21ODMPT655-Research Methods and Statistics –I

Sampling Design and Data Collection 145 population. A defined target population consists of the complete group of elements (people or objects) that are specifically identified for investigation according to the objectives of the research project. A precise definition of the target population is usually done in terms of elements, sampling units and time frames. An element is a single member of the population. It is a person or object from which the data/ information is sought. Elements must be unique be countable and when added together make up the whole of the target population. If 250 workers in a concern happen to the population of interest to the researcher, then each worker therein is an element. The population frame is listing of all elements in the population from which the sample is drawn. The nominal roll of class students could be the population frame for the study of students in a class. A subject is a single member of the sample, just as an element is a single member of the population. If 200 members from the total population of 500 workers form the sample for the study, then each worker in the sample is a subject. The confidence level which is also termed as reliability is expressed in terms of percentage of times that an actual value will fall within the prescribed precision limit. For example, if we have to consider a confidence level of 90% which will imply that if we repeat a particular exercise 100 times, 90 times the parameters of population under consideration will lie within the prescribed limit. The significant level, on the other hand, indicates the likelihood of the observation falling outside the prescribed range. Therefore, if the confidence level (CL) is 90%, then significance level (SL) would be 10%. If the confidence level is 98%, then significance level is 2%. Therefore, significance level can mathematically be expressed as SL% = 100% – CL%. A sample design is a definite plan for obtaining a sample from a given population Sample constitutes a certain portion of the population or universe. Sampling design refers to the technique or the procedure the researcher adopts for selecting items for the sample from the population or universe. CU IDOL SELF LEARNING MATERIAL (SLM)

146 Research Methods and Statistics - I A sample design helps to decide the number of items to be included in the sample, i.e., the size of the sample. The sample design should be determined prior to data collection. Data is the facts in raw or unorganised form such as alphabets, numbers or symbols that refer to or represent conditions, ideas or objects. This represents facts and statistics which are collected together for reference or analysis. Primary data refers to information gathered first hand by the researcher for the specific purpose of the study. It is raw data without interpretation and represents the personal or official opinion or position. Primary sources are most authoritative since the information is not filtered or tampered. Some examples of the sources of primary data are individuals, focus groups and panel of respondents. Data collection from individuals can be made through interviews, observation, etc. Secondary data refers to the information gathered from already existing sources. Secondary data may be either published or unpublished data. Tertiary sources are an interpretation of a secondary source. It is generally represented by index, bibliographies, dictionaries, encyclopedias, handbooks, directories and other finding aids like the internet search engines. Data collection method is an integral part of the research design. There are various methods of data collection. Each method has its own advantages and disadvantages. Selection of an appropriate method of data collection may enhance the value of research and at the same time the wrong choice may lead to questionable research findings. Data collection methods include interviews, self- administered questionnaires, observations and other methods. Questionnaire is a list of questions or statements pertaining to an issue or program. It is used for studying the opinions of people. It is commonly used in opinion polls. People are asked to express their responses to the listed or reactions to the listed statements. Data editing is the activity aimed at detecting and correcting errors (logical inconsistencies) in data. Editing techniques refers to a range of procedures and processes used for detecting and handling errors in data. CU IDOL SELF LEARNING MATERIAL (SLM)

Sampling Design and Data Collection 147 Tabulation may be defined as the systematic presentation of numerical data in rows or/and columns according to certain characteristics. It expresses the data in concise and attractive form which can be easily understood, and used to compare numerical figures. Tabulation is the process of arranging data into rows and column. Rows are horizontal arrangements whereas columns are vertical arrangements. Tabulation may be simple, double or complex depending upon the type of classification. 6.26 Key Words/Abbreviations  Sampling: Sampling is the process of learning about the population on the basis of a sample drawn from it.  Laws of Sampling: The law of statistical regularity and the law of inertia of large numbers.  Sampling Method: A sampling design is a definite plan for obtaining a sample from the sampling frame.  Sampling Design: A sample design is a definite plan for obtaining a sample from a given population.  Data Collection: Data collection is the process of gathering and measuring information on variables of interest.  Data Transcription: Data transcription would probably be outsourced to foreign countries with low labour costs. 6.27 Learning Activity 1. You are required to find out most suitable sampling method for data collection with respect to FMCG. _________________________________________________________________ _________________________________________________________________ CU IDOL SELF LEARNING MATERIAL (SLM)

148 Research Methods and Statistics - I 2. You are suggested to find psychological and certain behavioural problem. Prepare a questionnaire related to new startups in India. _________________________________________________________________ _________________________________________________________________ 3. You are requested to identify the data editing procedure and prepare the report based on those issues. _________________________________________________________________ _________________________________________________________________ 6.28 Unit End Exercises (MCQs and Descriptive) Descriptive Type Questions 1. Give the meaning of Sampling. 2. Discuss the purpose of Sampling. 3. Explain the various steps in developing a Sampling Plan. 4. Discuss the various Sampling Techniques. 5. What is Sampling Design? Discuss the characteristics of a Good Sample Design. 6. What is Data Collection? Explain the various methods of Data Collection. 7. What is Questionnaire? Explain the procedure of design the Questionnaire. 8. What is Tabulation of Data? Explain the various types of Tabulation. 9. What is Editing Data? Explain the various types of Editing of data. 10. What is Data Transcription? Discuss the functions of Data Transcription. CU IDOL SELF LEARNING MATERIAL (SLM)

Sampling Design and Data Collection 149 Multiple Choice Questions 1. Which is a key term in sampling? (b) Firm (a) Population (d) None of the above (c) Country 2. Quota sampling is the non-probability version of __________. (a) Stratified Sampling (b) Random Sampling (c) Convenience Sampling (d) Snowball Sampling 3. Which gives probability of being selected as sample is equal? (b) Project (b) Factor Analysis (c) Randomness (d) None 4. What is distribution of infinite number of sample of population called? (a) Sampling Distribution (b) Data Collection (c) Research (d) None 5. Which is rarely done in sampling distribution? (a) Research (b) Construction (c) Distribution (d) None 6. Greater the sample size, what is standard error? (a) Smaller (b) Average (c) Larger (d) None CU IDOL SELF LEARNING MATERIAL (SLM)

150 Research Methods and Statistics - I 7. What is also called random sampling? (a) None (b) Both (d) Probability (c) Theory Answers: 1. (a), 2. (a), 3. (c), 4. (a), 5. (b) 6. (a), 7. (d). 6.29 References References of this unit have been given at the end of the book.  CU IDOL SELF LEARNING MATERIAL (SLM)

Reliability and Validity 151 UNIT 7 RELIABILITYAND VALIDITY Structure: 7.0 Learning Objectives 7.1 Introduction 7.2 Concept of Reliability 7.3 Types of Reliability 7.4 Concept of Validity 7.5 Types of Validity in Research 7.6 Factors that Impact Validity 7.7 Summary 7.8 Key Words/Abbreviations 7.9 LearningActivity 7.10 Unit End Exercises (MCQs and Descriptive) 7.11 References 7.0 Learning Objectives After studying this unit, you will be able to:  Describe the concepts and explore the types  Explain the various types of validity in research CU IDOL SELF LEARNING MATERIAL (SLM)

