SLM_(1)

194 Business Mathematics and Statistics unmarried and 10% are widowers. Of the males in T, the corresponding figures of married, unmarried and widowers are 50%, 25% and 25% respectively, in both towns the number of married women is exactly equal to the number of married men. Further the number of widows and unmarried females is the same in S and as well as in T.” z Draft a tabular form representing the information contained in the following extract relating to industrial accidents: “In 1929 there were 19,077 non-fatal compensated cases which might be classed as permanent injury, while there were 80,107 classed as temporary injury. For 1930 these figures were 22,434 and 86,106 respectively. For 1931 there were 21,761 and 70,0 respectively. The total accidents for the three years were 523,604, 471,510, 419,072 respectively. The total deaths (included in the former accident totals) were 2,093, 2,006, 1,793 respectively of which total compensation was paid to 1,217, 1,308 and 1,241 respectively. The number of total accidents for which compensation was paid was 100,401 in 1929, 109,848 in 1930 and 102,985 in 1931.” z Tabulate the following data: A survey was conducted amongst one lakh spectators visiting on a particular day cinema houses showing criminal, social, historical, comic and mythological films. The proportion of male to female spectators under survey was three to two. It indicated that while the respective percentages of spectators seeing criminal, social and historical films was sixteen, twenty-six and eighteen, the actual number of female viewers seeing these types was four thousand six hundred, twelve thousand two hundred, and seven thousand eight hundred respectively. The remaining two types of films namely comic and mythological were seen by forty percent and one percent of the male spectators. The number of female spectators seeing mythological films was four thousand four hundred. z Present the following data in tabular form:

“The total strength of a college in 1960-61 was 1,100 students distributes as under: F.Y. Com Class-400: 1. Com. Class-300; Jr. B. Com. Class-200; Sr. B. Com. Class- 200. In 1961-62 there was a fall of 10% in the strength of each of the first three classes as compared with their strength in 1960-61 and the Sr. B. Com. Class had a strength of 200. In 1962-63, there was an increase of 10% in each of the four classes as compared CU IDOL SELF LEARNING MATERIAL (SLM)

Classification and Tabulation of Data 195 with their strength in 1961-62. In 1963-64 the total strength of the college was 1,250. The number of students in F.Y. Com Class was double the number of students in Sr. B.Com. class. The number of students in 1. Com. class was 330.” © Represent the information given below by a suitable table. “The number of students in a college in the year 1941 was 510. Of these 480 were boys and the rest girls. In 1951, the number of boys increased by 100% and that of girls increased by 300% as compared to their strengths in 1941. In 1961, the total number of students in the college was 1200, the number of boys being double than the number of girls.” © Present the following information in a concise tabular form and indicate with type of lamp shows the greatest wastage during manufacture: ‘Lamps are rejected at several manufacturing stages for different fault. 12,000 glass tubes are supplied to make 40-watt, 60-watt and 100-watt lamps in the ratio 1:2:3. At the stage I, 10 per cent of the 40-watt, 4 per cent of the 60-watt and 5 per cent of the 100-watt bulbs are broken. At the stage II, about 1 percent of the remainder of the lamps have broken filaments, At the stage III 100 100-watt lamps have badly soldered caps, and half as many have crooked caps; twice as many 40-watt and 60-watt lamps have these faults. In the stage IV, about 3 percent are rejected for bad type-marking and 1 in every 100 are broken in the packing which follows. 9.13 Unit End Questions (MCQ and Descriptive) z Descriptive Type Define ‘classification’ and ‘tabulation’ and explain their importance in statistics. What are the different types of tables that you know? Explain them with suitable illustrations.

