, informants will almost always be somewhat “marginal” or atypi- When field research involves the researcher’s cal within their group. Sometimes this is obvious. attempt to understand some social setting—a Other times, however, you’ll learn about their juvenile gang or local neighborhood, for exam- marginality only in the course of your research. ple—much of that understanding will come from a collaboration with some members of the group be- In Jeffrey Johnson’s study, the county agent ing studied. Whereas social researchers speak of re- identified one fisherman who seemed squarely in spondents as people who provide information about the mainstream of the community. Moreover, he themselves, allowing the researcher to construct a was cooperative and helpful to Johnson’s research. composite picture of the group those respondents The more Johnson worked with the fisherman, represent, an informant is a member of the group however, the more he found the man to be a mar- who can talk directly about the group per se. ginal member of the fishing community. Especially important to anthropologists, infor- First, he was a Yankee in a southern town. mants are important to other social researchers as Second, he had a pension from the Navy [so he well. If you wanted to learn about informal social was not seen as a “serious fisherman” by others networks in a local public-housing project, for in the community]. . . . Third, he was a major example, you would do well to locate individuals Republican activist in a mostly Democratic who could understand what you were looking for village. Finally, he kept his boat in an isolated and help you find it. anchorage, far from the community harbor. When Jeffrey Johnson (1990) set out to study (1990: 56) a salmon-fishing community in North Carolina, he used several criteria to evaluate potential infor- Informants’ marginality may not only bias the view mants. Did their positions allow them to interact you get, but their marginal status may also limit regularly with other members of the camp, for their access (and hence yours) to the different sec- example, or were they isolated? (He found that the tors of the community you wish to study. carpenter had a wider range of interactions than the boat captain did.) Was their information about These comments should give you some sense of the camp pretty much limited to their specific the concerns involved in nonprobability sampling, jobs, or did it cover many aspects of the operation? typically used in qualitative research projects. I These and other criteria helped determine how conclude with the following injunction: useful the potential informants might be. Your overall goal is to collect the richest possible Usually, you’ll want to select informants some- data. By rich data, we mean a wide and diverse what typical of the groups you’re studying. Oth- range of information collected over a relatively erwise, their observations and opinions may be prolonged period of time in a persistent and misleading. Interviewing only physicians will not systematic manner. Ideally, such data enable give you a well-rounded view of how a community you to grasp the meanings associated with medical clinic is working, for example. Along the the actions of those you are studying and to same lines, an anthropologist who interviews only understand the contexts in which those actions men in a society where women are sheltered from are embedded. outsiders will get a biased view. Similarly, although informants fluent in English are convenient for (Lofland et al. 2006: 15) English-speaking researchers from the United States, they do not typify the members of many informant Someone who is well versed in the social phenomenon that you wish to study and who is willing to tell you what he or she knows about it. Not to be confused with a respondent.

196 ■ Chapter 7: The Logic of Sampling In other words, nonprobability sampling does have its uses, particularly in qualitative research projects. But researchers must take care to acknowledge the limitations of nonprobability sampling, especially regarding accurate and precise representations of populations. This point will be- come clearer as we discuss the logic and techniques of probability sampling. The Theory and Logic of FIGURE 7-2 Probability Sampling A Population of 100 Folks. Typically, sampling aims to reflect the characteristics and dynamics of large populations. For the However appropriate to some research purposes, purpose of some simple illustrations, let’s assume our total nonprobability sampling methods cannot guaran- population only has 100 members. tee that the sample we observed is representative of the whole population. When researchers want pre- tion: The 100 members of this small population cise, statistical descriptions of large populations— differ by gender and race. We’ll use this hypotheti- for example, the percentage of the population cal micropopulation to illustrate various aspects of who are unemployed, plan to vote for Candidate probability sampling. X, or feel a rape victim should have the right to an abortion—they turn to probability sampling. The fundamental idea behind probability All large-scale surveys use probability-sampling sampling is this: To provide useful descriptions of methods. the total population, a sample of individuals from a population must contain essentially the same Although the application of probability sam- variations that exist in the population. This isn’t as pling involves some sophisticated use of statistics, simple as it might seem, however. Let’s take a min- the basic logic of probability sampling is not difficult ute to look at some of the ways researchers might to understand. If all members of a population were go astray. Then, we’ll see how probability sampling identical in all respects—all demographic charac- provides an efficient method for selecting a sample teristics, attitudes, experiences, behaviors, and so that should adequately reflect variations that exist on—there would be no need for careful sampling in the population. procedures. In this extreme case of perfect ho- mogeneity, in fact, any single case would suffice Conscious and Unconscious as a sample to study characteristics of the whole Sampling Bias population. At first glance, it may look as though sampling is In fact, of course, the human beings who pretty straightforward. To select a sample of 100 compose any real population are quite heteroge- university students, you might simply interview the neous, varying in many ways. Figure 7-2 offers a first 100 students you find walking around campus. simplified illustration of a heterogeneous popula- This kind of sampling method is often used by untrained researchers, but it runs a high risk of probability sampling The general term for introducing biases into the samples. samples selected in accord with probability theory, typically involving some random-selection mecha- nism. Specific types of probability sampling include EPSEM, PPS, simple random sampling, and system- atic sampling.

The Theory and Logic of Probability Sampling ■ 197 FIGURE 7-3 A Sample of Convenience: Easy, but Not Representative. Simply selecting and observing those people who are most readily at hand is the simplest method, perhaps, but it’s unlikely to provide a sample that accurately reflects the total population. In connection with sampling, bias simply you might feel that the attitudes of “super-straight- means that those selected are not typical nor repre- looking” students would be irrelevant to your sentative of the larger populations they have been research purposes and so avoid interviewing them. chosen from. This kind of bias does not have to be intentional. In fact, it is virtually inevitable when Even if you sought to interview a “balanced” you pick people by the seat of your pants. group of students, you wouldn’t know the exact proportions of different types of students making Figure 7-3 illustrates what can happen when up such a balance, and you wouldn’t always be researchers simply select people who are conve- able to identify the different types just by watching nient for study. Although women are only 50 per- them walk by. cent of our micropopulation, the people closest to the researcher (in the lower right corner) happen Even if you made a conscientious effort to to be 70 percent women, and although the popula- interview, say, every tenth student entering the tion is 12 percent African American, none was university library, you could not be sure of a selected into the sample. representative sample, because different types of students visit the library with different frequencies. Beyond the risks inherent in simply study- Your sample would overrepresent students who ing people who are convenient, other problems visit the library more often than others do. can arise. To begin with, the researcher’s personal leanings may affect the sample to the point where Similarly, the “public opinion” call-in polls—in it does not truly represent the student population. which radio stations or newspapers ask people to Suppose you’re a little intimidated by students who call specified telephone numbers to register their look particularly “cool,” feeling they might ridicule opinions—cannot be trusted to represent general your research effort. You might consciously or populations. At the very least, not everyone in unconsciously avoid interviewing such people. Or, the population will even be aware of the poll. This problem also invalidates polls by magazines and

198 ■ Chapter 7: The Logic of Sampling newspapers that publish coupons for readers to purpose, a sample is representative of the popula- complete and mail in. Even among those who are tion from which it is selected if the aggregate char- aware of such polls, not all will express an opinion, acteristics of the sample closely approximate those especially if doing so will cost them a stamp, an same aggregate characteristics in the population. envelope, or a telephone charge. Similar consider- If, for example, the population contains 50 percent ations apply to polls taken over the Internet. women, then a sample must contain “close to” 50 percent women to be representative. Later, we’ll Ironically, the failure of such polls to represent discuss “how close” in detail. all opinions equally was inadvertently acknowl- edged by Phillip Perinelli (1986), a staff manager Note that samples need not be representative of AT&T Communications’ DIAL-IT 900 Service, in all respects; representativeness is limited to those which offers a call-in poll facility to organizations. characteristics that are relevant to the substantive Perinelli attempted to counter criticisms by saying, interests of the study. However, you may not know “The 50-cent charge assures that only interested in advance which characteristics are relevant. parties respond and helps assure also that no indi- vidual ’stuffs’ the ballot box.” We cannot determine A basic principle of probability sampling is general public opinion while considering “only in- that a sample will be representative of the popula- terested parties.” This excludes those who don’t care tion from which it is selected if all members of the 50-cents’ worth, as well as those who recognize that population have an equal chance of being selected such polls are not valid. Both types of people may in the sample. (We’ll see shortly that the size of the have opinions and may even vote on election day. sample selected also affects the degree of represen- Perinelli’s assertion that the 50-cent charge will pre- tativeness.) Samples that have this quality are often vent ballot stuffing actually means that only those labeled EPSEM samples (EPSEM stands for “equal who can afford it will engage in ballot stuffing. probability of selection method”). Later, we’ll discuss variations of this principle, which forms the The possibilities for inadvertent sampling bias basis of probability sampling. are endless and not always obvious. Fortunately, many techniques can help us avoid bias. Moving beyond this basic principle, we must realize that samples—even carefully selected Representativeness and EPSEM samples—seldom if ever perfectly repre- Probability of Selection sent the populations from which they are drawn. Nevertheless, probability sampling offers two spe- Although the term representativeness has no cial advantages. precise, scientific meaning, it carries a common- sense meaning that makes it useful here. For our First, probability samples, although never perfectly representative, are typically more repre- representativeness That quality of a sample of sentative than other types of samples, because the having the same distribution of characteristics as the biases previously discussed are avoided. In practice, population from which it was selected. By implica- a probability sample is more likely than a nonprob- tion, descriptions and explanations derived from an ability sample to be representative of the popula- analysis of the sample may be assumed to represent tion from which it is drawn. similar ones in the population. Representativeness is enhanced by probability sampling and provides for Second, and more important, probability generalizability and the use of inferential statistics. theory permits us to estimate the accuracy or representativeness of the sample. Conceivably, an EPSEM (equal probability of selection method) uninformed researcher might, through wholly A sample design in which each member of a popula- haphazard means, select a sample that nearly tion has the same chance of being selected into the perfectly represents the larger population. The odds sample. are against doing so, however, and we would be unable to estimate the likelihood that he or she has achieved representativeness. The probability sam- pler, on the other hand, can provide an accurate