152 Research Methods and Statistics - I 7.1 Introduction Reliability in statistics and psychometrics is the overall consistency of a measure. A measure is said to have a high reliability if it produces similar results under consistent conditions. It is the characteristic of a set of test scores that relates to the amount of random error from the measurement process that might be embedded in the scores. Scores that are highly reliable are accurate, reproducible, and consistent from one testing occasion to another. That is, if the testing process were repeated with a group of test takers, essentially the same results would be obtained. Validity is the extent to which a concept, conclusion or measurement is well-founded and likely corresponds accurately to the real world. The validity of a measurement tool for example, a test in education) is the degree to which the tool measures what it claims to measure. Validity has a particular application known as test validity: “the degree to which evidence and theory support the interpretations of test scores”. It is generally accepted that the concept of scientific validity addresses the nature of reality in terms of statistical measures and as such is an epistemological and philosophical issue as well as a question of measurement. 7.2 Concept of Reliability The term reliability in research refers to the consistency of a research study or measuring test. For example, if a person weighs themselves during the course of a day, they would expect to see a similar reading. Scales which measured weight differently each time would be of little use. The same analogy could be applied to a tape measure which measures inches differently each time it was used. It would not be considered reliable. If findings from research are replicated consistently they are reliable. A correlation coefficient can be used to assess the degree of reliability. If a test is reliable, it should show a high positive correlation. Of course, it is unlikely the exact same results will be obtained each time as participants and situations vary, but a strong positive correlation between the results of the same test indicates reliability. CU IDOL SELF LEARNING MATERIAL (SLM)

Reliability and Validity 153 7.3 Types of Reliability Various types of reliability are as follows: 1. Test-Retest Reliability 2. Internal Consistency 3. Inter-rater Reliability 4. Parallel Forms Reliability 1. Test-Retest Reliability When researchers measure a construct that they assume to be consistent across time, the scores they obtain should also be consistent across time. Test-retest reliability is the extent to which this is actually the case. For example, intelligence is generally thought to be consistent across time. A person who is highly intelligent today will be highly intelligent next week. This means that any good measure of intelligence should produce roughly the same scores for this individual next week as it does today. Clearly, a measure that produces highly inconsistent scores over time cannot be a very good measure of a construct that is supposed to be consistent. Assessing test-retest reliability requires using the measure on a group of people at one time, using it again on the same group of people at a later time, and then looking at test-retest correlation between the two sets of scores. This is typically done by graphing the data in a scatterplot and computing Pearson’s r. Figure below shows the correlation between two sets of scores of several university students on the Rosenberg Self-esteem Scale, administered two times, a week apart. Pearson’s r for these data is +.95. In general, a test-retest correlation of +.80 or greater is considered to indicate good reliability. CU IDOL SELF LEARNING MATERIAL (SLM)

154 Research Methods and Statistics - I Figure 7.1: Test-Retest Reliability Again, high test-retest correlations make sense when the construct being measured is assumed to be consistent over time, which is the case for intelligence, self-esteem, and the Big Five personality dimensions. But other constructs are not assumed to be stable over time. The very nature of mood, for example, is that it changes. So, a measure of mood that produced a low test-retest correlation over a period of a month would not be a cause for concern. 2. Internal Consistency A second kind of reliability is internal consistency, which is the consistency of people’s responses across the items on a multiple-item measure. In general, all the items on such measures are supposed to reflect the same underlying construct, so people’s scores on those items should be correlated with each other. On the Rosenberg Self-esteem Scale, people who agree that they are a person of worth should tend to agree that that they have a number of good qualities. If people’s responses to the different items are not correlated with each other, then it would no longer make sense to claim that CU IDOL SELF LEARNING MATERIAL (SLM)

Reliability and Validity 155 they are all measuring the same underlying construct. This is as true for behavioural and physiological measures as for self-report measures. For example, people might make a series of bets in a simulated game of roulette as a measure of their level of risk seeking. This measure would be internally consistent to the extent that individual participants’ bets were consistently high or low across trials. Like test-retest reliability, internal consistency can only be assessed by collecting and analysing data. One approach is to look at a split-half correlation. This involves splitting the items into two sets, such as the first and second halves of the items or the even- and odd-numbered items. Then a score is computed for each set of items, and the relationship between the two sets of scores is examined. For example, Figure below shows the split-half correlation between several university students’ scores on the even-numbered items and their scores on the odd-numbered items of the Rosenberg Self- esteem Scale. Pearson’s r for these data is +.88. A split-half correlation of +.80 or greater is generally considered good internal consistency. Figure 7.2: Internal Consistency CU IDOL SELF LEARNING MATERIAL (SLM)

156 Research Methods and Statistics - I Perhaps, the most common measure of internal consistency used by researchers in psychology is a statistic called Cronbach’s a (the Greek letter alpha). Conceptually, a is the mean of all possible split-half correlations for a set of items. For example, there are 252 ways to split a set of 10 items into two sets of five. Cronbach’s a would be the mean of the 252 split-half correlations. Note that this is not how a is actually computed, but it is a correct way of interpreting the meaning of this statistic. Again, a value of +.80 or greater is generally taken to indicate good internal consistency. 3. Inter-rater Reliability Many behavioural measures involve significant judgment on the part of an observer or a rater. Inter-rater reliability is the extent to which different observers are consistent in their judgments. For example, if you were interested in measuring university students’ social skills, you could make video recordings of them as they interacted with another student whom they are meeting for the first time. Then you could have two or more observers watch the videos and rate each student’s level of social skills. To the extent that each participant does in fact have some level of social skills that can be detected by an attentive observer, different observers’ ratings should be highly correlated with each other. Inter-rater reliability would also have been measured in Bandura’s Bobo doll study. In this case, the observers’ ratings of how many acts of aggression a particular child committed while playing with the Bobo doll should have been highly positively correlated. Inter-rater reliability is often assessed using Cronbach’s a when the judgments are quantitative or an analogous statistic called Cohen’s d (the Greek letter kappa) when they are categorical. 4. Parallel Forms Reliability Parallel forms reliability measures the correlation between two equivalent versions of a test. You use it when you have two different assessment tools or sets of questions designed to measure the same thing. Parallel forms reliability also called equivalent forms reliability uses one set of questions divided into two equivalent sets (“forms”), where both sets contain questions that measure the same construct, knowledge or skill. The two sets of questions are given to the same sample of people within a short period of time and an estimate of reliability is calculated from the two sets. Put simply, you are trying to find out if test A measures the same thing as test B. In other words, you want to know if test scores stay the same when you use different instruments. CU IDOL SELF LEARNING MATERIAL (SLM)