State the precautions that you would take in tabulating statistical data. CU IDOL SELF LEARNING MATERIAL (SLM)

196 Business Mathematics and Statistics 4. Following are the marks secured by 50 candidates at an examination (out of 100 marks). 60 50 49 40 31 23 8 22 0 30 41 40 56 69 66 52 43 41 34 24 19 18 26 35 42 51 53 42 49 53 44 36 25 17 10 29 39 46 44 58 46 38 31 14 21 20 32 33 30 47 Classify the above data in intervals 0-10, 10-20 and so on. 12. Draw up a blank table, with suitable headings, in which the number of workers occupied in five factories, distinguishing males from females and skilled from unskilled, can be entered. 13. Present the following information in a suitable tabular form: TISCOs production for the whole financial year — that is 1963-64 ending March 1964 — of saleable steel, steel ingots and pig iron stood at 1,506,400 metric tonnes, 1,891,700 metric tonnes and 1,811,200 metric tonnes in that order. The corresponding figures for the preceding year were 1,317,700 metric tonnes, 1,799,300 metric tonnes and 1,766,000 metric tonnes, respectively. Source: Quarterly Bulletin of the Eastern Economist Vol. 15, No. 3. 14. Explain the purpose and importance of classification and tabulation of statistical data. What are specific purpose and general purpose-tables?

15. What do you understand by classification of data? Discuss its importance in statistical analysis. 16. From the following observations prepare a frequency distribution table in ascending order starting with 5-10 (exclusive method). CU IDOL SELF LEARNING MATERIAL (SLM)

Classification and Tabulation of Data 197 Marks in English 20 19 19 7 12 36 40 30 28 17 37 19 27 15 26 20 26 37 5 20 11 10 10 16 45 33 21 30 20 5 5. From the following observations prepare a frequency distribution table in ascending order starting with 100-110 (exclusive method). 125 108 112 126 110 132 120 130 136 138 125 111 147 137 145 150 142 135 136 130 149 155 119 125 140 148 137 132 165 154

z The following are the marks of 100 examinees in statistics out of a maximum of 100. Group them into classes with an interval of 10. 57 44 8 75 0 18 45 14 0 4 64 66 72 51 69 34 56 22 34 8 58 83 20 70 57 28 22 38 5 45 51 88 17 93 64 36 34 37 58 32 64 30 80 73 24 46 48 a 16 65 96 56 20 64 50 63 47 4 32 10 78 48 55 52 66 8 53 50 0 35 28 54 38 33 20 54 52 48 84 50 94 90 38 84 30 58 20 0 99 42 79 33 38 60 61 36 10 34 2 80 CU IDOL SELF LEARNING MATERIAL (SLM)

198 Business Mathematics and Statistics 12. The following are the weekly wages (in rupees) of 40 labourers in a certain factory: 22 21 23 21 19 25 18 20 22 24 28 29 30 35 36 38 34 30 28 26 35 31 34 25 27 31 33 26 27 32 25 26 28 32 24 28 27 31 29 29 Classify the data by taking a class-interval of ` 3. If the class-marks of the first three classes of a frequency distribution are 32.5, 37.5 and 42.5, what are the class boundaries of these classes? If the observations are taken as correct to the nearest integer, what are the class-limits? A record of marks of 180 candidates shows marks ranging in value from 12 to 79. In how many classes would you classify this data? What are the boundaries of the first three classes? What are the class-limits of the first three classes? What are the class- marks of the last three classes? B. Multiple Choice/Objective Type Questions Classifying data under several heads (attributes) is different classes is ________. (a) Simple classification (b) Manifold classification (c) Multiple classification (d) Complex classification None of these G The basic requirement of good classifiction is ________. (a) Incomparability (b) Instability (c) Non-flexibility (d) Non-ambiguity

(e) None of these 17. ________________ is meant to present the entire original data on a subject. (a) Special purpose table (b) General purpose table (c) Statistical table (d) None of these 4. A one-way table gives answers to questions about ________________ characteristics of the data. (a) Two (b) Three (c) One (d) Four Answers (1) (b); (2) (c); (3) (b); (4) (c) 9.14 References References of this unit have been given at the end of the book. ˆˆˆ CU IDOL SELF LEARNING MATERIAL (SLM)