The Theory and Logic of Probability Sampling ■ 199 estimate of success or failure. We’ll shortly see has a chance of being selected in the sample. Even exactly how this estimate can be achieved. where lists of elements exist for sampling purposes, the lists are usually somewhat incomplete. Some I’ve said that probability sampling ensures that students are always inadvertently omitted from samples are representative of the population we student rosters. Some telephone subscribers request wish to study. As we’ll see in a moment, probability that their names and numbers be unlisted. sampling rests on the use of a random-selection procedure. To develop this idea, though, we need Often, researchers decide to limit their study to give more-precise meaning to two important populations more severely than indicated in the terms: element and population.* preceding examples. National polling firms may limit their national samples to the 48 adjacent An element is that unit about which infor- states, omitting Alaska and Hawaii for practical rea- mation is collected and that provides the basis of sons. A researcher wishing to sample psychology analysis. Typically, in survey research, elements are professors may limit the study population to those people or certain types of people. However, other in psychology departments, omitting those in other kinds of units can constitute the elements for social departments. Whenever the population under research: Families, social clubs, or corporations examination is altered in such fashions, you must might be the elements of a study. In a given study, make the revisions clear to your readers. elements are often the same as units of analysis, though the former are used in sample selection and Random Selection the latter in data analysis. With these definitions in hand, we can define the Up to now we’ve used the term population to ultimate purpose of sampling: to select a set of ele- mean the group or collection that we’re interested ments from a population in such a way that descrip- in generalizing about. More formally, a population tions of those elements accurately portray the total is the theoretically specified aggregation of study population from which the elements are selected. elements. Whereas the vague term Americans Probability sampling enhances the likelihood of ac- might be the target for a study, the delineation of complishing this aim and also provides methods for the population would include the definition of the estimating the degree of probable success. element Americans (for example, citizenship, resi- dence) and the time referent for the study (Ameri- Random selection is the key to this process. cans as of when?). Translating the abstract “adult In random selection, each element has an equal New Yorkers” into a workable population would chance of selection independent of any other event require a specification of the age defining adult in the selection process. Flipping a coin is the most and the boundaries of New York. Specifying the frequently cited example: Provided that the coin is term college student would include a consideration of perfect (that is, not biased in terms of coming up full- and part-time students, degree candidates and nondegree candidates, undergraduate and graduate element That unit of which a population is com- students, and so forth. posed and which is selected in a sample. Distin- guished from units of analysis, which are used in data A study population is that aggregation of ele- analysis. ments from which the sample is actually selected. population The theoretically specified aggregation As a practical matter, researchers are seldom in a of the elements in a study. position to guarantee that every element meet- study population That aggregation of elements ing the theoretical definitions laid down actually from which a sample is actually selected. random selection A sampling method in which *I would like to acknowledge a debt to Leslie Kish and each element has an equal chance of selection inde- his excellent textbook Survey Sampling. Although I’ve pendent of any other event in the selection process. modified some of the conventions used by Kish, his presentation is easily the most important source of this discussion.

200 ■ Chapter 7: The Logic of Sampling heads or tails), the “selection” of a head or a tail statistically. More formally, probability theory pro- is independent of previous selections of heads or vides the basis for estimating the parameters of a tails. No matter how many heads turn up in a row, population. A parameter is the summary descrip- the chance that the next flip will produce “heads” tion of a given variable in a population. The mean is exactly 50–50. Rolling a perfect set of dice is income of all families in a city is a parameter; so is another example. the age distribution of the city’s population. When researchers generalize from a sample, they’re us- Such images of random selection, although ing sample observations to estimate population useful, seldom apply directly to sampling methods parameters. Probability theory enables them to in social research. More typically, social researchers both make these estimates and arrive at a judgment use tables of random numbers or computer pro- of how likely the estimates will accurately repre- grams that provide a random selection of sampling sent the actual parameters in the population. For units. A sampling unit is that element or set of example, probability theory allows pollsters to infer elements considered for selection in some stage of from a sample of 2,000 voters how a population of sampling. In Chapter 9, on survey research, we’ll 100 million voters is likely to vote—and to specify see how computers are used to select random exactly what the probable margin of error of the telephone numbers for interviewing, a technique estimates is. called random-digit dialing. Probability theory accomplishes these seem- The reasons for using random-selection meth- ingly magical feats by way of the concept of sam- ods are twofold. First, this procedure serves as a pling distributions. A single sample selected from a check on conscious or unconscious bias on the part population will give an estimate of the population of the researcher. The researcher who selects cases parameter. Other samples would give the same or on an intuitive basis might very well select cases slightly different estimates. Probability theory tells that would support his or her research expecta- us about the distribution of estimates that would tions or hypotheses. Random selection erases this be produced by a large number of such samples. danger. More importantly, random selection offers To see how this works, we’ll look at two examples access to the body of probability theory, which of sampling distributions, beginning with a simple provides the basis for estimating the characteristics example in which our population consists of just of the population as well as estimating the accuracy ten cases, then moving on to a case of percentages of samples. Let’s now examine probability theory in that allows a clear illustration of probable margin of greater detail. error. Probability Theory, Sampling The Sampling Distribution of Ten Cases Distributions, and Estimates of Sampling Error Suppose there are ten people in a group, and each has a certain amount of money in his or her Probability theory is a branch of mathematics that pocket. To simplify, let’s assume that one person provides the tools researchers need to devise has no money, another has one dollar, another sampling techniques that produce representative has two dollars, and so forth up to the person with samples and to analyze the results of their sampling nine dollars. Figure 7-4 presents the population of ten people.* sampling unit That element or set of elements considered for selection in some stage of sampling. Our task is to determine the average amount parameter The summary description of a given of money one person has: specifically, the mean variable in a population. number of dollars. If you simply add up the money *I want to thank Hanan Selvin for suggesting this method of introducing probability sampling.

The Theory and Logic of Probability Sampling ■ 201 FIGURE 7-4 A Population of 10 People with $0–$9. Let’s simplify matters even more now by imagining a population of only 10 people with differing amounts of money in their pockets—ranging from $0 to $9. shown in Figure 7-4, you’ll find that the total is sible samples: [$0 $1], [$0 $2], . . . [$7 $8], [$8 $9]. $45, so the mean is $4.50. Our purpose in the rest Moreover, some of those samples produce the same of this exercise is to estimate that mean without means. For example, [$0 $6], [$1 $5], and [$2 $4] actually observing all ten individuals. We’ll do that all produce means of $3. In Figure 7-6, the three by selecting random samples from the population dots shown above the $3 mean represent those and using the means of those samples to estimate three samples. the mean of the whole population. Moreover, the 45 samples are not evenly To start, suppose we were to select—at distributed, as they were when the sample size random—a sample of only one person from the was only one. Rather, they’re somewhat clustered ten. Our ten possible samples thus consist of the around the true value of $4.50. Only two possible ten cases shown in Figure 7-4. samples deviate by as much as $4 from the true value ([$0 $1] and [$8 $9]), whereas five of the The ten dots shown on the graph in Figure 7-5 samples would give the true estimate of $4.50; an- represent these ten samples. Because we’re tak- other eight samples miss the mark by only 50 cents ing samples of only one, they also represent the (plus or minus). “means” we would get as estimates of the popu- lation. The distribution of the dots on the graph Now suppose we select even larger samples. is called the sampling distribution. Obviously, it What do you think that will do to our estimates of wouldn’t be a very good idea to select a sample of the mean? Figure 7-7 presents the sampling distri- only one, because the chances are great that we’ll butions of samples of 3, 4, 5, and 6. miss the true mean of $4.50 by quite a bit. The progression of sampling distributions is Now suppose we take a sample of two. As clear. Every increase in sample size improves the shown in Figure 7-6, increasing the sample size distribution of estimates of the mean. The limit- improves our estimations. There are now 45 pos- ing case in this procedure, of course, is to select a