Reliability and Validity 157 Example You want to find the reliability for a test of mathematics comprehension. So, you create a set of 100 questions that measure that construct. You randomly split the questions into two sets of 50 (set A and set B), and administer those questions to the same group of students a week apart. Steps Step 1: Give test A to a group of 50 students on a Monday. Step 2: Give test B to the same group of students that Friday. Step 3: Correlate the scores from test A and test B. In order to call the forms “parallel”, the observed score must have the same mean and variances. If the tests are merely different versions (without the “sameness” of observed scores), they are called alternate forms. If you want to use multiple different versions of a test (e.g., to avoid respondents repeating the same answers from memory), you first need to make sure that all the sets of questions or measurements give reliable results. In educational assessment, it is often necessary to create different versions of tests to ensure that students do not have access to the questions in advance. Parallel forms reliability means that, if the same students take two different versions of a reading comprehension test, they should get similar results in both tests. How to Measure it? The most common way to measure parallel forms reliability is to produce a large set of questions to evaluate the same thing, then divide these randomly into two question sets. The same group of respondents answers both sets, and you calculate the correlation between the results. High correlation between the two indicates high parallel forms reliability. CU IDOL SELF LEARNING MATERIAL (SLM)

158 Research Methods and Statistics - I Parallel Forms Reliability Example A set of questions is formulated to measure financial risk aversion in a group of respondents. The questions are randomly divided into two sets, and the respondents are randomly divided into two groups. Both groups take both tests: group A takes test A first, and group B takes test B first. The results of the two tests are compared, and the results are almost identical, indicating high parallel forms reliability. Ensure that all questions or test items are based on the same theory and formulated to measure the same thing. 7.4 Concept of Validity The concept of validity was formulated by Kelly (1927) who stated that a test is valid if it measures what it claims to measure. The validity of an instrument is the idea that the instrument measures what it intends to measure. Validity pertains to the connection between the purpose of the research and which data the researcher chooses to quantify that purpose. Example Imagine a researcher who decides to measure the intelligence of a sample of students. Some measures, like physical strength, possess no natural connection to intelligence. Thus, a test of physical strength, like how many push-ups a student could do, would be an invalid test of intelligence. A test of intelligence should measure intelligence and not something else (such as memory). A distinction can be made between internal and external validity. These types of validity are relevant to evaluating the validity of a research study/procedure. Validity is the extent to which the scores from a measure represent the variable they are intended to. When a measure has good test-retest reliability and internal consistency, researchers should be more confident that the scores represent what they are supposed to. There has to be more to it, however, because a measure can be extremely reliable but have no validity whatsoever. As an absurd example, imagine someone who believes that people’s index finger length reflects their self- esteem and therefore tries to measure self-esteem by holding a ruler up to people’s index fingers. CU IDOL SELF LEARNING MATERIAL (SLM)

Reliability and Validity 159 Although this measure would have extremely good test-retest reliability, it would have absolutely no validity. The fact that one person’s index finger is a centimeter longer than another’s would indicate nothing about which one higher self-esteem had. Internal validity refers to whether the effects observed in a study are due to the manipulation of the independent variable and not some other factor. In other words, there is a causal relationship between the independent and dependent variable. Internal validity can be improved by controlling extraneous variables, using standardised instructions, counter-balancing, and eliminating demand characteristics and investigator effects. External validity refers to the extent to which the results of a study can be generalised to other settings (ecological validity), other people (population validity) and over time (historical validity). External validity can be improved by setting experiments in a more natural setting and using random sampling to select participants. 7.5 Types of Validity in Research Various types of Validity in Research are as follows: 1. Construct Validity Construct validity occurs when the theoretical constructs of cause and effect accurately represent the real-world situations they are intended to model. This is related to how well the experiment is operationalised. A good experiment turns the theory (constructs) into actual things you can measure. Sometimes, just finding out more about the construct (which itself must be valid) can be helpful. Construct validity is thus an assessment of the quality of an instrument or experimental design. It says 'Does it measure the construct it is supposed to measure'. If you do not have construct validity, you will likely draw incorrect conclusions from the experiment (garbage in, garbage out). Example There is no objective, observable entity called “depression” that we can measure directly. But based on existing psychological research and theory, we can measure depression based on a collection of symptoms and indicators, such as low self-confidence and low energy levels. CU IDOL SELF LEARNING MATERIAL (SLM)

160 Research Methods and Statistics - I Construct Validity can be classified into the following categories: (a) Convergent Validity: Convergent validity occurs where measures of constructs that are expected to correlate do so. This is similar to concurrent validity (which looks for correlation with other tests). (b) Discriminant Validity: Discriminant validity occurs where constructs that are expected not to relate do not, such that it is possible to discriminate between these constructs. Convergence and discrimination are often demonstrated by correlation of the measures used within constructs. Convergent validity and Discriminant validity together demonstrate construct validity. (c) Nomological Network: Defined by Cronbach and Meehl, this is the set of relationships between constructs and between consequent measures. The relationships between constructs should be reflected in the relationships between measures or observations. (d) Multitrait-Multimethod Matrix (MTMM): Defined by Campbell and Fiske, this demonstrates construct validity by using multiple methods (e.g., survey, observation, test, etc.) to measure the same set of ‘traits’ and showing correlations in a matrix, where blocks and diagonals have special meaning. 2. Content Validity Content validity refers to the extent to which a measure represents all facets of a given construct. For example, a depression scale may lack content validity if it only assesses the affective dimension of depression but fails to take into account the behavioural dimension. An element of subjectivity exists in relation to determining content validity, which requires a degree of agreement about what a particular personality trait such as extraversion represents. A disagreement about a personality trait will prevent the gain of a high content validity. Content validity is different from face validity, which refers not to what the test actually measures, but to what it superficially appears to measure. Face validity assesses whether the test “looks valid” CU IDOL SELF LEARNING MATERIAL (SLM)

Reliability and Validity 161 to the examinees who take it, the administrative personnel who decide on its use, and other technically untrained observers. Content validity requires the use of recognised subject matter experts to evaluate whether test items assess defined content and more rigorous statistical tests than does the assessment of face validity. Content validity is most often addressed in academic and vocational testing, where test items need to reflect the knowledge actually required for a given topic area (e.g., history) or job skill (e.g., accounting). In clinical settings, content validity refers to the correspondence between test items and the symptom content of a syndrome. 3. Internal Validity Internal validity occurs when it can be concluded that there is a causal relationship between the variables being studied. A danger is that changes might be caused by other factors. It is related to the design of the experiment, such as in the use of random assignment of treatments. It is the extent to which a piece of evidence supports a claim about cause and effect, within the context of a particular study. It is one of the most important properties of scientific studies, and is an important concept in reasoning about evidence more generally. Internal validity is determined by how well a study can rule out alternative explanations for its findings (usually, sources of systematic error or ‘bias’). It contrasts with external validity, the extent to which results can justify conclusions about other contexts (i.e., the extent to which results can be generalised). 4. Conclusion Validity Conclusion validity occurs when you can conclude that there is a relationship of some kind between the two variables being examined. This may be positive or negative correlation. It is important to realise that conclusion validity is an issue whenever you conclude there is a relationship, even when the relationship is between some program (or treatment) and some outcome. In other words, conclusion validity also pertains to causal relationships. How do we distinguish it from internal validity which is also involved with causal relationships? Conclusion validity is only concerned with whether there is a relationship. For instance, in a program evaluation, we might CU IDOL SELF LEARNING MATERIAL (SLM)