Measures of Central Tendency 199 UNIT 10 MEASURES OF CENTRAL TENDENCY Structure 10.0 Learning Objectives 10.1 Introduction 10.2 Characteristics of an ideal average 10.3 Types of Statistical Averages 10.4 Arithmetic Mean 10.5 Weighted Arithmetic Mean 10.6 Median 10.7 Advantages and Disadvantages of Median 10.8 Mode

10.9 Advantages and Disadvantages of Mode 10.10 Geometric Mean 10.11 Advantages and Disadvantages of Geometric Mean 10.12 Quartiles, Deciles and Percentiles 10.13 Additional Problems (Solved) 10.14 Summary 10.15 Key Words/Abbreviations 10.16 Learning Activity 10.17 Unit End Questions (MCQ and Descriptive) 10.18 References

CU IDOL SELF LEARNING MATERIAL (SLM) 200 Business Mathematics and Statistics 10.0 Learning Objectives After studying this unit, you will be able to:  Explain the definition of arithmetic mean, median, mode and the advantages and disadvantages in respect of each of these.  Describe the meaning of dispersion and the type of measures of dispersion that include: Range, Mean Deviation, Quartile Deviation and Standard Deviation.   Grasp the procedure of the calculation of range, mean deviation, Q.D., S.D. and the coefficient of variation in respect of data both discrete and continuous. 10.1 Introduction When statistical data is properly classified and condensed into a frequency distribution, then it is easy for us to study the different characteristics of that data. Further, graphs and diagrams may also be drawn so as to convey a better impression to the mind about the data. But graphs and diagrams are only visual aids which appeal more to the eye than to the mind. Therefore to know more about the data, we feel the necessity of some descriptive measures. Such measures are known as measures of location.

Definition Measures which describe, characterise or represent a given complex data, are known as ‘statistical averages’. In other words, every single expression that sums up the characteristics of an entire group of figures or complex data. Descriptive measures such as these generally throw light on the central characteristics of the data aid are therefore present in the central parts. Hence, they are also known as ‘measures of central tendency.’ Main Objects of a Statistical Average The two main objects in calculating a statistical average. Firstly, to indicate precisely the sum and substance of an entire mass of numerical data. Secondly, to serve as a means as well as a measure of comparison with other similar groups of numerical data.

CU IDOL SELF LEARNING MATERIAL (SLM)

Measures of Central Tendency 201 10.2 Characteristics of an Ideal Average  It should be easily intelligible and workable in the sense that its calculation should be simple.   Its definition should be clearcut, rigid and most appropriate.   It should take into account all the items of the data.   It should not be influenced due to fluctuations of sampling.   It should be useful for the purposes of further algebraic analysis. 10.3 Types of Statistical Averages The different types of Statistical Averages –  Arithmetic Mean,   Median,   Mode,   Geometric Mean,   Harmonic Mean,   Quadratic Mean,   Moving Average,   Progressive Average.

These are grouped into three separate groups as: mathematical averages, averages of position, and business averages, arithmetic mean, geometric mean, harmonic mean and quadratic mean are known as mathematical averages since they are calculated by using mainly certain mathematical procedures. It is necessary to consider every item in the data while calculating these averages. Mode and median are known as averages of position since the position of certain items in the data are taken into consideration while finding these. For instance, the middle item in a series of ordered items is a positional item since it occupies the central position. Similarly, an item that is repeated a number of times has a certain position when items are arranged with their corresponding frequencies. CU IDOL SELF LEARNING MATERIAL (SLM)