202 ■ Chapter 7: The Logic of Sampling FIGURE 7-5 FIGURE 7-6 The Sampling Distribution of Samples of 1. In this simple example, the mean amount of money these people have is The Sampling Distribution of Samples of 2. By merely increas- $4.50 ($45/10). If we picked 10 different samples of 1 person ing our sample size to 2, we get possible samples that provide each, our “estimates” of the mean would range all across the somewhat better estimates of the mean. We couldn’t get either board. $0 or $9, and the estimates are beginning to cluster around the true value of the mean: $4.50. sample of ten. There would be only one possible sample (everyone) and it would give us the true purposes of estimating the entire student body. The mean of $4.50. As we’ll see shortly, this principle variable under consideration will be attitudes toward applies to actual sampling of meaningful popula- the code, a binomial variable: approve and disapprove. tions. The larger the sample selected, the more ac- (The logic of probability sampling applies to the curate it is as an estimation of the population from examination of other types of variables, such as which it was drawn. mean income, but the computations are somewhat more complicated. Consequently, this introduction Sampling Distribution focuses on binomials.) and Estimates of Sampling Error The horizontal axis of Figure 7-8 presents all Let’s turn now to a more realistic sampling situa- possible values of this parameter in the popula- tion involving a much larger population and see tion—from 0 percent to 100 percent approval. The how the notion of sampling distribution applies. midpoint of the axis—50 percent—represents half Assume that we wish to study the student popu- the students approving of the code and the other lation of State University (SU) to determine the half disapproving. percentage of students who approve or disap- prove of a student-conduct code proposed by the To choose our sample, we give each student on administration. The study population will be the the student roster a number and select 100 random aggregation of, say, 20,000 students contained in a numbers from a table of random numbers. Then student roster: the sampling frame. The elements we interview the 100 students whose numbers will be the individual students at SU. We’ll select have been selected and ask for their attitudes a random sample of, say, 100 students for the toward the student code: whether they approve or disapprove. Suppose this operation gives us 48 statistic The summary description of a variable in a students who approve of the code and 52 who sample, used to estimate a population parameter. disapprove. This summary description of a variable in a sample is called a statistic. We present this statistic by placing a dot on the x axis at the point representing 48 percent.

FIGURE 7-7 The Sampling Distributions of Samples of 3, 4, 5, and 6. As we increase the sample size, the possible samples cluster ever more tightly around the true value of the mean. The chance of extremely inaccurate estimates is reduced at the two ends of the distribu- tion, and the percentage of the samples near the true value keeps increasing.

204 ■ Chapter 7: The Logic of Sampling FIGURE 7-8 FIGURE 7-9 Range of Possible Sample Study Results. Shifting to a more Results Produced by Three Hypothetical Studies. Assuming a realistic example, let’s assume that we want to sample student large student body, let’s suppose that we selected three dif- attitudes concerning a proposed conduct code. Let’s assume ferent samples, each of substantial size. We would not neces- that 50 percent of the whole student body approves and sarily expect those samples to perfectly reflect attitudes in the 50 percent disapproves—though the researcher doesn’t whole student body, but they should come reasonably close. know that. the sampling distribution of, say, hundreds of Now let’s suppose we select another sample of samples. This is often referred to as a normal curve. 100 students in exactly the same fashion and mea- sure their approval or disapproval of the student Note that by increasing the number of samples code. Perhaps 51 students in the second sample ap- selected and interviewed, we’ve also increased the prove of the code. We place another dot in the ap- range of estimates provided by the sampling opera- propriate place on the x axis. Repeating this process tion. In one sense we’ve increased our dilemma in once more, we may discover that 52 students in the attempting to guess the parameter in the popula- third sample approve of the code. tion. Probability theory, however, provides certain important rules regarding the sampling distribution Figure 7-9 presents the three different sample presented in Figure 7-10. statistics representing the percentages of students in each of the three random samples who approved First, if many independent random samples of the student code. The basic rule of random sam- are selected from a population, the sample statistics pling is that such samples drawn from a popula- provided by those samples will be distributed tion give estimates of the parameter that exists in around the population parameter in a known way. the total population. Each of the random samples, Thus, although Figure 7-10 shows a wide range then, gives us an estimate of the percentage of of estimates, more of them are in the vicinity of students in the total student body who approve of 50 percent than elsewhere in the graph. Probability the student code. Unhappily, however, we have se- theory tells us, then, that the true value is in the lected three samples and now have three separate vicinity of 50 percent. estimates. Second, probability theory gives us a formula To retrieve ourselves from this problem, let’s for estimating how closely the sample statistics draw more and more samples of 100 students each, are clustered around the true value. To put it question each of the samples concerning their another way, probability theory enables us to approval or disapproval of the code, and plot the estimate the sampling error—the degree of er- new sample statistics on our summary graph. In ror to be expected for a given sample design. This drawing many such samples, we discover that some formula contains three factors: the parameter, the of the new samples provide duplicate estimates, as sample size, and the standard error (a measure of in the illustration of ten cases. Figure 7-10 shows sampling error): sampling error The degree of error to be expected sϭ P ϫQ by virtue of studying a sample instead of everyone. n For probability sampling, the maximum error de- pends on three factors: the sample size, the diversity of the population, and the confidence level.

The Theory and Logic of Probability Sampling ■ 205 FIGURE 7-10 The Sampling Distribution. If we were to select a large number of good samples, we would expect them to cluster around the true value (50 percent), but given enough such samples, a few would fall far from the mark. The symbols P and Q in the formula equal the dard error below the parameter. In our example, population parameters for the binomial: If 60 per- the standard error increment is 5 percent, so we cent of the student body approve of the code and know that 34 percent of our samples will give 40 percent disapprove, P and Q are 60 percent and estimates of student approval between 50 percent 40 percent, respectively, or 0.6 and 0.4. Note that (the parameter) and 55 percent (one standard error Q ϭ 1 Ϫ P and P ϭ 1 Ϫ Q. The symbol n equals above); another 34 percent of the samples will give the number of cases in each sample, and s is the estimates between 50 percent and 45 percent standard error. (one standard error below the parameter). Taken together, then, we know that roughly two-thirds Let’s assume that the population parameter in (68 percent) of the samples will give estimates the student example is 50 percent approving of within 5 percent of the parameter. the code and 50 percent disapproving. Recall that we’ve been selecting samples of 100 cases each. Moreover, probability theory dictates that When these numbers are put into the formula, roughly 95 percent of the samples will fall within we find that the standard error equals 0.05, or plus or minus two standard errors of the true value, 5 percent. and 99.9 percent of the samples will fall within plus or minus three standard errors. In our present In probability theory, the standard error is a example, then, we know that only one sample out valuable piece of information because it indicates of a thousand would give an estimate lower than the extent to which the sample estimates will be 35 percent approval or higher than 65 percent. distributed around the population parameter. (If you’re familiar with the standard deviation in statis- The proportion of samples falling within one, tics, you may recognize that the standard error, in two, or three standard errors of the parameter this case, is the standard deviation of the sampling is constant for any random sampling procedure distribution.) Specifically, probability theory indi- such as the one just described, providing that a cates that certain proportions of the sample esti- large number of samples are selected. The size of mates will fall within specified increments—each the standard error in any given case, however, is equal to one standard error—from the population a function of the population parameter and the parameter. Approximately 34 percent (0.3413) of sample size. If we return to the formula for a mo- the sample estimates will fall within one standard ment, we note that the standard error will increase error increment above the population parameter, as a function of an increase in the quantity P and another 34 percent will fall within one stan- times Q. Note further that this quantity reaches its

206 ■ Chapter 7: The Logic of Sampling maximum in the situation of an even split in the ity theory provides the basis for inferences about population. If P ϭ 0.5, PQ ϭ 0.25; if P ϭ 0.6, PQ ϭ the typical social research situation. Knowing what 0.24; if P ϭ 0.8, PQ ϭ 0.16; if P ϭ 0.99, PQ ϭ it would be like to select thousands of samples al- 0.0099. By extension, if P is either 0.0 or 1.0 (either lows us to make assumptions about the one sample 0 percent or 100 percent approve of the student we do select and study. code), the standard error will be 0. If everyone in the population has the same attitude (no variation), Confidence Levels then every sample will give exactly that estimate. and Confidence Intervals The standard error is also a function of the Whereas probability theory specifies that 68 per- sample size—an inverse function. As the sample cent of that fictitious large number of samples size increases, the standard error decreases. As the would produce estimates falling within one sample size increases, the several samples will be standard error of the parameter, we can turn the clustered nearer to the true value. Another general logic around and infer that any single random guideline is evident in the formula: Because of sample estimate has a 68 percent chance of falling the square root formula, the standard error is within that range. This observation leads us to the reduced by half if the sample size is quadrupled. two key components of sampling error estimates: In our present example, samples of 100 produce a confidence level and confidence interval. We standard error of 5 percent; to reduce the standard express the accuracy of our sample statistics in error to 2.5 percent, we must increase the sample terms of a level of confidence that the statistics fall size to 400. within a specified interval from the parameter. For example, we may say we are 95 percent confident All of this information is provided by es- that our sample statistics (for example, 50 per- tablished probability theory in reference to the cent favor the new student code) are within plus selection of large numbers of random samples. (If or minus 5 percentage points of the population you’ve taken a statistics course, you may know parameter. As the confidence interval is expanded this as the Central Tendency Theorem.) If the for a given statistic, our confidence increases. For population parameter is known and many random example, we may say that we are 99.9 percent samples are selected, we can predict how many confident that our statistic falls within three stan- of the sample estimates will fall within specified dard errors of the true value. (Now perhaps you intervals from the parameter. can appreciate the humorous quip of unknown origin: Statistics means never having to say you Recognize that this discussion illustrates only are certain.) the logic of probability sampling; it does not describe the way research is actually conducted. Although we may be confident (at some level) Usually, we don’t know the parameter: The very of being within a certain range of the parameter, reason we conduct a sample survey is to estimate we’ve already noted that we seldom know what that value. Moreover, we don’t actually select large the parameter is. To resolve this problem, we numbers of samples: We select only one sample. substitute our sample estimate for the parameter Nevertheless, the preceding discussion of probabil- in the formula; that is, lacking the true value, we substitute the best available guess. confidence level The estimated probability that a population parameter lies within a given confidence The result of these inferences and estimations interval. Thus, we might be 95 percent confident is that we can estimate a population parameter that between 35 and 45 percent of all voters favor and also the expected degree of error on the basis Candidate A. of one sample drawn from a population. Begin- ning with the question “What percentage of the confidence interval The range of values within student body approves of the student code?” you which a population parameter is estimated to lie. could select a random sample of 100 students and