162 Research Methods and Statistics - I conclude that there is a positive relationship between our educational program and achievement test scores -- students in the program get higher scores and students not in the program get lower ones. Conclusion validity is essentially whether that relationship is a reasonable one or not, given the data. But it is possible that we will conclude that, while there is a relationship between the program and outcome, the program did not cause the outcome. Perhaps some other factor, and not our program, was responsible for the outcome in this study. 5. External Validity External validity occurs when the causal relationship discovered can be generalised to other people, times and contexts. Correct sampling will allow generalisation and hence give external validity. External validity is the validity of applying the conclusions of a scientific study outside the context of that study. In other words, it is the extent to which the results of a study can be generalised to and across other situations, people, stimuli, and times. In contrast, internal validity is the validity of conclusions drawn within the context of a particular study. Because general conclusions are almost always a goal in research, external validity is an important property of any study. Mathematical analysis of external validity concerns a determination of whether generalisation across heterogeneous populations is feasible and devising statistical and computational methods that produce valid generalisations. 6. Criterion-related Validity Criterion validity is the extent to which people’s scores on a measure are correlated with other variables (known as criteria) that one would expect them to be correlated with. For example, people’s scores on a new measure of test anxiety should be negatively correlated with their performance on an important school exam. If it were found that people’s scores were in fact negatively correlated with their exam performance, then this would be a piece of evidence that these scores really represent people’s test anxiety. But if it were found that people scored equally well on the exam regardless of their test anxiety scores, and then this would cast doubt on the validity of the measure. CU IDOL SELF LEARNING MATERIAL (SLM)

Reliability and Validity 163 A criterion can be any variable that one has reason to think should be correlated with the construct being measured, and there will usually be many of them. For example, one would expect test anxiety scores to be negatively correlated with exam performance and course grades and positively correlated with general anxiety and with blood pressure during an exam. Or imagine that a researcher develops a new measure of physical risk taking. People’s scores on this measure should be correlated with their participation in “extreme” activities such as snowboarding and rock climbing, the number of speeding tickets they have received, and even the number of broken bones they have had over the years. When the criterion is measured at the same time as the construct, criterion validity is referred to as concurrent validity; however, when the criterion is measured at some point in the future (after the construct has been measured), it is referred to as predictive validity (because scores on the measure have “predicted” a future outcome). Criteria can also include other measures of the same construct. For example, one would expect new measures of test anxiety or physical risk taking to be positively correlated with existing established measures of the same constructs. This is known as convergent validity. 7. Face Validity Face validity occurs where something appears to be valid. This of course depends very much on the judgment of the observer. In any case, it is never sufficient and requires more solid validity to enable acceptable conclusions to be drawn. Measures often start out with face validity as the researcher selects those which seem likely prove the point. Face validity is the extent to which a test is subjectively viewed as covering the concept it purports to measure. It refers to the transparency or relevance of a test as it appears to test participants. In other words, a test can be said to have face validity if it “looks like” it is going to measure what it is supposed to measure. For instance, if a test is prepared to measure whether students can perform multiplication, and the people to whom it is shown all agree that it looks like a good test of multiplication ability, this demonstrates face validity of the test. It is often contrasted with content validity and construct validity. CU IDOL SELF LEARNING MATERIAL (SLM)

164 Research Methods and Statistics - I Some people use the term face validity to refer only to the validity of a test to observers who are not expert in testing methodologies. For instance, if a test is designed to measure whether children are good spellers, and parents are asked whether the test is a good test, this measures the face validity of the test. If an expert is asked instead, some people would argue that this does not measure face validity. This distinction seems too careful for most applications. Generally, face validity means that the test “looks like” it will work, as opposed to “has been shown to work”. 7.6 Factors that Impact Validity The validity is measured and differentiating between the different types of validity, it is important to understand how external and internal factors impact validity. A student's reading ability can have an impact on the validity of an assessment. For example, if a student has a hard time comprehending what a question is asking, a test will not be an accurate assessment of what the student truly knows about a subject. Educators should ensure that an assessment is at the correct reading level of the student. Student self-efficacy can also impact validity of an assessment. If students have low self- efficacy or beliefs about their abilities in the particular area they are being tested in, they will typically perform lower. Their own doubts hinder their ability to accurately demonstrate knowledge and comprehension. Student test anxiety level is also a factor to be aware of. Students with high test anxiety will underperform due to emotional and physiological factors, such as upset stomach, sweating, and increased heart rate, which leads to a misrepresentation of student knowledge. Validity is measured using a coefficient. Typically, two scores from two assessments or measures are calculated to determine a number between 0 and 1. Higher coefficients indicate higher validity. Generally, assessments with a coefficient of .60 and above are considered acceptable or highly valid. 7.7 Summary Reliability in statistics and psychometrics is the overall consistency of a measure. A measure is said to have a high reliability if it produces similar results under consistent conditions. It is the characteristic of a set of test scores that relates to the amount of random error from the measurement CU IDOL SELF LEARNING MATERIAL (SLM)

Reliability and Validity 165 process that might be embedded in the scores. Scores that are highly reliable are accurate, reproducible and consistent from one testing occasion to another. That is, if the testing process were repeated with a group of test takers, essentially the same results would be obtained. Validity is the extent to which a concept, conclusion or measurement is well-founded and likely corresponds accurately to the real world. The validity of a measurement tool for example, a test in education) is the degree to which the tool measures what it claims to measure. Validity has a particular application known as test validity: “the degree to which evidence and theory support the interpretations of test scores”. It is generally accepted that the concept of scientific validity addresses the nature of reality in terms of statistical measures and as such is an epistemological and philosophical issue as well as a question of measurement. The term reliability in research refers to the consistency of a research study or measuring test. For example, if a person weighs themselves during the course of a day they would expect to see a similar reading. Scales which measured weight differently each time would be of little use. The same analogy could be applied to a tape measure which measures inches differently each time it was used. It would not be considered reliable. If findings from research are replicated consistently, they are reliable. A correlation coefficient can be used to assess the degree of reliability. If a test is reliable, it should show a high positive correlation. Of course, it is unlikely the exact same results will be obtained each time as participants and situations vary, but a strong positive correlation between the results of the same test indicates reliability. The concept of validity was formulated by Kelly (1927) who stated that a test is valid if it measures what it claims to measure. The validity of an instrument is the idea that the instrument measures what it intends to measure. Validity pertains to the connection between the purpose of the research and which data the researcher chooses to quantify that purpose. A test of intelligence should measure intelligence and not something else (such as memory). A distinction can be made between internal and external validity. These types of validity are relevant to evaluating the validity of a research study/procedure. CU IDOL SELF LEARNING MATERIAL (SLM)