202 Business Mathematics and Statistics Moving averages and progressive averages are known as business averages since they are of immense use in ascertaining the progress of business. For instance, three-year moving averages of profits or progressive averages of sales may be calculated to determine the trend and progress in business. 10.4 Arithmetic Mean The arithmetic mean (AM.) is the quantity obtained by summing up the values of items in a variable and dividing the sum by the number of items. Individual Items If xl + x2 + x3 … + xn (items) are n items with frequencies. x1x2x3 xn n AM = ¦x AM = n Where x = xl + x2 + x3 … + xn (items) S = ‘Sum of’ or Summation N = Number of Items

This method of finding the A.M. is called the direct method. It is very simple and does not involve any mathematical skill, but it is useful only when the items are we and the size of the figure is small. Individual items are many and their size is large it would be difficult to calculate the A.M. with this method. To remove this difficulty a shortcut method is used. We can assume one of the items as an arbitrary (assumed) mean and in how much the items deviate room this assumed mean. We then divide the total number of deviations by the number of items and get a certain quantity. By adding this quantity to the assumed average we obtain the actual A.M. Formula: Arithmetic Mean = {Assumed mean} + {Sum of the deviations of items} From the assumed mean [Total number of items] CU IDOL SELF LEARNING MATERIAL (SLM)

Measures of Central Tendency 203 Symbolically, a + ¦D AM. = n Problem 1: Using the shortcut method, in the A.M. of 215, 218, 222, 235 and 270. Let 222 be the assumed mean. Item Deviation (d) A=222 6fd– 50 215 –7 n=5 218 –4 222 235 0 270 + 13 + 48 Total + 50 AM. = 222 + 50/5 = 222 + 10 = 232. Discrete Series (Direct Method) If x1 + x2 + x3 … xn (items) are n items with frequencies f1 f2, f3, … fn respectively, then AM = x1f1 x2 f2 x3 f3 xn fn

f1 f2 f3 n fn ¦n xf AM = i1 ii n ¦n f i 1i ¦x AM = ¦f ¦fx AM = n where n = Sfn 6 is the summation notation and is pronounced as ‘sigma’. CU IDOL SELF LEARNING MATERIAL (SLM)

204 Business Mathematics and Statistics Direct Method: This method of ending the A.M. of the frequency distribution is called the direct method. Here, the A.M. is obtained by taking the sum of the products of the items and the corresponding frequencies and dividing this sum by the total number of frequencies. Problem 2: Calculate the AM. of the following frequency distribution: Weekly Wages (`) No of Labourers 20 3 25 5 30 8 35 14 40 10 45 6 50 4 Weekly Wages (`) No. of Labourers x X 20 3 60 25 5 125 30 8 240 35 14 490 40 10 400 45 6 270

50 4 200 Total 50 1785 ¦ fx AM = n where n = Sfn Average weekly wages = ` 35.70 Shortcut Method: To minimize calculations in the case of a frequency distribution, it is better to use the shortcut method. In this method the deviations of the quantities room an assumed mean are first calculated; these deviations are then multiplied by their respective frequencies. Then the quantity obtained by dividing the sum of these products (deviation X frequency) by the total number of frequencies is added to the assumed mean. The resulting figure is the actual A.M. CU IDOL SELF LEARNING MATERIAL (SLM)

Measures of Central Tendency 205 Symbolically, AM = ¦fd n Where, D = x – a = deviation, a = assumed mean n = 6f = number of items = total of frequencies The previous example can be worked out using the shortcut method: Weekly No. of Deviation room D Wages (`) X Labourers Assumed Mean D .45 20 3 –15. –50 25 5 –10 –40 30 8 [35] 14 –5 0 40 10 0 +50 45 6 +5 +60 50 4 + 10 +60 + 15 Total 50 +35 Let 35 be the assumed mean. (iii) fd AM = n