The Theory and Logic of Probability Sampling ■ 207 interview them. You might then report that your intuitive fact is that the equations for calculating best estimate is that 50 percent of the student body sampling error all assume that the populations approves of the code and that you are 95 percent being sampled are infinitely large, so every sample confident that between 40 and 60 percent (plus would equal 0 percent of the whole. or minus two standard errors) approve. The range from 40 to 60 percent is the confidence interval. Of course, this is not literally true in practice. (At the 68 percent confidence level, the confidence However, a sample of 2,000 represents only interval would be 45–55 percent.) 0.68 percent of the Vermonters who voted for pres- ident in the 2000 election, and a sample of 2,000 The logic of confidence levels and confidence U.S. voters represents a mere 0.002 percent of the intervals also provides the basis for determining the national electorate. Both of these proportions are appropriate sample size for a study. Once you’ve sufficiently small as to approach the situation with decided on the degree of sampling error you can infinitely large populations. tolerate, you’ll be able to calculate the number of cases needed in your sample. Thus, for example, Unless a sample represents, say, 5 percent if you want to be 95 percent confident that your or more of the population it’s drawn from, that study findings are accurate within plus or proportion is irrelevant. In those rare cases of minus 5 percentage points of the population large proportions being selected, a “finite popula- parameters, you should select a sample of at least tion correction” can be calculated to adjust the 400. (Appendix F is a convenient guide in this confidence intervals. The following formula cal- regard.) culates the proportion to be multiplied against the calculated error. This, then, is the basic logic of probability sampling. Random selection permits the researcher finite population correction ϭ N Ϫ n to link findings from a sample to the body of N Ϫ1 probability theory so as to estimate the accuracy of those findings. All statements of accuracy in In the formula, N is the population size and n sampling must specify both a confidence level is the size of the sample. Notice that in the extreme and a confidence interval. The researcher must case where you studied the whole population report that he or she is x percent confident that (hence N ϭ n), the formula would yield zero as the the population parameter is between two specific finite population correction. Multiplying zero times values. In this example, I’ve demonstrated the the sampling error calculated by the earlier formula logic of sampling error using a variable analyzed in would give a final sampling error of zero, which percentages. A different statistical procedure would would, of course, be precisely the case since you be required to calculate the standard error for wouldn’t have sampled at all. a mean, for example, but the overall logic is the same. Two cautions are in order before we conclude this discussion of the basic logic of probability Notice that nowhere in this discussion of sam- sampling. First, the survey uses of probability ple size and accuracy of estimates did we consider theory as discussed here are technically not wholly the size of the population being studied. This is justified. The theory of sampling distribution because the population size is almost always irrel- makes assumptions that almost never apply in evant. A sample of 2,000 respondents drawn prop- survey conditions. The exact proportion of samples erly to represent Vermont voters will be no more contained within specified increments of standard accurate than a sample of 2,000 drawn properly errors, for example, mathematically assumes to represent all voters in the United States, even an infinitely large population, an infinite number though the Vermont sample would be a substan- of samples, and sampling with replacement—that tially larger proportion of that small state’s voters is, every sampling unit selected is “thrown back than would the same number chosen to represent into the pot” and could be selected again. Sec- the nation’s voters. The reason for this counter- ond, our discussion has greatly oversimplified the

208 ■ Chapter 7: The Logic of Sampling inferential jump from the distribution of several at one aspect of field conditions that requires a samples to the probable characteristics of one compromise with idealized theoretical conditions sample. and assumptions: the congruence of or disparity between populations of sampling frames. I offer these cautions to provide perspective on the uses of probability theory in sampling. Social Simply put, a sampling frame is the list or researchers often appear to overestimate the preci- quasi list of elements from which a probability sion of estimates produced by the use of probability sample is selected. If a sample of students is selected theory. As I’ll mention elsewhere in this chapter from a student roster, the roster is the sampling and throughout the book, variations in sampling frame. If the primary sampling unit for a complex techniques and nonsampling factors may further population sample is the census block, the list of reduce the legitimacy of such estimates. For ex- census blocks composes the sampling frame—in ample, those selected in a sample who fail or refuse the form of a printed booklet or, better, some digi- to participate detract further from the representa- tal format permitting computer manipulation. Here tiveness of the sample. are some reports of sampling frames appearing in research journals. In each example I’ve italicized Nevertheless, the calculations discussed in the actual sampling frames. this section can be extremely valuable to you in understanding and evaluating your data. Although The data for this research were obtained from a the calculations do not provide as precise estimates random sample of parents of children in the third as some researchers might assume, they can be grade in public and parochial schools in Yakima quite valid for practical purposes. They are unques- County, Washington. tionably more valid than less rigorously derived estimates based on less-rigorous sampling methods. (Petersen and Maynard 1981: 92) Most important, being familiar with the basic logic underlying the calculations can help you react sen- The sample at Time 1 consisted of 160 names sibly both to your own data and to those reported drawn randomly from the telephone directory of by others. Lubbock, Texas. Populations (Tan 1980: 242) and Sampling Frames The data reported in this paper . . . were gath- The preceding section introduced the theoretical ered from a probability sample of adults aged 18 model for social research sampling. Although as and over residing in households in the 48 contiguous students, research consumers, and researchers we United States. Personal interviews with 1,914 need to understand that theory, it’s no less impor- respondents were conducted by the Survey tant to appreciate the less-than-perfect conditions Research Center of the University of Michigan that exist in the field. In this section we’ll look during the fall of 1975. sampling frame That list or quasi list of units com- (Jackman and Senter 1980: 345) posing a population from which a sample is selected. If the sample is to be representative of the popula- Properly drawn samples provide information tion, it is essential that the sampling frame include appropriate for describing the population of ele- all (or nearly all) members of the population. ments composing the sampling frame—nothing more. I emphasize this point in view of the all-too- common tendency for researchers to select samples from a given sampling frame and then make assertions about a population similar to, but not identical to, the population defined by the sampling frame. For example, take a look at this report, which discusses the drugs most frequently prescribed by U.S. physicians:

Populations and Sampling Frames ■ 209 Information on prescription drug sales is not Studies of organizations are often the simplest easy to obtain. But Rinaldo V. DeNuzzo, a pro- from a sampling standpoint because organizations fessor of pharmacy at the Albany College typically have membership lists. In such cases, the of Pharmacy, Union University, Albany, NY, list of members constitutes an excellent sampling has been tracking prescription drug sales for frame. If a random sample is selected from a mem- 25 years by polling nearby drugstores. He pub- bership list, the data collected from that sample lishes the results in an industry trade magazine, may be taken as representative of all members—if MM&M. all members are included in the list. DeNuzzo’s latest survey, covering 1980, Populations that can be sampled from good is based on reports from 66 pharmacies in 48 organizational lists include elementary school, high communities in New York and New Jersey. school, and university students and faculty; church Unless there is something peculiar about that members; factory workers; fraternity or sorority part of the country, his findings can be taken members; members of social, service, or political as representative of what happens across the clubs; and members of professional associations. country. The preceding comments apply primarily to (Moskowitz 1981: 33) local organizations. Often, statewide or national organizations do not have a single membership list. What is striking in the excerpt is the casual com- There is, for example, no single list of Episcopalian ment about whether there is anything peculiar church members. However, a slightly more com- about New York and New Jersey. There is. The plex sample design could take advantage of local lifestyle in these two states hardly typifies the other church membership lists by first sampling churches 48. We cannot assume that residents in these large, and then subsampling the membership lists of urbanized, eastern seaboard states necessarily have those churches selected. (More about that later.) the same drug-use patterns that residents of Missis- sippi, Nebraska, or Vermont do. Other lists of individuals may be especially relevant to the research needs of a particular study. Does the survey even represent prescription Government agencies maintain lists of registered patterns in New York and New Jersey? To deter- voters, for example, and some political pollsters use mine that, we would have to know something registration-based sampling (RBS), using those lists. about the way the 48 communities and the In some cases, there may be delays in keeping such 66 pharmacies were selected. We should be wary files up-to-date, and a person who is registered in this regard, in view of the reference to “polling to vote may not actually do so in the election of nearby drugstores.” As we’ll see, there are sev- interest. eral methods for selecting samples that ensure representativeness, and unless they’re used, we Other lists that may be available contain the shouldn’t generalize from the study findings. names of automobile owners, welfare recipients, taxpayers, business permit holders, licensed profes- A sampling frame, then, must be consonant sionals, and so forth. Although it may be difficult with the population we wish to study. In the to gain access to some of these lists, they provide simplest sample design, the sampling frame is a excellent sampling frames for specialized research list of the elements composing the study popula- purposes. tion. In practice, though, existing sampling frames often define the study population rather than the Of course, the sampling elements in a study other way around. That is, we often begin with a need not be individuals. Social researchers might population in mind for our study; then we search use lists of universities, businesses, cities, academic for possible sampling frames. Having examined and journals, newspapers, unions, political clubs, pro- evaluated the frames available for our use, we de- fessional associations, and so forth. cide which frame presents a study population most appropriate to our needs. Telephone directories are frequently used for “quick-and-dirty” public opinion polls. They’re easy and inexpensive to use—no doubt the reason for