166 Research Methods and Statistics - I Validity is the extent to which the scores from a measure represent the variable they are intended to. When a measure has good test-retest reliability and internal consistency, researchers should be more confident that the scores represent what they are supposed to. There has to be more to it, however, because a measure can be extremely reliable but have no validity whatsoever. As an absurd example, imagine someone who believes that people’s index finger length reflects their self- esteem and therefore tries to measure self-esteem by holding a ruler up to people’s index fingers. Although this measure would have extremely good test-retest reliability, it would have absolutely no validity. The fact that one person’s index finger is a centimeter longer than another’s would indicate nothing about which one higher self-esteem had. Construct validity occurs when the theoretical constructs of cause and effect accurately represent the real-world situations they are intended to model. This is related to how well the experiment is operationalised. A good experiment turns the theory (constructs) into actual things you can measure. Sometimes just finding out more about the construct (which itself must be valid) can be helpful. Content validity refers to the extent to which a measure represents all facets of a given construct. For example, a depression scale may lack content validity if it only assesses the affective dimension of depression but fails to take into account the behavioural dimension. An element of subjectivity exists in relation to determining content validity, which requires a degree of agreement about what a particular personality trait such as extraversion represents. A disagreement about a personality trait will prevent the gain of a high content validity Internal validity occurs when it can be concluded that there is a causal relationship between the variables being studied. A danger is that changes might be caused by other factors. It is related to the design of the experiment, such as in the use of random assignment of treatments. Internal validity is the extent to which a piece of evidence supports a claim about cause and effect, within the context of a particular study. It is one of the most important properties of scientific studies, and is an important concept in reasoning about evidence more generally. Internal validity is determined by how well a study can rule out alternative explanations for its findings (usually, sources of systematic error or ‘bias’). Conclusion validity occurs when you can conclude that there is a relationship of some kind between the two variables being examined. This may be positive or negative correlation. It's important CU IDOL SELF LEARNING MATERIAL (SLM)

Reliability and Validity 167 to realise that conclusion validity is an issue whenever you conclude there is a relationship, even when the relationship is between some program (or treatment) and some outcome. In other words, conclusion validity also pertains to causal relationships. 7.8 Key Words/Abbreviations  Reliability: Reliability in statistics and psychometrics is the overall consistency of a measure.  Test-Retest Reliability: When researchers measure a construct that they assume to be consistent across time, the scores they obtain should also be consistent across time.  Internal Consistency: A second kind of reliability is internal consistency.  Inter-rater Reliability: Many behavioural measures involve significant judgment on the part of an observer or a rater.  Parallel Forms Reliability: Parallel forms reliability measures the correlation between two equivalent versions of a test.  Validity: The concept of validity was formulated by Kelly (1927) who stated that a test is valid if it measures what it claims to measure.  Construct Validity: Construct validity occurs when the theoretical constructs of cause and effect accurately represent the real world.  Content Validity: Content validity refers to the extent to which a measure represents all facets of a given construct.  Internal Validity: Internal validity occurs when it can be concluded that there is a causal relationship between the variables being studied.  Conclusion Validity: Conclusion validity occurs when you can conclude that there is a relationship of some kind between the two variables being examined. CU IDOL SELF LEARNING MATERIAL (SLM)

168 Research Methods and Statistics - I  External Validity: External validity occurs when the causal relationship discovered can be generalised to other people, times and contexts.  Criterion-related Validity: Criterion validity is the extent to which people’s scores on a measure are correlated with other variables.  Face Validity: Face validity occurs where something appears to be valid. 7.9 Learning Activity 1. You are required to prepare the data sets for reliability test and show the report. _________________________________________________________________ _________________________________________________________________ 2. You are suggested to demonstrate the procedure and report of validity test. _________________________________________________________________ _________________________________________________________________ 3. You are requested to find the best type of validity and reliability test and give the valid reasons. _________________________________________________________________ _________________________________________________________________ 7.10 Unit End Exercises (MCQs and Descriptive) Descriptive Type Questions 1. What is Reliability? 2. Discuss the concept Reliability. 3. What is Test-Retest Reliability? Explain its applications. CU IDOL SELF LEARNING MATERIAL (SLM)

Reliability and Validity 169 4. What is internal consistency? 5. What is Inter-rater Reliability? 6. What is Parallel Form Reliability? 7. What is Validity? Explain the concept of Validity. 8. Explain various types of Validity. 9. Discuss in details about Construct Validity and Content Validity. 10. Explain in brief about Internal Validity and Conclusion Validity. Multiple Choice Questions 1. Which of the following in research refers to the consistency of a research study or measuring test? (a) Reliability (b) Validity (c) Consistency (d) All the above 2. Which of the following is the consistency of people’s responses across the items on a multiple-item measure? (a) Test-Retest Reliability (b) Internal Consistency (c) Interrater Reliability (d) Parallel Forms Reliability 3. Which of the following measures the correlation between two equivalent versions of a test? (a) Test-Retest Reliability (b) Internal Consistency (c) Interrater Reliability (d) Parallel Forms Reliability CU IDOL SELF LEARNING MATERIAL (SLM)

170 Research Methods and Statistics - I 4. Which of the following is the idea that the instrument measures what it intends to measure? (a) Reliability (b) Validity (c) Consistency (d) All the above 5. Which of the following is the type of validity? (a) Construct Validity (b) Content validity (c) Internal validity (d) All the above Answers: 1. (a), 2. (b), 3. (d), 4. (b), 5. (d) 7.11 References References of this unit have been given at the end of the book.  CU IDOL SELF LEARNING MATERIAL (SLM)

Data Analysis 171 UNIT 8 DATA ANALYSIS Structure: 8.0 Learning Objectives 8.1 Introduction 8.2 Descriptive Statistics 8.3 Mean (Arithmetic Mean) 8.4 Median 8.5 Mode 8.6 Standard Deviation and Variance 8.7 Skewness 8.8 Kurtosis 8.9 Range 8.10 Summary 8.11 Key Words/Abbreviations 8.12 LearningActivity 8.13 Unit End Exercises (MCQs and Descriptive) 8.14 References CU IDOL SELF LEARNING MATERIAL (SLM)

172 Research Methods and Statistics - I 8.0 Learning Objectives After studying this unit, you will be able to:  Describe the data analysis  Explain standard deviation and variance 8.1 Introduction Data analysis is a process of inspecting, cleansing, transforming and modelling data with the goal of discovering useful information, informing conclusion and supporting decision-making. Data analysis has multiple facets and approaches, encompassing diverse techniques under a variety of names, and is used in different business, science, and social science domains. In today’s business world, data analysis plays a role in making decisions more scientific and helping businesses operate more effectively. Data initially obtained must be processed or organised for analysis. For instance, these may involve placing data into rows and columns in a table format (i.e., structured data) for further analysis, such as within a spreadsheet or statistical software. Once processed and organised, the data may be incomplete, contain duplicates, or contain errors. The need for data cleaning will arise from problems in the way that data are entered and stored. Data cleaning is the process of preventing and correcting these errors. Common tasks include record matching, identifying inaccuracy of data, and overall quality of existing data, deduplication and column segmentation. Such data problems can also be identified through a variety of analytical techniques. 8.2 Descriptive Statistics Descriptive statistics is the type of statistics that probably springs to most people’s minds when they hear the word “statistics.” Here, the goal is to describe. Numerical measures are used to tell about features of a set of data. There are a number of items that belong in this portion of statistics, such as: 1. The average or measure of the center of a data set, consisting of the mean, median, mode or mid-range. 2. The spread of a data set, which can be measured with the range or standard deviation. 3. Overall descriptions of data such as the five number summary. CU IDOL SELF LEARNING MATERIAL (SLM)