35 (B) 35+ 50 (C) 35 + 0.70 (D) 35.70. Average weekly wages = ` 35.70 The calculation can be simplified further by dividing the deviations (D) room the assumed mean by a common actor. The deviations that are divided by the common actor become step deviations (denoted by d). The sum of the products of these step deviations and their respective frequencies (Sfd) are multiplied by this common actor (I). The resulting numerical expression (6fdxi) divided by the total frequencies (n) gives a certain quantity. This quantity, when added to the assumed mean, gives the actual A.M. CU IDOL SELF LEARNING MATERIAL (SLM)

206 Business Mathematics and Statistics ¦fd *i Symbolically, Mean = n where a = assumed mean. 6fd = sum of the products of the step deviations and their respective frequencies, i = common factor, n = total number of frequencies. In the case of the previous example the calculation is simplified as shown below: Weekly Wages No. of Deviation room d /d ` X Labourers Assumed Mean D –3 –9 20 3 –15 –2 –10 25 5 –10 –1 30 8 0 –8 35 14 –5 +1 0 40 10 0 +2 + 10 45 6 +5 +3 + 12 50 4 + 10 + 12 + 15 Total 50 +7 ¦fd Mean = *i 7

5 35+ 50 ×5 5 35 + 0.70 = 35.70 Average weekly wages = ` 35.70 Continuous Series In the case of a continuous series, where items are grouped in certain class intervals, the common factor is the length of the class interval, or instance, the length of the class interval 10-15 is 5. The procedure or calculating the A.M. here is: firstly, finding the mid-values of the respective class intervals, so that these mid-values represent the respective classes. Next, one of these mid-values is taken as an assumed mean and deviations (step) of other mid-values room the assumed mean are found, room this step onwards the working procedure and the formula of calculating the A.M. is exactly the same as in the case of the discrete series. CU IDOL SELF LEARNING MATERIAL (SLM)

Measures of Central Tendency 207 Formula: ¦fd AM = 50n *i Problem 3: Calculate the mean room the following data: Marks No. of Students 0-10 6 10-20 11 20-30 14 30-40 20 40-50 15 50-60 9 60-70 5 80 Class Mid values / D D /d 0-10 5 6 –30 –3 –18 10-20 15 11 –20 –2 –22 20-30 25 14 –10 –1 –14 30-40 35 20 00 40-50 45 15 0 +1 + 15 + 10

50-60 55 9 +20 +2 + 18 60-70 65 +30 +3 + 15 Total 80 –6 Here n = 80. ¦fd ui Mean = a + n §6· ¨¸ 9. 35 + ©80 ¹ ×10 10. 35 – 0.75 = 34.25 CU IDOL SELF LEARNING MATERIAL (SLM)

208 Business Mathematics and Statistics Advantages of A.M. Disadvantages of A.M. 1. The common man can easily understand its 1. Omission of even a single item of the data gives meaning and use. an incorrect value of the A.M., unlike median 2. Since its definition is precise and clear, and mode where extreme items can simply be calculation is easy and its value always discarded. determinate. 2. It may not be identical with anyone of the items 3. It satisfies most of the conditions laid down or of the data. That is, it may not be one of the an ideal average. figures that comprise the data. 4. Its chief merit consists in the act that it takes into 3. The act that it gives a large weight to extreme consideration every item in the data. items is its handicap since it then ails to be a 5. No special arrangement of the data is necessary good representative. That is, the value of the while calculating it, unlike median and mode A.M. of the data consisting of very large and where we have to arrange and group the data in a very small items may lead to. certain manner. 4. Unlike median and mode, it cannot be observed 6. It is of much significance in manipulations either by a mere inspection of the data but has to be of an algebraic or arithmetic nature. calculated. 7. It forms a good basis of comparison while 5. It cannot be used in qualitative (or descriptive) comparing different group’s of numerical data. studies, unlike the median. 8. Given the A.M. and the number of items, the total of the items can be calculated at once (by multiplication), and given the total of the items and their number, the A.M. can be found easily. 10.5 Weighted Arithmetic Mean