210 ■ Chapter 7: The Logic of Sampling their popularity. And, if you want to make asser- ample, the government maintains quite accurate tions about telephone subscribers, the directory is a population registration lists. Moreover, citizens are fairly good sampling frame. (Realize, of course, that required by law to keep their information up-to- a given directory will not include new subscribers or date, such as changes in residence or births and those who have requested unlisted numbers. Sam- deaths in the household. As a consequence, you pling is further complicated by the directories’ inclu- can select simple random samples of the popula- sion of nonresidential listings.) Unfortunately, tele- tion more easily in Japan than in the United States. phone directories are all too often used as a listing Such a registration list in the United States would of a city’s population or of its voters. Of the many conflict directly with this country’s norms regard- defects in this reasoning, the chief one involves a ing individual privacy. bias, as we have seen. Poor people are less likely to have telephones; rich people may have more than In recent years, American researchers have one line. A telephone directory sample, therefore, is begun experimenting with address files maintained likely to have a middle- or upper-class bias. by the U.S. Postal Service, such as the Special De- livery Sequence File. As problems have increasingly The class bias inherent in telephone directory arisen with regard to the sampling of telephone samples is often hidden. Preelection polls con- numbers (discussed further in Chapter 9), address- ducted in this fashion are sometimes quite accurate, based sampling (ABS) for use in mail surveys has perhaps because of the class bias evident in voting been improving (Link et al. 2008). itself: Poor people are less likely to vote. Frequently, then, these two biases nearly coincide, so that the Review of Populations results of a telephone poll may come very close to and Sampling Frames the final election outcome. Unhappily, you never know for sure until after the election. And some- Because social research literature gives surprisingly times, as in the case of the 1936 Literary Digest poll, little attention to the issues of populations and you may discover that the voters have not acted sampling frames, I’ve devoted special attention to according to the expected class biases. The ultimate them. Here is a summary of the main guidelines to disadvantage of this method, then, is the research- remember: er’s inability to estimate the degree of error to be expected in the sample findings. 1. Findings based on a sample can be taken as representing only the aggregation of elements In Chapter 9 we’ll return to the matter of that compose the sampling frame. sampling telephones, in connection with survey research. We’ll examine random-digit dialing, 2. Often, sampling frames do not truly include which was developed to resolve some of the prob- all the elements their names might imply. lems just discussed, and we’ll see that the growth in Omissions are almost inevitable. Thus, a first popularity of cell phones has further complicated concern of the researcher must be to assess the matters. extent of the omissions and to correct them if possible. (Of course, the researcher may feel Street directories and tax maps are sometimes that he or she can safely ignore a small number used for easy samples of households, but they may of omissions that cannot easily be corrected.) present incompleteness and bias. For example, in strictly zoned urban regions, illegal housing units 3. To be generalized even to the population are unlikely to appear on official records. As a composing the sampling frame, all elements result, such units could not be selected, and sample must have equal representation in the frame. findings could not be representative of those units, Typically, each element should appear only which are often poorer and more crowded than the once. Elements that appear more than once average. will have a greater probability of selection, and the sample will, overall, overrepresent those The preceding comments apply to the United elements. States but not to all countries. In Japan, for ex-

Types of Sampling Designs ■ 211 Other, more practical matters relating to popu- If your sampling frame is in a machine-read- lations and sampling frames will be treated else- able form, such as CD-ROM or magnetic tape, a where in this book. For example, the form of the computer can automatically select a simple random sampling frame—such as a list in a publication, a sample. (In effect, the computer program numbers 3-by-5 card file, CD-ROM, or magnetic tape—can the elements in the sampling frame, generates its affect how easy it is to use. And ease of use may own series of random numbers, and prints out the often take priority over scientific considerations: list of elements selected.) An “easier” list may be chosen over a “harder” one, even though the latter is more appropriate to the Figure 7-11 offers a graphic illustration of target population. We should not take a dogmatic simple random sampling. Note that the members position in this regard, but every researcher should of our hypothetical micropopulation have been carefully weigh the relative advantages and disad- numbered from 1 to 100. Moving to Appendix C, vantages of such alternatives. we decide to use the last two digits of the first col- umn and to begin with the third number from the Types of Sampling Designs top. This yields person number 30 as the first one selected into the sample. Number 67 is next, and so Up to this point, we’ve focused on simple random forth. (Person 100 would have been selected if “00” sampling. Indeed, the body of statistics typically had come up in the list.) used by social researchers assumes such a sample. As you’ll see shortly, however, you have several Systematic Sampling options in choosing your sampling method, and you’ll seldom if ever choose simple random sam- Simple random sampling is seldom used in prac- pling. There are two reasons for this. First, with all tice. As you’ll see, it’s not usually the most efficient but the simplest sampling frame, simple random method, and it can be laborious if done manually. sampling is not feasible. Second, and probably Typically, simple random sampling requires a list of surprisingly, simple random sampling may not be elements. When such a list is available, researchers the most accurate method available. Let’s turn now usually employ systematic sampling instead. to a discussion of simple random sampling and the other options available. In systematic sampling, every kth element in the total list is chosen (systematically) for inclusion Simple Random Sampling in the sample. If the list contained 10,000 elements and you wanted a sample of 1,000, you would As noted, simple random sampling (SRS) is the select every tenth element for your sample. To en- basic sampling method assumed in the statistical computations of social research. Because the math- simple random sampling (SRS) A type of prob- ematics of random sampling are especially com- ability sampling in which the units composing a plex, we’ll detour around them in favor of describ- population are assigned numbers. A set of random ing the ways of employing this method in the field. numbers is then generated, and the units having those numbers are included in the sample. Once a sampling frame has been properly established, to use simple random sampling the systematic sampling A type of probability sam- researcher assigns a single number to each element pling in which every kth unit in a list is selected for in the list, not skipping any number in the process. inclusion in the sample—for example, every 25th A table of random numbers (Appendix C) is then student in the college directory of students. You used to select elements for the sample. See “How to compute k by dividing the size of the population Do It: Using a Table of Random Numbers” for more. by the desired sample size; k is called the sampling interval. Within certain constraints, systematic sam- pling is a functional equivalent of simple random sampling and usually easier to do. Typically, the first unit is selected at random.

212 ■ Chapter 7: The Logic of Sampling How to Do It: Using a Table of Random Numbers In social research,it’s often appropriate to select a set of random num- agree to take the digits farthest to the right,480,or the middle bers from a table such as the one in Appendix C.Here’s how to do that. three digits,048,and any of these plans would work.) They key Suppose you want to select a simple random sample of 100 people is to make a plan and stick with it.For convenience,let’s use the (or other units) out of a population totaling 980. left-most three digits. 5. We can also choose to progress through the tables any way we 1. To begin,number the members of the population:in this case,from want:down the columns,up them,across to the right or to the left, 1 to 980.Now the task is to select 100 random numbers.Once or diagonally.Again,any of these plans will work just fine as long you’ve done that,your sample will consist of the people having as we stick to it.For convenience,let’s agree to move down the the numbers you’ve selected.(Note: It’s not essential to actually columns.When we get to the bottom of one column,we’ll go to number them,as long as you’re sure of the total.If you have them the top of the next. in a list,for example,you can always count through the list after 6. Now,where do we start? You can close your eyes and stick a pencil you’ve selected the numbers.) into the table and start wherever the pencil point lands.(I know it doesn’t sound scientific,but it works.) Or,if you’re afraid you’ll 2. The next step is to determine the number of digits you’ll need in hurt the book or miss it altogether,close your eyes and make up the random numbers you select.In our example,there are 980 a column number and a row number.(“I’ll pick the number in the members of the population,so you’ll need three-digit numbers to fifth row of column 2.”) Start with that number. give everyone a chance of selection.(If there were 11,825 mem- 7. Let’s suppose we decide to start with the fifth number in column bers of the population,you’d need to select five-digit numbers.) 2.If you look on the first page of Appendix C,you’ll see that the Thus,we want to select 100 random numbers in the range from starting number is 39975.We’ve selected 399 as our first random 001 to 980. number,and we have 99 more to go.Moving down the second column,we select 069,729,919,143,368,695,409,939,and so 3. Now turn to the first page of Appendix C.Notice there are several forth.At the bottom of column 2 (on the second page of this table), rows and columns of five-digit numbers,with the columns we select number 017 and continue to the top of column 3:015, continuing from the first page to the second.The table represents 255,and so on. a series of random numbers in the range from 00001 to 99999. 8. See how easy it is? But trouble lies ahead.When we reach column To use the table for your hypothetical sample,you have to answer 5,we’re speeding along,selecting 816,309,763,078,061,277,988 these questions: ...Wait a minute! There are only 980 students in the senior class. a. How will you create three-digit numbers out of five-digit How can we pick number 988? The solution is simple:Ignore it. numbers? Any time you come across a number that lies outside your range, b. What pattern will you follow in moving through the table to skip it and continue on your way:188,174,and so forth.The same select your numbers? solution applies if the same number comes up more than once.If c. Where will you start? you select 399 again,for example,just ignore it the second time. 9. That’s it.You keep up the procedure until you’ve selected 100 Each of these questions has several satisfactory answers.The key is to random numbers.Returning to your list,your sample consists of create a plan and follow it.Here’s an example. person number 399,person number 69,person number 729,and so forth. 4. To create three-digit numbers from five-digit numbers,let’s agree to select five-digit numbers from the table but consider only the left-most three digits in each case.If we picked the first number on the first page—10480—we’d consider only the 104.(We could