Data Analysis 173 4. Other measurements such as skewness and kurtosis. 5. The exploration of relationships and correlation between paired data. 6. The presentation of statistical results in graphical form. 8.3 Mean (Arithmetic Mean) Arithmetic mean is defined as the value obtained by dividing the total values of all items in the series by their number. In other words, it is defined as the sum of the given observations divided by the number of observations, i.e., add values of all items together and divide this sum by the number of observations. Symbolically x  x  x  x  ............. x n 1 2 3 n Merits of Arithmetic Mean 1. It is rigidly defined. Its value is always definite. 2. It is easy to calculate and easy to understand. Hence, it is very popular. 3. It is based on all the observations, so that it becomes a good representative. 4. It can be easily used for comparison. 5. It is capable of further algebraic treatment such as finding the sum of the values of the observations, if the mean and the total number of the observations are given; finding the combined arithmetic mean when different groups are given, etc. 6. It is not affected much by sampling fluctuations. Demerits of Arithmetic Mean 1. Arithmetic mean is affected very much by extreme values. 2. It cannot be determined by inspection nor it can be located graphically. 3. Arithmetic mean cannot be obtained if a single observation is missing or lost. 4. We cannot calculate it when open-end class intervals are present in the data. CU IDOL SELF LEARNING MATERIAL (SLM)

174 Research Methods and Statistics - I Properties of Arithmetic Mean 1. The sum of the deviations, of all the values of x, from their arithmetic mean, is zero. 2. The product of the arithmetic mean and the number of items gives the total of all items. 3. Finding the combined arithmetic mean when different groups are given. Arithmetic Mean for Ungrouped Data A. Individual Series Direct Method The following steps are involved in calculating arithmetic mean under individual series using direct method: 1. Add up all the values of all the observations in the series. 2. Divide the sum of the values by the number of observations. The result is the arithmetic mean. The following formula is used: x where, X = Arithmetic mean X= n åx = Sum of the values n = Number of items Illustration 1 Value (x) 125 128 132 135 140 148 155 157 159 161 Solution: Calculation of arithmetic mean: Sl. No. Value (x) A 125 B 128 C 132 D 135 CU IDOL SELF LEARNING MATERIAL (SLM)

Data Analysis 175 E 140 F 148 G 155 H 157 I 159 J 161 n = 10 åx = 1,440 X  x  1,440 = 144 n 10 Short-cut Method or Indirect Method The following steps are involved in calculating arithmetic mean under individual series using short-cut or indirect method: 1. Assume one of the values in the series as an average. It is called as working mean or assumed average. 2. Find out the deviation of each value from the assumed average. 3. Add up the deviations 4. Apply the following formula: X= A   dx X= n A= where, Arithmetic mean Assumed average ådx = Sum of the deviations n = Number of items CU IDOL SELF LEARNING MATERIAL (SLM)

176 Research Methods and Statistics - I Illustration 2 Calculate the arithmetic average of the data given below using short-cut method. Roll No. 1 2 3 4 5 6 7 8 9 10 Marks Obtained 43 48 65 57 31 60 37 48 78 59 Solution: Calculation of Arithmetic mean: Roll No. Marks obtained (x) dx = (x – A) 1 43 –17 2 48 –12 3 65 +5 4 57 –3 5 31 –29 6 60 (A) 0 7 37 –23 8 48 –12 9 78 +18 10 59 –1 ådx = –74 X = A   dx  60    74  = 60 – 7.4 = 52.6 marks n  10  B. Discrete Series In the discrete series, every term (i.e., value of x) is multiplied by its corresponding frequency (fx) and then their total (sum)  fx is divided by the total frequency (N) or  f . CU IDOL SELF LEARNING MATERIAL (SLM)

Data Analysis 177 Direct Method Formula: X   fx N Illustration 3 Following are the marks obtained by students of a class in statistics. Calculate arithmetic mean. Marks 35 40 45 50 55 60 65 70 75 80 85 90 95 No. of Students 3 8 12 9 4 7 15 5 10 7 5 3 2 Solution: Marks No. of Students fx (x) (f) 35 3 105 40 8 320 45 12 540 50 9 450 55 4 220 60 7 420 65 15 975 70 5 350 75 10 750 80 7 560 85 5 425 90 3 270 95 2 190  fx = 5,575 N = 90 CU IDOL SELF LEARNING MATERIAL (SLM)

178 Research Methods and Statistics - I X   fx = 5,575 = 61.94 marks N 90 Indirect Method or Short-cut Method Formula: X  A   fdx N Illustration 4 Following are the marks obtained by students of a class in statistics. Calculate arithmetic mean. Marks 35 40 45 50 55 60 65 70 75 80 85 90 95 No. of Students 3 8 12 9 4 7 15 5 10 7 5 3 2 Solution: Marks No. of Students (x – A) = dx fdx (x) (f) 35 3 –30 –90 40 8 –25 –200 45 12 –20 –240 50 9 –15 –135 55 4 –10 –40 60 7 –5 –35 65 (A) 15 00 70 5 5 25 75 10 10 100 80 7 15 105 85 5 20 100 CU IDOL SELF LEARNING MATERIAL (SLM)

Data Analysis 179 90 3 25 75 95 2 30 60 N = 90  fdx = –275 X  A   fdx = 65 + -275 N 90 = 65 – 3.06 = 61.94 marks Step Deviation Method Formula: = A   fdx   c N X The following steps are involved in computing mean under step deviation method: 1. Find out the mid-value of each group or class. 2. Assume one of the mid-values as an average. 3. Find out the deviation of each mid-value from the assumed average in terms of class interval. 4. Multiplying the deviation of each class by its frequency. 5. Add up the products. 6. Apply the following formula: X= A   fdx   c N where, X = Arithmetic mean A = Assumed average åfd x = Sum of the deviations in terms of class interval N = Total frequency c = Class interval CU IDOL SELF LEARNING MATERIAL (SLM)

