MCM 602 Quantitative Techniques for Managers

Charts and Graphs 45 4. Cumulative frequency is __________. (a) Increasing frequency (b) decreasing frequency (c) Fixed frequency (d) None of these 5. The average value of the lower and upper limit of a class is called __________. (a) Class Boundary (b) Mid point (b) Class-interval (d) Class-frequency Answers 1. (c), 2. (c), 3. (a), 4. (a), 5 (b). 2.8 References References of this unit have been given at the end of the book. CU IDOL SELF LEARNING MATERIAL (SLM)

46 Quantitative Techniques for Managers UNIT 3 DESCRIPTIVE SUMMARY MEASURES - I Structure: 3.0 Learning Objectives 3.1 Introduction 3.2 Measure of CentralTendency and Location 3.3 Mathematical Means 3.4 Calculation of Median 3.5 Calculation of Mode 3.6 Empirical Relation among Mean, Mode and Median 3.7 Percentiles 3.8 Quartiles 3.9 Solved Problems 3.10 Summary 3.11 Key Words/Abbreviations 3.12 LearningActivity 3.13 Unit End Questions (MCQ and Descriptive) 3.14 References CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - I 47 3.0 Learning Objectives After studying this unit, you will be able to:  Define the concept of Arithmetic Mean.  Discuss the concept of Median and Mode.  Describe the use of data for calculation of centre tendency.  Measure the capability assessment through Self-Assessment problems. 3.1 Introduction In the previous chapter we elaborated as to how the statistical data can be tabulated and presented in a form to draw a meaning inference at a glance. The analysis of the data, therefore, becomes easier and can be made use of in decision making process. The objective of the statistical analysis is to determine various numerical measures and thus summarising the values of a variable by representing the number of values of one single value. One such measure is called ‘Average or Mean’. The single value is termed as average or the expected value of the variable and it reduces a large group of data into a single value that can be used in decision making. Averages are the values lying between the largest and the smallest value of the observations and denote the central part of the distribution of the data. These are called ‘Measure of Central Tendency’. Averages are also sometimes called as ‘Measure of Location’ as they enable us to locate the position or place of the distribution in question. 3.2 Measure of Central Tendency and Location In the discussion so far we elaborated as to how the statistical data can be tabulated and presented in a form to draw a meaningful inference at a glance. The analysis of the data, therefore, becomes easier and can be made use of in decision making process. The objective of the statistical analysis is to determine various numerical measures and summarising the values of a variable by representing the number of values of one single variable. Two of these characteristics are “Central Tendency” and “Dispersion”. Whereas central tendency is the central or middle value of the distribution, the dispersion is the spread of the data in the distribution. It can be clearly seen from CU IDOL SELF LEARNING MATERIAL (SLM)

48 Quantitative Techniques for Managers fig. 3.1 that central location of curve A and B are same whereas for curve C, it is different and to the right of curve A and B. Fig. 3.1: Comparison ofcentral location The measure for curve A and B having the same central location is called average i.e. the average value of the statistic for curve A and B are the same. This measure reduces a large group of data into a single value that can be used in decision making. Averages are the values lying between the largest and the smallest value of the observations and denote the central part of the distribution of the data. These are called ‘Measure of Central Tendency’. Averages are also sometimes called as ‘Measure of Location’ as they enable us to locate the position or place of the distribution in question. There are five measures of average — Arithmetic, Geometric and Harmonic means, median and mode. Requisites of a Good Average In general, the averages should satisfy following conditions : 1. It should be rigidly defined. 2. It should be based on all the relevant observations 3. It should be easily understandable. 4. It should be suitable for further mathematical treatment 5. It should not be affected much by extreme observations. 6. It should be least affected by data fluctuations. CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - I 49 3.3 Mathematical Means Calculation ofArithmetic Mean The arithmetic mean of a given set of data is their sum divided by the number of observations. Thus the arithmetic mean of 5. 10, 12, 18, 22, 25, 28, 30, 31, 35 can be calculated as under. 5  10  12  18  22  25  28  30  31  35 Mean or Average = 10 216 = 10 = 21.6 Calculation of Mean for ungrouped data We can see that if the daily wages of 10 workers are given as ` 5, 10, 12, 18, 22, 25, 28, 30, 31, 35, then on the average, a worker gets ` 21.60 per day. The manager of this group would have a reasonable single value for a meaningful usage for his work force. Writing the same relationship in general. Arthmetic mean or Average = x = x1  x2  x3 ....... xn n where x1, x2, x3 etc are the individual values of n observations and the average value is represented by x (x - bar)and is mean of n number of observations. The individual values of observations used from a large poulation are sample values and computed values are called “statistics”. If we compute this value for the entire population (and not for a limited number, called sample), the mean value is represented by a greek letter  (pronounced as mu). The symbol for the total number of observations or element in a population is denoted by N (as against n for a sample). Thus we can distinguish the sample mean from the population mean as follows : x x= n x and = N The greek letter  (sigma) is used for summation of all the values of x together, The method is called “calculation of Mean or Average value of ungrouped data”. CU IDOL SELF LEARNING MATERIAL (SLM)