Types of Sampling Designs ■ 213 FIGURE 7-11 A Simple Random Sample. Having numbered everyone in the population, we can use a table of random numbers to select a repre- sentative sample from the overall population. Anyone whose number is chosen from the table is in the sample. sure against any possible human bias in using this sampling. The sampling interval is the standard method, you should select the first element at ran- distance between elements selected in the sample: dom. Thus, in the preceding example, you would ten in the preceding sample. The sampling ratio begin by selecting a random number between one and ten. The element having that number is sampling interval The standard distance between included in the sample, plus every tenth element elements selected from a population for a sample. following it. This method is technically referred to sampling ratio The proportion of elements in the as a systematic sample with a random start. Two terms population that are selected to be in a sample. are frequently used in connection with systematic

214 ■ Chapter 7: The Logic of Sampling is the proportion of elements in the population that As another example, suppose we select a are selected: 1⁄10 in the example. sample of apartments in an apartment building. If the sample is drawn from a list of apartments ar- population size ranged in numerical order (for example, 101, 102, sampling interval ϭ sample size 103, 104, 201, 202, and so on), there is a danger of the sampling interval coinciding with the number sampling ratio ϭ sample size of apartments on a floor or some multiple thereof. population size Then the samples might include only northwest- corner apartments or only apartments near the In practice, systematic sampling is virtually elevator. If these types of apartments have some identical to simple random sampling. If the list of other particular characteristic in common (for ex- elements is indeed randomized before sampling, ample, higher rent), the sample will be biased. The one might argue that a systematic sample drawn same danger would appear in a systematic sample from that list is in fact a simple random sample. of houses in a subdivision arranged with the same By now, debates over the relative merits of simple number of houses on a block. random sampling and systematic sampling have been resolved largely in favor of the latter, simpler In considering a systematic sample from a list, method. Empirically, the results are virtually identi- then, you should carefully examine the nature of cal. And, as you’ll see in a later section, systematic that list. If the elements are arranged in any par- sampling, in some instances, is slightly more ac- ticular order, you should figure out whether that curate than simple random sampling. order will bias the sample to be selected, then you should take steps to counteract any possible bias There is one danger involved in systematic (for example, take a simple random sample from sampling. The arrangement of elements in the list cyclical portions). can make systematic sampling unwise, especially an arrangement usually called periodicity. If the list Usually, however, systematic sampling is supe- of elements is arranged in a cyclical pattern that co- rior to simple random sampling, in convenience if incides with the sampling interval, a grossly biased nothing else. Problems in the ordering of elements sample might be drawn. Here are two examples in the sampling frame can usually be remedied that illustrate this danger. quite easily. In a classic study of soldiers during World War Stratified Sampling II, the researchers selected a systematic sample from unit rosters. Every tenth soldier on the roster So far we’ve discussed two methods of sample was selected for the study. The rosters, however, selection from a list: random and systematic. were arranged in a table of organizations: sergeants Stratification is not an alternative to these meth- first, then corporals and privates, squad by squad. ods; rather, it represents a possible modification of Each squad had ten members. As a result, every their use. tenth person on the roster was a squad sergeant. The systematic sample selected contained only ser- Simple random sampling and systematic geants. It could, of course, have been the case that sampling both ensure a degree of representative- no sergeants were selected for the same reason. ness and permit an estimate of the error present. Stratified sampling is a method for obtaining a stratification The grouping of the units composing greater degree of representativeness by decreasing a population into homogeneous groups (or strata) the probable sampling error. To understand this before sampling. This procedure, which may be used method, we must return briefly to the basic theory in conjunction with simple random, systematic, or of sampling distribution. cluster sampling, improves the representativeness of a sample, at least in terms of the stratification Recall that sampling error is reduced by two variables. factors in the sample design. First, a large sample produces a smaller sampling error than a small

Types of Sampling Designs ■ 215 sample does. Second, a homogeneous popula- versity lists are typically arranged by class. Lists of tion produces samples with smaller sampling faculty members may indicate their departmental errors than a heterogeneous population does. If affiliation. Government agency files may be ar- 99 percent of the population agrees with a certain ranged by geographic region. Voter registration lists statement, it’s extremely unlikely that any probabil- are arranged according to precinct. ity sample will greatly misrepresent the extent of agreement. If the population is split 50–50 on the In selecting stratification variables from among statement, then the sampling error will be much those available, however, you should be concerned greater. primarily with those that are presumably related to variables you want to represent accurately. Because Stratified sampling is based on this second gender is related to many variables and is often factor in sampling theory. Rather than selecting available for stratification, it’s often used. Education a sample from the total population at large, the is related to many variables, but it’s often not avail- researcher ensures that appropriate numbers of able for stratification. Geographic location within elements are drawn from homogeneous subsets of a city, state, or nation is related to many things. that population. To get a stratified sample of uni- Within a city, stratification by geographic loca- versity students, for example, you would first orga- tion usually increases representativeness in social nize your population by college class and then draw class, ethnic group, and so forth. Within a nation, appropriate numbers of freshmen, sophomores, it increases representativeness in a broad range of juniors, and seniors. In a nonstratified sample, attitudes as well as in social class and ethnicity. representation by class would be subjected to the same sampling error as other variables would. In a When you’re working with a simple list of sample stratified by class, the sampling error on this all elements in the population, two methods of variable is reduced to zero. stratification predominate. In one method, you sort the population elements into discrete groups based More-complex stratification methods are also on whatever stratification variables are being used. possible. In addition to stratifying by class, you On the basis of the relative proportion of the popu- might also stratify by gender, by GPA, and so forth. lation represented by a given group, you select— In this fashion you might be able to ensure that randomly or systematically—several elements from your sample would contain the proper numbers that group constituting the same proportion of of male sophomores with a 3.5 average, of female your desired sample size. For example, if sopho- sophomores with a 4.0 average, and so forth. more men with a 4.0 average compose 1 percent of the student population and you desire a sample The ultimate function of stratification, then, of 1,000 students, you would select 10 sophomore is to organize the population into homogeneous men with a 4.0 average. subsets (with heterogeneity between subsets) and to select the appropriate number of elements from The other method is to group students as each. To the extent that the subsets are homoge- described and then put those groups together in neous on the stratification variables, they may be a continuous list, beginning with all freshmen homogeneous on other variables as well. Because men with a 4.0 average and ending with all senior age is related to college class, a sample stratified by women with a 1.0 or below. You would then select class will be more representative in terms of age a systematic sample, with a random start, from as well, compared with an unstratified sample. the entire list. Given the arrangement of the list, Because occupational aspirations still seem to be a systematic sample would select proper num- related to gender, a sample stratified by gender will bers (within an error range of 1 or 2) from each be more representative in terms of occupational subgroup. (Note: A simple random sample drawn aspirations. from such a composite list would cancel out the stratification.) The choice of stratification variables typically depends on what variables are available. Gender Figure 7-12 offers a graphic illustration of can often be determined in a list of names. Uni- stratified, systematic sampling. As you can see, we

216 ■ Chapter 7: The Logic of Sampling FIGURE 7-12 A Stratified, Systematic Sample with a Random Start. A stratified, systematic sample involves two stages. First the members of the population are gathered into homogeneous strata; this simple example merely uses gender and race as stratification variables, but more could be used. Then every kth (in this case, every 10th) person in the stratified arrangement is selected into the sample. lined up our micropopulation according to gender Implicit Stratification in and race. Then, beginning with a random start Systematic Sampling of “3,” we’ve taken every tenth person thereafter: 3, 13, 23, . . . , 93. I mentioned that systematic sampling can, under certain conditions, be more accurate than simple Stratified sampling ensures the proper repre- random sampling. This is the case whenever sentation of the stratification variables; this, in turn, the arrangement of the list creates an implicit enhances the representation of other variables stratification. As already noted, if a list of university related to them. Taken as a whole, then, a stratified students is arranged by class, then a systematic sample is more likely than a simple random sample sample provides a stratification by class where a to be more representative on several variables. Al- simple random sample would not. though the simple random sample is still regarded as somewhat sacred, it should now be clear that In a study of students at the University of you can often do better. Hawaii, after stratification by school class, the stu-