When the different items in a data are assigned weights according to their significance (relatively) the average then calculated is called :a weighted arithmetic average. Symbolically, the weighted average of n items x1, x2, x3 …xn whose weights are w1, w2, w3 … wn respectively is: x1w1 x2 w2 x3 w3 xn wn = w1 w2 w3 wn For example, a person buys three different books or ` 12, ` 18 and ` 30 respectively then the average price of a book that he purchases is: 121830 3 = `20 On the other hand, he purchases five copies of the book at ` 12 each, three copies at ` 18 each and two copies at ` 30 each 10 books in all the average price of a book would be: CU IDOL SELF LEARNING MATERIAL (SLM)

Measures of Central Tendency 209 12*518*330*2 10 This is called the weighted average since it has been calculated by giving the weights 5, 3 and 2 to the respective prices of the books. 10.6 Median The value of the middle item of a series that is ordered either in the ascending or descending order of magnitude is called the median. In other words, to determine the median it is necessary to arrange the items first either in ascending order or in descending order, for example, the median of the following 11 numbers 12, 15, 16, 21, 14, 22, 31, 32, 30, 28, 19, is 21 since 21 is the middle number when the numbers are written in ascending order as under: 12, 14, 15, 16, 19, 21, 22, 28, 30, 31, 32. ª n 1º « »th In symbols, i n stands or number of items in a data then the median is the value of the ¬2¼ item, when the numbers are written either in ascending or descending order 0 magnitude. I n is an odd number, then the median can be located at once. But i n is an even number, then the median is equal to the value of the A.M. of the two central items. For example, the median of the numbers 17, 18, 21, 26, 28, 31, 33, 36 is: (26 + 28)/2. The median can be obtained in the following way also: Here n = – 8, therefore

10. n 1º Median = the value of « »th item ¬ 2¼ Median = the value of (4.5)th item Median = the value of 4th item +1/2 (5th item – 4th item) Median = the value of 26 + 1/2(28 – 26) Median = the value of 26 + 1 Median = 27. CU IDOL SELF LEARNING MATERIAL (SLM)

210 Business Mathematics and Statistics Discrete Series The following is the procedure or finding the median in the case of a discrete series: ? Items are arranged either in ascending or descending order of magnitude and then their respective frequencies are written against them. ? These frequencies are cumulated. = n 1º The value of the « »th item, where n = total number of frequencies, is determined by ¬ 2¼ mere inspection of the cumulative frequencies. Problem 4: In the median room the following data: Age 18 19 20 21 22 23 24 25 26 27 28 29 No. of Persons 4 57 9 11 16 22 18 15 12 10 6 The following Problem explains the above procedure: Cumulative Frequency Age Frequency 4 9 18 4 16 19 5 25 20 7 36 21 9 52 22 11 23 16

24 22 74 25 18 92 26 15 107 27 12 119 28 10 129 29 6 135 135 ? n 1º Median = the value of « »th item ¬ 2¼ ª 135 2º Median = the value of « »th item ¬ 2¼ Median = the value of 68th item Median = 24. CU IDOL SELF LEARNING MATERIAL (SLM)

Measures of Central Tendency 211 Continuous Series The technique of interpolation is used to in the median in a continuous frequency distribution. Interpolation is the process or technique of finding out the appropriate value or a missing or unknown value. Depending on the given data, the technique of interpolation also enables us to in missing or unknown values in the data. Interpolation may not necessarily give the exact value, only an approximate or as far as possible accurate value. The method of interpolation or finding out the median is given by the following formula: §N · CF¸ ¨ Median = L + ¹ ©2 Where L = the lower limit of the class in which the median item is situated. i = length of the class interval. F = frequency of the median class. N = the total of the frequencies. CF = the cumulative frequency of the class immediately preceding the median class. Procedure: The given frequencies of a continuous series are to be cumulated and these cumulated frequencies are to be written against the respective classes. The class in which the value of the n/2 item aIls is to be determined by inspecting the cumulative frequencies. The class determined in such a manner is called the median class since the median exists in that class. The next step is to in the values of F and CF and then use the given formula.