180 Research Methods and Statistics - I Discrete Series Illustration 5 Following are the marks obtained by students of a class in statistics. Marks 35 40 45 50 55 60 65 70 75 80 85 90 95 No. of Students 3 8 12 9 4 7 15 5 10 7 5 3 2 Calculate arithmetic mean. Solution: x f dx = xA (c = 5) fdx  c –18 35 3 –30/5 = –6 40 8 –25/5 = –5 –40 45 12 –20/5 = –4 –48 50 9 –15/5 = –3 –27 55 4 –10/5 = –2 –8 60 7 –5/5 = –1 –7 65 15 0/5 = 0 0 70 5 5/5 = 1 5 75 10 10/5 = 2 20 80 7 15/5 = 3 21 85 5 20/5 = 4 20 90 3 25/5 = 5 15 95 2 30/5 = 6 12 N = 90 åfd x = –55 X = A  fdx   c = 65 + 55 ×5 = 65 +  275 = 65  3.05 = 61.95 marks N 90 90 CU IDOL SELF LEARNING MATERIAL (SLM)

Data Analysis 181 Continuous Series In continuous series, variables are represented by class-interval. Each class intervals has its own frequency. Midpoint (class marks) of each class should be ascertained first. Then the procedure of finding the arithmetic mean is the same as used in the discrete series. Exclusive Class Interval Direct Method Illustration 6 From the following figures, find the mean using indirect method. Marks No. of Persons 0 – 10 5 10 – 20 10 20 – 30 20 30 – 40 40 40 – 50 30 50 – 60 20 60 – 70 10 70 – 80 4 Solution: Calculation of mean: Marks x f dx = (x – A) fdx  0 – 10 5 5 –40 –200 10 – 20 15 10 –30 –300 20 – 30 25 20 –20 –400 CU IDOL SELF LEARNING MATERIAL (SLM)

182 Research Methods and Statistics - I 30 – 40 35 40 –10 –400 40 – 50 50 – 60 45 (A) 30 0 0 60 – 70 70 – 80 55 20 +10 +200 8.4 Median 65 10 +20 +200 75 4 +30 +120 N = 139 åfd x = –780 = A  fdx   c  45    780  = 45 + (–5.6) = 39.4 marks N  139  X Median is defined as the value of that item which divides the series into two equal halves, one half contains all values less than (or equal to) it and the other half containing all values greater than (or equal to) it. It is also defined as the “central value of the variable”. It should be noted that the value of items should be arranged in order of their magnitude or size to find out the median). The median is that value of the variable which divides the group into two equal parts, one part comprising all values greater and the other all values lesser than the median. Median is a positional average. The term position refers to the place of a value in a series the place of median in a series is such that an equal number of items lie on either side. Therefore, it is also called a locative average. Individual Series The following steps are involved in calculating median in individual series: 1. Arrange the values in the group either in ascending order or descending order. 2. Find out the value of the middle item by applying the following formula.  N  1 th  M = Size of  2  item where, M = Median N = Number of items CU IDOL SELF LEARNING MATERIAL (SLM)

Data Analysis 183 Illustration 1 Determine the median from the following: 25, 15, 23, 40, 27, 25, 23, 25, 20 Solution: The figures given must be arranged either in ascending order (or descending order) of their magnitude. Calculation of median: Sl. No. Value or Size 1 15 2 20 3 23 4 23 5 25 6 25 7 25 8 27 9 40  N  1 th 10  Median = Size of  2  item = 2 = 5th item, i.e., 25 Discrete Series Illustration 2 Calculate the median for the following data: Size of item: 4 6 8 10 12 14 16 Frequency: 2 4 5 3 2 1 4 CU IDOL SELF LEARNING MATERIAL (SLM)

184 Research Methods and Statistics - I Solution: Calculation of median: Size of item Frequency Cumulative frequency 4 22 6 46 8 5 11 10 3 14 12 2 16 14 1 17 16 4 21  N  1 th 21  1 22  Median is the size of  2  item = 2 = 2 = 11th item 8 lies between 11. So, M = 8 Continuous Series The following steps are involved in calculating median in continuous series: 1. Find out the cumulative frequency.  N th  2. Find out the median class, i.e.,  2  item. 3. Find out the group or class containing the median 4. Estimate the median applying the following formula. l N  cf c 2 f M=  CU IDOL SELF LEARNING MATERIAL (SLM)

Data Analysis 185 where, M = Median l = Lower limit of the median class cf = Cumulative frequency of the class preceding the median class c = Size of class interval f = Frequency of the median class. Illustration 3 Calculate the median of the following distribution. Length (in inches) No. of units 0 – 20 1 20 – 40 14 40 – 60 35 60 – 80 85 80 – 100 90 100 – 120 15 Solution: Calculation of median: Length (in inches) f cf 0 – 20 1 1 20 – 40 14 15 40 – 60 35 50 (cf) 60 – 80 85 (f) 135 80 – 100 90 225 100 – 120 15 240 CU IDOL SELF LEARNING MATERIAL (SLM)

186 Research Methods and Statistics - I N 240 M = 2  2 = 120th item, i.e., 60 – 80 group. M= l  N  cf c = 60  120  50  20 2 f 85 = 60  1400 = 60 + 16.47 = 76.47 inches 85 Illustration 4 Calculate median for the following data Wages (x) 1-5 6-10 11-15 16-20 21-25 26-30 31-35 36-40 41-45 46-50 f 5 7 9 15 12 10 9 8 4 2 Solution: Class (x) f cf 0.5-5.5 5 5 5.5-10.5 7 12 10.5-15.5 9 21 15.5-20.5 15 36 (cf) 20.5-25.5 12 (f) 48 25.5-30.5 10 58 30.5-35.5 9 67 35.5-40.5 8 75 40.5-45.5 4 79 45.5-50.5 2 81 N = 81 CU IDOL SELF LEARNING MATERIAL (SLM)

Data Analysis 187 Step 1: Inclusive class convert into exclusive continuous class. For this, we require a correction 1 factor. Correction factor = 2 (the upper limit of a class – the lower limit of the next class,) 1 which is generally 0.5. Example: 5 – 6 = 1 × 2 = .5. Step 2: Subtract it (0.5) for the lower limits and add it (0.5) to the upper limits of the class given in the inclusive method, for convert in to exclusive continuous class (1 – 0.5 = 0.5 and 5 + 0.5 = 5.5), i.e., exclusive class is 0.5-5.5 ... and so on. Step 3: Convert simple frequencies into cumulative frequency.  N th  Step 4: Determine median class. Use  2  item = 81/2 = 40.5 terms. In the order of the cumulative frequency, the 40.50th item is greater than 36. Hence, the 40.5th term is present in the 48th cumulative frequency. The median class is 20.5-25.5. l1 = 20.5, l2 = 25.5, cf = 36 (preceding the median class) and f = 12 (Normal frequency of median class). Step 5: Apply formula M = l  N  cf c 2 f = 20.5 +  40.5  36   5  12  = 20.5 +  4.5  5= 20.5 +  22.5  = 20.5 + 1.875  12   12  M = 22.375 8.5 Mode The word “mode” is derived from the French word “1a mode” meaning fashion. So, it can be regarded as the most fashionable item in the series or the group. Croxtan and Cowden regard mode as “the most typical of a series of values”. As a result, it can sum up the characteristics of a group more satisfactorily than the arithmetic mean or median. CU IDOL SELF LEARNING MATERIAL (SLM)