Types of Sampling Designs ■ 217 dents were arranged by their student identification university, including all colleges and departments, numbers. These numbers, however, were their both undergraduate and graduate students, and social security numbers. The first three digits of the both U.S. and foreign students. The computer pro- social security number indicate the state in which gram used for sampling then limited consideration the number was issued. As a result, within a class, to students fitting this definition. students were arranged by the state in which they were issued a social security number, providing a Stratification rough stratification by geographic origin. The sampling program also permitted stratification An ordered list of elements, therefore, may be of students before sample selection. The researchers more useful to you than an unordered, randomized decided that stratification by college class would be list. I’ve stressed this point in view of the unfortu- sufficient, although the students might have been nate belief that lists should be randomized before further stratified within class, if desired, by gender, systematic sampling. Only if the arrangement college, major, and so forth. presents the problems discussed earlier should the list be rearranged. Sample Selection Illustration: Sampling Once the students had been arranged by class, a University Students systematic sample was selected across the entire rearranged list. The sample size for the study was Let’s put these principles into practice by looking at initially set at 1,100. To achieve this sample, the an actual sampling design used to select a sample sampling program was set for a 1⁄14 sampling ratio. of university students. The purpose of the study The program generated a random number between was to survey, with a mail-out questionnaire, a 1 and 14; the student having that number and representative cross section of students attending every 14th student thereafter was selected in the the main campus of the University of Hawaii. The sample. following sections describe the steps and decisions involved in selecting that sample. Once the sample had been selected, the com- puter was instructed to print each student’s name Study Population and Sampling Frame and mailing address on self-adhesive mailing labels. These labels were then simply transferred to enve- The obvious sampling frame available for use in this lopes for mailing the questionnaires. sample selection was the computerized file main- tained by the university administration. The file Sample Modification contained students’ names, local and permanent addresses, and social security numbers, as well as a This initial design of the sample had to be modified. variety of other information such as field of study, Before the mailing of questionnaires, the re- class, age, and gender. searchers discovered that, because of unexpected expenses in the production of the questionnaires, The computer database, however, contained they couldn’t cover the costs of mailing to all 1,100 entries on all people who could, by any conceiv- students. As a result, one-third of the mailing labels able definition, be called students, many of whom were systematically selected (with a random start) seemed inappropriate for the purposes of the study. for exclusion from the sample. The final sample for As a result, researchers needed to define the study the study was thereby reduced to 733 students. population in a somewhat more restricted fashion. The final definition included those 15,225 day- I mention this modification in order to il- program degree candidates who were registered lustrate the frequent need to alter a study plan in for the fall semester on the Manoa campus of the midstream. Because the excluded students were systematically omitted from the initial systematic sample, the remaining 733 students could still be

218 ■ Chapter 7: The Logic of Sampling taken as reasonably representing the study popula- Another typical situation concerns sampling tion. The reduction in sample size did, of course, among population areas such as a city. Although increase the range of sampling error. there is no single list of a city’s population, citizens reside on discrete city blocks or census blocks. Re- Multistage Cluster Sampling searchers can, therefore, select a sample of blocks initially, create a list of people living on each of the The preceding sections have dealt with reason- selected blocks, and take a subsample of the people ably simple procedures for sampling from lists of on each block. elements. Such a situation is ideal. Unfortunately, however, much interesting social research requires In a more complex design, researchers might the selection of samples from populations that sample blocks, list the households on each selected cannot easily be listed for sampling purposes: the block, sample the households, list the people resid- population of a city, state, or nation; all univer- ing in each household, and, finally, sample the peo- sity students in the United States; and so forth. In ple within each selected household. This multistage such cases, the sample design must be much more sample design leads ultimately to a selection of complex. Such a design typically involves the initial a sample of individuals but does not require the ini- sampling of groups of elements—clusters—followed tial listing of all individuals in the city’s population. by the selection of elements within each of the selected clusters. Multistage cluster sampling, then, involves the repetition of two basic steps: listing and sampling. Cluster sampling may be used when it’s The list of primary sampling units (churches, either impossible or impractical to compile an blocks) is compiled and, perhaps, stratified for sam- exhaustive list of the elements composing the pling. Then a sample of those units is selected. The target population, such as all church members in selected primary sampling units are then listed and the United States. Often, however, the population perhaps stratified. The list of secondary sampling elements are already grouped into subpopulations, units is then sampled, and so forth. and a list of those subpopulations either exists or can be created practically. For example, church The listing of households on even the selected members in the United States belong to discrete blocks is, of course, a labor-intensive and costly churches, which are either listed or could be. Fol- activity— one of the elements making face-to- lowing a cluster sample format, then, researchers face, household surveys quite expensive. Vincent could sample the list of churches in some manner Iannacchione, Jennifer Staab, and David Redden (for example, a stratified, systematic sample). Next, (2003) report some initial success using postal mail- they would obtain lists of members from each of ing lists for this purpose. Although the lists are not the selected churches. Each of the lists would then perfect, they may be close enough to warrant the be sampled, to provide samples of church members significant savings in cost. for study. Multistage cluster sampling makes possible those studies that would otherwise be impossible. Specific research circumstances often call for spe- cial designs, as “Sampling Iran” demonstrates. cluster sampling A multistage sampling in which Multistage Designs natural groups (clusters) are sampled initially, with and Sampling Error the members of each selected group being sub- sampled afterward. For example, you might select Although cluster sampling is highly efficient, the a sample of U.S. colleges and universities from a price of that efficiency is a less-accurate sample. directory, get lists of the students at all the selected A simple random sample drawn from a popula- schools, then draw samples of students from each. tion list is subject to a single sampling error, but a two-stage cluster sample is subject to two sampling

Multistage Cluster Sampling ■ 219 Sampling Iran Whereas most of the examples given in this textbook are taken 6. The eastern provinces including Khorasan and Semnan from its country of origin,the United States,the basic methods 7. The northern provinces including Gilan,Mazandran and of sampling would apply in other national settings as well.At the same time,researchers may need to make modifications appropriate to local Golestan conditions.In selecting a national sample of Iran,for example,Hamid 8. Systan Abdollahyan and Taghi Azadarmaki (2000:21) from the University of 9. Kurdistan Tehran began by stratifying the nation on the basis of cultural differences, dividing the country into nine cultural zones as follows: Within each of these cultural areas,the researchers selected samples of census blocks and,on each selected block,a sample 1. Tehran of households.Their sample design made provisions for getting 2. Central region including Isfahan,Arak,Qum,Yazd and Kerman the proper numbers of men and women as respondents within 3. The southern provinces including Hormozgan,Khuzistan,Bushehr households and provisions for replacing those households where no one was at home. and Fars 4. The marginal western region including Lorestan,Charmahal and Source: Hamid Abdollahyan and Taghi Azadarmaki,Sampling Design in a Survey Research:The Sampling Practice in Iran, paper presented to the meetings of the Bakhtiari,Kogiluyeh and Eelam American Sociological Association,August 12–16,2000,Washington,DC. 5. The western provinces including western and eastern Azarbaijan, Zanjan,Ghazvin and Ardebil errors. First, the initial sample of clusters will Recall that sampling error is reduced by two represent the population of clusters only within a factors: an increase in the sample size and increased range of sampling error. Second, the sample of ele- homogeneity of the elements being sampled. These ments selected within a given cluster will represent factors operate at each level of a multistage sample all the elements in that cluster only within a range design. A sample of clusters will best represent all of sampling error. Thus, for example, a researcher clusters if a large number are selected and if all runs a certain risk of selecting a sample of dispro- clusters are very much alike. A sample of elements portionately wealthy city blocks, plus a sample will best represent all elements in a given cluster of disproportionately wealthy households within if a large number are selected from the cluster and those blocks. The best solution to this problem lies if all the elements in the cluster are very much in the number of clusters selected initially and the alike. number of elements within each cluster. With a given total sample size, however, if the Typically, researchers are restricted to a total number of clusters is increased, the number of sample size; for example, you may be limited to elements within a cluster must be decreased. In conducting 2,000 interviews in a city. Given this this respect, the representativeness of the clusters is broad limitation, however, you have several options increased at the expense of more poorly represent- in designing your cluster sample. At the extremes ing the elements composing each cluster, or vice you could choose one cluster and select 2,000 ele- versa. Fortunately, homogeneity can be used to ments within that cluster, or you could select 2,000 ease this dilemma. clusters with one element selected within each. Of course, neither approach is advisable, but a broad Typically, the elements composing a given range of choices lies between them. Fortunately, natural cluster within a population are more ho- the logic of sampling distributions provides a gen- mogeneous than all elements composing the total eral guideline for this task. population are. The members of a given church are more alike than all church members are; the