The Problem worked below makes it clearer. Problem 5: Calculate the median room of the following data: Class Frequency Cumulative frequency 0-10 6 6 10-20 11 17 20-30 14 31 30-40 20 51 40-50 15 66 50-60 9 75 60-70 5 80 80 CU IDOL SELF LEARNING MATERIAL (SLM)

212 Business Mathematics and Statistics Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 No. of students 6 11 14 20 15 9 5 Here, N = ?, f = 80, N/2 = 40. By observing the cumulative frequencies it is clear that the median class is 30-40 ? L = 30, i = 10, =20, CF = 31 Median = i§N · L + ¨ CF¸ Median = f© 2 ¹ Median = 30+^[40–31] Median = 30 ± 30 + 4.5 Median = 34.5 10.7 Advantages and Disadvantages of the Median Advantages of the Median Disadvantages of the Median 1. Its meaning is clear and intelligible. 1. An irregular series is characterized by extreme variations between the items; in the case of such a series the median ails to be

representative. 2. Most of the conditions of an ideal average are 2. Arrangement of the data either in the satisfied by it. ascending order of magnitude or descending order of magnitude is the chief requisite. This process is tedious when the data is vast, etc. 3. It is very easy to in it in the case of a discrete 3. It absolutely ails to be of any use in cases series. where it is necessary to give large weights to extreme items. 4. The median is of immense use while estimating 4. Sometimes the median may exist between two qualities such as honesty, intelligence, virtue, values, thus involving the work of i to morality etc. and proves be a good estimation. representative. 5. It’s possible to in the median by knowing only 5. Unlike the arithmetic mean, it is not possible the values of the central items and the number to in the total value of all items if we know of items. That is, the values of the extreme items the value of the median and the number of are not necessary or finding the median. items. 6. It can also be used algebraically to some extent. 6. Its field of application through algebraic processes is restricted. CU IDOL SELF LEARNING MATERIAL (SLM)

Measures of Central Tendency 213 10.8 Mode The value of the item in a variable that is repeated the greatest number of times is called the mode. In other words, it is the value of the item that occurs most frequently in the data. It is said to be the most prominent item as well as a typical measurement, for instance, when we speak of modal income or modal marks we mean average income or average marks respectively. It is easy to determine the mode in the case of a discrete series that is regular — that is, where frequencies are not distributed irregularly. But i the frequency table has a number of irregularities, the location of the mode becomes mid cult. To overcome this difficultly it is necessary to adjust the frequencies through the grouping process. It’s then possible to locate the mode at once. In the case of a continuous series that is regular the modal class can be determined at once and the interpolation formula can be used. But I the continuous series is irregular it is necessary to adjust the frequencies by grouping, which involves widening the groups into which the frequencies all until the modal group is traced. After tracing the modal class the following interpolation formula is used: Mode = L + f1 f0 *i 2f f f 10 2 Where L = lower limit of the modal class. F1 = frequency of the modal class. F0 = frequency of the class immediately lower to the modal class. F2 = frequency of the class immediately higher to the modal class. I = length of the class interval.

Problems: (Discrete series) in the mode or the following data: X 21 6 22 9 23 12 24 16 25 22 26 22 CU IDOL SELF LEARNING MATERIAL (SLM)

214 Business Mathematics and Statistics 27 24 28 18 29 17 30 15 31 12 32 7 Solution: 2 3 45 6 21 27 x1 15 21 6 46 22 9 23 12 37 28 50 24 16 60 38 68 64 25 22 44 26 22

27 24 59 42 50 28 18 44 35 34 29 17 32 30 15 27 31 12 19 32 7 CU IDOL SELF LEARNING MATERIAL (SLM)