188 Research Methods and Statistics - I Mode is defined as the value of the variable which occurs most frequently in a distribution. In other words, it is the most frequent size of item in a series. Merits of Mode The following are the merits of mode: 1. The most important advantage of mode is that it is usually on actual value. 2. It can be easily located by inspection in the case of discrete series. 3. It is not affected by extreme values. 4. It can be determined even if extreme values are not given. 5. It is easy to understand. This average is used by people in their every day speech. Demerits of Mode The following are the demerits of mode: 1. It is not based on all the observation of the data. 2. It is not suitable for further mathematical treatment. 3. In a number of cases, there will be more than one mode in the series. 4. If mode is multiplied by the number of items, the product will not be equal to the total value of the items. 5. It will not truly represent the group if there are a small number of items of the same size in a large group of items of different sizes. Ungrouped Data Individual Series The mode of this series can be obtained by mere inspection. The number which occurs most often is the mode. CU IDOL SELF LEARNING MATERIAL (SLM)

Data Analysis 189 Illustration 1 Locate mode in the data 7, 12, 8, 5, 9, 6, 10, 9, 4, 9, 9. Solution: On inspection, it is observed that the number 9 has maximum frequency, i.e., repeated maximum of 4 times than any other number. Therefore, mode (Z) = 9. Discrete Series The mode is calculated by applying grouping and analysis table. (i) Grouping Table: Consisting of six columns including frequency column, 1st column is the frequency 2nd and 3rd column is the grouping two way frequencies and 4th, 5th and 6th column is the grouping three way frequencies. (ii) Analysis table: consisting of 2 columns namely tally bar and frequency Steps in Calculating Mode in Discrete Series The following steps are involved in calculating mode in discrete series: 1. Group the frequencies by 2’s. 2. Leave the first frequency and group the other frequencies in 2’s. 3. Leave the first two frequencies and moved the frequency in 2’s. 4. Group the frequencies in 3’s. 5. Leave the first frequency of the first size and add the frequencies of other sizes in 3’s. 6. Leave the first frequencies and add the frequencies of the other sizes in 3’s. 7. Prepare an analysis table to know the size occurring the maximum number of times. Find out the size, which occurs the largest number of times. That particular size is the mode. CU IDOL SELF LEARNING MATERIAL (SLM)

190 Research Methods and Statistics - I Illustration 2 Value (x): 270 275 278 280 282 284 285 286 287 290 292 f: 8 12 24 32 93 56 48 29 24 32 30 Solution: In the analysis table, the highest frequency is 6 which corresponding to 282 value of Z = 282. Step 1: Grouping two frequencies for 2nd column 8 + 12 = 20, 24 + 32 = 56, 93 + 56 = 149, 48 + 29 = 77, 24 + 32 = 56 and just leave the last one frequency it cannot group. Grouping 3rd column two frequencies leave the first one and start from 2nd frequency, i.e., 12 + 24 = 36, 32 + 93 = 125, 56 + 48 = 104, 29 + 24 = 53, 32 + 30 = 62 Grouping 4th column three frequencies start from beginning, i.e., 8 + 12 + 24 = 44, 32 + 93 + 56 = 181, 48 + 29 + 24 = 101, and leave the last two frequencies as they cannot be grouping into threes. Grouping 5th column start from second frequency for grouping. Grouping 6th column start from third frequencies for grouping. CU IDOL SELF LEARNING MATERIAL (SLM)

Data Analysis 191 Step 2: After the grouping is done, the next step is to identify the highest frequency in each column and encircle it. Step 3: Mark the tally bar for highest frequency against the values. Step 4: In the analysis table, identify the highest frequency, the value of mode is corresponding to highest frequency. In case of more than one highest frequencies, the value of mode is ill- defined. Continuous Series The following steps are involved in calculating mode in continuous series. 1. Find out the modal class. Modal class can be easily found out by inspection. The group containing maximum frequency is the modal group. Where two or more classes appear to be a modal class group, it can be decided by grouping process and preparing an analysed table as was discussed in question number 2.102. 2. The actual value of mode is calculated by applying the following formula: Z = l+ fm  f1 c 2fm  f1  f2 where, Z = Mode l = Lower limit fm = Highest frequency f1 = Preceding value of highest frequency f2 = Succeeding value of highest frequency c = Size of class interval Illustration 3 Calculate the modal wages. Daily wages in ` (x) : 20-25 25-30 30-35 35-40 40-45 45-50 No. of workers (f) : 1 3 8 12 7 5 CU IDOL SELF LEARNING MATERIAL (SLM)

192 Research Methods and Statistics - I Solution: Here, the maximum frequency is 12, corresponding to the class interval (35 - 40) which is the modal class. xf 20-25 1 25-30 3 30-35 8 f1 35-40 12 fm 40-45 7 f2 45-50 5 Mode = l+ fm  f1 c = 35 +  12  8 7   5 2fm  f1  f2  2(12)  8     = 35 +  4 5 = 35 +  20  = 35 + 2.22 = 37.22  24 -15   9   Mode in Inclusive Method: Illustration 4 Calculation of mode from the following data: x: 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 f: 5 12 22 25 14 10 84 Solution: xf 10-19 10.5-19.5 5 20-29 20.5-29.5 12 CU IDOL SELF LEARNING MATERIAL (SLM)

Data Analysis 193 20-29 20.5-29.5 22 f1 40-49 40.5-49.5 25 fm 50-59 50.5-59.5 14 f2 60-69 60.5-69.5 10 70-79 70.5-79.5 8 80-89 80.5-89.5 4 Mode = l+ fm  f1 c = 39.5 +  25  22 14   10 2fm  f1  f2  2(25)  22     = 39.5 +  3 10 = 39.5 +  30  = 39.5 + 2.143 Z = 41.64  50-36   14   Calculation of Mode When Mid-values are Given: Illustration 5 Calculation mode from the following data: MP Class 120 115-125 130 125-135 140 135-145 150 145-155 160 155-165 170 165-175 180 175-185 190 185-195 CU IDOL SELF LEARNING MATERIAL (SLM)

194 Research Methods and Statistics - I Solution: MP Class f 120 115-125 5 130 125-135 12 140 135-145 32 150 145-155 40 160 155-165 20 170 165-175 10 180 175-185 9 190 185-195 11 fm  f1 c = 145 +  40  32   10 Z=l+ 2fm  f1  f2    2(40)  32  20  = 145 +  80 8 52  10 = 145 +  80  = 145 + 2.85 Z = 147.85  -   28    Mid-point can be converted into exclusive class (Class length is 130 – 120 = 10) = 10/2 = 5; lower limit = 120 - 5 = 115; upper limit = 120 + 5 = 125). Calculate Mode When Less Than and More Than Classes are Given Illustration 6 Less than: 10 20 30 40 50 60 70 80 Frequency: 4 16 40 76 96 112 120 125 Solution: Need to ascertain lower limit of the continuous class (LL = UL – CL) Class length (CL) = 20 – 10 = 10, i.e., (10 – 10 = 0…………) CU IDOL SELF LEARNING MATERIAL (SLM)