220 ■ Chapter 7: The Logic of Sampling residents of a given city block are more alike than essarily smaller at each stage than the total sample the residents of a whole city are. As a result, rela- size, the sampling error at each stage will be greater tively few elements may be needed to represent a than would be the case for a single-stage random given natural cluster adequately, although a larger sample of elements. Second, sampling error is esti- number of clusters may be needed to represent mated on the basis of observed variance among the adequately the diversity found among the clusters. sample elements. When those elements are drawn This fact is most clearly seen in the extreme case from among relatively homogeneous clusters, the of very different clusters composed of identical estimated sampling error will be too optimistic and elements within each. In such a situation, a large must be corrected in the light of the cluster sample number of clusters would adequately represent all design. its members. Although this extreme situation never exists in reality, it’s closer to the truth in most cases Stratification in Multistage than its opposite: identical clusters composed of Cluster Sampling grossly divergent elements. Thus far, we’ve looked at cluster sampling as The general guideline for cluster design, then, is though a simple random sample were selected to maximize the number of clusters selected while at each stage of the design. In fact, stratification decreasing the number of elements within each techniques can be used to refine and improve the cluster. However, this scientific guideline must be sample being selected. balanced against an administrative constraint. The efficiency of cluster sampling is based on the ability The basic options here are essentially the same to minimize the listing of population elements. By as those in single-stage sampling from a list. In se- initially selecting clusters, you need only list the lecting a national sample of churches, for example, elements composing the selected clusters, not all you might initially stratify your list of churches by elements in the entire population. Increasing the denomination, geographic region, size, rural or number of clusters, however, goes directly against urban location, and perhaps by some measure of this efficiency factor. A small number of clusters social class. may be listed more quickly and more cheaply than a large number. (Remember that all the elements in Once the primary sampling units (churches, a selected cluster must be listed even if only a few blocks) have been grouped according to the rele- are to be chosen in the sample.) vant, available stratification variables, either simple random or systematic-sampling techniques can The final sample design will reflect these two be used to select the sample. You might select constraints. In effect, you’ll probably select as many a specified number of units from each group, clusters as you can afford. Lest this issue be left too or stratum, or you might arrange the stratified clus- open-ended at this point, here’s one general guide- ters in a continuous list and systematically sample line. Population researchers conventionally aim at that list. the selection of 5 households per census block. If a total of 2,000 households are to be interviewed, To the extent that clusters are combined you would aim at 400 blocks with 5 household into homogeneous strata, the sampling error at interviews on each. Figure 7-13 presents a graphic this stage will be reduced. The primary goal of overview of this process. stratification, as before, is homogeneity. Before we turn to other, more detailed proce- There’s no reason why stratification couldn’t dures available to cluster sampling, let me reiterate take place at each level of sampling. The elements that this method almost inevitably involves a loss of listed within a selected cluster might be stratified accuracy. The manner in which this appears, how- before the next stage of sampling. Typically, how- ever, is somewhat complex. First, as noted earlier, ever, this is not done. (Recall the assumption of a multistage sample design is subject to a sampling relative homogeneity within clusters.) error at each stage. Because the sample size is nec-

Multistage Cluster Sampling ■ 221 FIGURE 7-13 Multistage Cluster Sampling. In multistage cluster sampling, we begin by selecting a sample of the clusters (in this case, city blocks). Then, we make a list of the elements (households, in this case) and select a sample of elements from each of the selected clusters. Probability Proportionate ing discussion, I talked about selecting a random or to Size (PPS) Sampling systematic sample of clusters and then a random or systematic sample of elements within each cluster This section introduces you to a more sophisticated selected. Notice that this produces an overall sam- form of cluster sampling, one that is used in many pling scheme in which every element in the whole large-scale survey-sampling projects. In the preced- population has the same probability of selection.

222 ■ Chapter 7: The Logic of Sampling Let’s say we’re selecting households within a As the name suggests, each cluster is given a city. If there are 1,000 city blocks and we initially chance of selection proportionate to its size. Thus, select a sample of 100, that means that each block a city block with 200 households has twice the has a 100⁄1,000 or 0.1 chance of being selected. If chance of selection as one with only 100 house- we next select 1 household in 10 from those resid- holds. Within each cluster, however, a fixed ing on the selected blocks, each household has a number of elements is selected, say, 5 households 0.1 chance of selection within its block. To calcu- per block. Notice how this procedure results in each late the overall probability of a household being household having the same probability of selection selected, we simply multiply the probabilities at the overall. individual steps in sampling. That is, each house- hold has a 1⁄10 chance of its block being selected Let’s look at households of two different city and a 1⁄10 chance of that specific household being blocks. Block A has 100 households; Block B has selected if the block is one of those chosen. Each only 10. In PPS sampling, we would give Block household, in this case, has a 1⁄10 ϫ 1⁄10 ϭ 1⁄100 A ten times as good a chance of being selected as chance of selection overall. Because each house- Block B. So if, in the overall sample design, Block hold would have the same chance of selection, the A has a 1⁄20 chance of being selected, that means sample so selected should be representative of all Block B would only have a 1⁄200 chance. Notice households in the city. that this means that all the households on Block A would have a 1⁄20 chance of having their block There are dangers in this procedure, however. selected; Block B households have only a 1⁄200 In particular, the variation in the size of blocks chance. (measured in numbers of households) presents a problem. Let’s suppose that half the city’s popula- If Block A is selected and we’re taking 5 house- tion resides in 10 densely packed blocks filled with holds from each selected block, then the house- high-rise apartment buildings, and suppose that holds on Block A have a 5⁄100 chance of being the rest of the population lives in single-family selected into the block’s sample. Because we can dwellings spread out over the remaining 900 multiply probabilities in a case like this, we see that blocks. When we first select our sample of 1⁄10 of every household on Block A has an overall chance the blocks, it’s quite possible that we’ll miss all of of selection equal to 1⁄20 ϫ 5⁄100 ϭ 5⁄2000 ϭ 1⁄400. the 10 densely packed high-rise blocks. No matter what happens in the second stage of sampling, our If Block B happens to be selected, on the final sample of households will be grossly unrepre- other hand, its households stand a much better sentative of the city, comprising only single-family chance of being among the 5 chosen there: 5⁄10. dwellings. When this is combined with their relatively poorer chance of having their block selected in the first Whenever the clusters sampled are of greatly place, however, they end up with the same chance differing sizes, it’s appropriate to use a modified of selection as those on Block A: 1⁄200 ϫ 5⁄10 ϭ sampling design called PPS (probability propor- 5⁄2000 ϭ 1⁄400. tionate to size). This design guards against the problem I’ve just described and still produces a final Further refinements to this design make it sample in which each element has the same chance a very efficient and effective method for select- of selection. ing large cluster samples. For now, however, it’s enough to understand the basic logic involved. PPS (probability proportionate to size) This Disproportionate Sampling refers to a type of multistage cluster sample in which and Weighting clusters are selected, not with equal probabilities (see EPSEM) but with probabilities proportionate to Ultimately, a probability sample is representative of their sizes—as measured by the number of units to a population if all elements in the population have be subsampled. an equal chance of selection in that sample. Thus,

Multistage Cluster Sampling ■ 223 in each of the preceding discussions, we’ve noted households should be given a weight of 3 n. This 4 that the various sampling procedures result in an weighting procedure could be simplified by merely equal chance of selection—even though the ulti- giving a weight of 3 to each of the households selected outside the suburban area. mate selection probability is the product of several Here’s an example of the problems that can partial probabilities. be created when disproportionate sampling is not accompanied by a weighting scheme. When More generally, however, a probability sample the Harvard Business Review decided to survey its subscribers on the issue of sexual harassment at is one in which each population element has a work, it seemed appropriate to oversample women because female subscribers were vastly outnum- known nonzero probability of selection—even bered by male subscribers. Here’s how G. C. Collins and Timothy Blodgett explained the matter: though different elements may have different We also skewed the sample another way: to probabilities. If controlled probability sampling ensure a representative response from women, we mailed a questionnaire to virtually every procedures have been used, any such sample may female subscriber, for a male/female ratio of 68% to 32%. This bias resulted in a response of be representative of the population from which it is 52% male and 44% female (and 4% who gave no indication of gender)—compared to HBR’s drawn if each sample element is assigned a weight U.S. subscriber proportion of 93% male and 7% female. equal to the inverse of its probability of selection. (1981: 78) Thus, where all sample elements have had the Notice a couple of things in this excerpt. First, same chance of selection, each is given the same it would be nice to know a little more about what “virtually every female” means. Evidently, the weight: 1. This is called a self-weighting sample. authors of the study didn’t send questionnaires to all female subscribers, but there’s no indication of Sometimes it’s appropriate to give some who was omitted and why. Second, they didn’t use the term representative with its normal social cases more weight than others, a process called science usage. What they mean, of course, is that they wanted to get a substantial or “large enough” weighting. Disproportionate sampling and weight- response from women, and oversampling is a perfectly acceptable way of accomplishing that. ing come into play in two basic ways. First, you By sampling more women than a straightfor- may sample subpopulations disproportionately ward probability sample would have produced, the authors were able to “select” enough women to ensure sufficient numbers of cases from each (812) to compare with the men (960). Thus, when for analysis. For example, a given city may have weighting Assigning different weights to cases that were selected into a sample with different probabili- a suburban area containing one-fourth of its total ties of selection. In the simplest scenario, each case is given a weight equal to the inverse of its probability population. Yet you might be especially interested of selection. When all cases have the same chance of selection, no weighting is necessary. in a detailed analysis of households in that area and may feel that one-fourth of this total sample size would be too few. As a result, you might decide to select the same number of households from the suburban area as from the remainder of the city. Households in the suburban area, then, are given a disproportionately better chance of selection than those located elsewhere in the city are. As long as you analyze the two area samples separately or comparatively, you need not worry about the differential sampling. If you want to combine the two samples to create a composite picture of the entire city, however, you must take the disproportionate sampling into account. If n is the number of households selected from each area, then the households in the suburban area had a chance of selection equal to n divided by one-fourth of the total city population. Because the total city population and the sample size are the same for both areas, the suburban-area households should be given a weight of 1 n, and the remaining 4


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