Measures of Central Tendency 215 Table of Analysis Column x with maximum frequency 1 27 2 3 25 26 4 5 26 27 6 No. of times 24 25 26 25 26 27 13 26 27 28 54 1 The modal value of x is 26 since it is repeated the largest number of times according to the table of analysis. By merely observing the given data one may infer that 27 is the mode since it is repeated 24 times. But the analysis proves that this is not correct. Hence 26 is the mode. Problem 7: (Continuous Series) Calculate the mode room the following data: Marks No. of Students 0-10 6 10-20 11 20-30 14 30-40 20 40-50 15

50-60 9 60-70 5 80 Solution: Since the given frequency distribution is regular, we in that the class 30-40 that has the maximum frequency is the modal class. Therefore, i = 10, F0 = 14, F2 = 15, F1 = 20, L = 30. Substituting these values in the formula or the mode, we get: § F1F0 · Mode = L¨ ¸*i ©2F1F0 F2 ¹ § 2014 · ¨ ¸ * 10 = 30+ 401415¹ © CU IDOL SELF LEARNING MATERIAL (SLM)

216 Business Mathematics and Statistics 6· ¨¸ ? 30 + ©11 ¹ * 10 ? 30 + 5.45 ? 35.45 Problem 8: The following data relates to the age distribution of 50 persons: Age 20-30 30-40 40-50 50-60 60-70 70-80 No. of Persons 3 7 14 16 8 2 Draw a histogram or the data and in the modal value. Check the value by direct calculation. Solution: The following is the procedure to in the mode graphically: … Draw the histogram or the data by representing age along the x-axis and the number of persons along the y-axis. … Draw two straight lines diagonally in the inside of the bar relating to the modal class. Join each upper corner of the modal bar to the upper corner of the adjacent bar. … From the point of intersection of the two diagonal lines, draw a perpendicular as shown in Fig. (1) to the x axis. The root of the perpendicular shows the modal value, in this case 52. 5 5

5 5 5 10-20 20-30 30-40 40-50 50-60 60-70 70-80 Age Mode 52 Direct calculation to verify the value of the mode is like this: (· F1 F0 ¸ *1 Mode = L + ¨ © 2F1 F0 F2 ¹ CU IDOL SELF LEARNING MATERIAL (SLM)

Measures of Central Tendency 217 ¨ 1614 · ¸ 50 + ©32148 ¹ * 10 2· ¨¸ 560 + ©10 ¹ * 10 50+2 (3) Mode =52; Hence verified. Problem 9: Calculate the mode room the following data: Class Frequency 0-5 10 5-10 13 10-15 16 15-20 17 20-25 18 25-30 16 30-35 11 35-40 14 40-45 9 124 Since the given frequency is not regular, we have to group the frequencies and prepare the analysis table or ending the modal class.

Class Frequency 0-5 1 23 456 5-10 10-15 10 23 15-20 20-25 13 29 39 25-30 30-35 16 33 35-40 40-45 17 35 46 18 34 51 16 11 25 27 51 14 9 34 41 CU IDOL SELF LEARNING MATERIAL (SLM)

218 Business Mathematics and Statistics Column Table of Analysis 1 2 Cases with Maximum Frequencies 3 4 15-20 20-25 25-30 5 15-20 20-25 20-25 6 15-20 15-20 25-30 20-25 Total No. of times 5-10 10-15 10-15 12 4 5 2 It follows from the above tables of analysis that the modal group is 20-25. Therefore, L = 20, i = 18, F0 = 17, F2 = 16, F1 = 5. 10.9 Advantages and Disadvantages of Mode Advantages of Mode Disadvantages of Mode 1. It is quickly intelligible and its calculation is 1. It does not satisfy some of the conditions laid not difficult. down or an ideal average. 2. It is the most predominant item in a discrete 2. It is quite possible that in certain types of data series. it may not be properly defined and hence it may be indeterminate and indefinite. 3. It can be located by a mere inspection of the