50 Quantitative Techniques for Managers Measurement of Mean from Grouped data If the value of the observation is repeated, then these are called frequencies of the value and are denoted by f Thus x = (x1  x1 .... f1times)  (x2  x2 .... f2 times)....(xn  xn .... fn times) f1  f2 .... fn = f1x1  f 2 x2 .... fn xn f1  f2 .... fn  fx = f  fx = n Thus sample arithmetic mean of a grouped data of n-observations can be written as  fx x= n where x = Mean value of the sample observation f = frequency of various values x = values of observations n = total number of observations  = symbol for the summation The same relationship can be extended to a grouped data arranged as class intervals and their relative frequency. In this case f = frequency of observations in each class x = mid point of each class CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - I 51 Step Deviation Method forArithmetic Mean When values of x and f are large, then the mean calculations become very cumbersome. Hence we can use the step-deviation method for the purpose by using deviations of the given observations from any arbitrary value say A. Then d =X–A and  fd = f (X – A) = fX – Af (A being constant) = fX – n . A  fd  f X or = – A nn = x–A  fd  x =A+ n If class intervals are of equal magnitude, the computation can be further simplified by using Hence X–A d = h (h = common magnitude of the class interval) hd = X – A and hfd = f X – A.f or h fd = fX – n.A h  fd  f X or n = n – A = x –A h  fd x =A+ n Weighted Average When all the observations in a data bank do not have the same importance, such as sal es of airconditioners in the month of March can have better bearing as the forecast sales of April (being CU IDOL SELF LEARNING MATERIAL (SLM)

52 Quantitative Techniques for Managers seasonal variation), then for average computation, more importance can be attached to the sales of March than that for February and January. If these importance levels (called weightages for the observations) are taken into account, the average so obtained is termed as Weighted Average. This can be calculated as follows. Let W1, w2, w3 .... wn be the weights attached to observations x1, x2, x3....xn. Then weighted average of the observations can be calculated as under. w1x1  w2 x2 ....wn xn xw = w1  w2  ....wn In case of frequency distribution, the relationship undergoes the change as follows : xw = w1( f1x1)  w2 ( f2 x2 )....wn ( fn xn ) w1  w2 ....wn  w( fx) = w Advantages of Arithmetic Mean 1. It is well known and established measure 2. It can be easily calculated 3. It is useful for many statistical procedures such as comparison of many sets of data 4. Every set of data has a mean value Disadvantages 1. It reflects all the values of the set of observations, but can be affected by extreme values, where may not be representing the set at best. 2. It is lengthy method for large number of observations. 3. For open ended data, it is difficult to compute Calculation of Geometric Mean The Geometric Mean of a set of n observations is the nth root of their product. If the observations are x1, x2, x3, .... xn. CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - I 53 Then Geometric Mean (GM) = n x1. x2 . x3..... xn = (x1, x2, x3, .... xn)1/n It is easy to calculate only if n = 2. If n is larger, then we have to use the logarithmic concept as follows 1 Log (GM) = n log (x1.x2.x3....xn) 1 = n (log x1 + log x2 + ....log xn) 1 = n  log x FGH KJIGM = Anti log Hence 1  log x n Calculation of Harmonic Mean If x1, x2, x3....xn is a set of n observations, their Harmonic Mean (HM) is given by 1 HM = LM PO1 1  1 .... 1 N Qn x1 x2 xn = 1 1LMN 1 PQO = NLMnX1 POQ n X1 Thus the Harmonic Mean is the reciprocal of the arithmetic mean of the reciprocals of the given observations. Relations Between Arithmetic Mean, Geometric Mean and Harmonic Mean The relationship of the three means is as follows : AM > GM > HM CU IDOL SELF LEARNING MATERIAL (SLM)

54 Quantitative Techniques for Managers The sign of equality holds good only if all the observations are equal.This can easily be proved for two numbers. There is yet another relationship of AM, GM and HM for two numbers only i.e., G2 = A × H where G = Geometic Mean A = Arithmetic Mean and H = Harmonic Mean 3.4 Calculation of Median The median is that value of the variable, which divides the group in two equal parts, one part comprising all the values greater and the other, all values less than the median. Thus median is only a positional average i.e. its value depends on the position occupied by a value in the frequency distribution. Calculation of Median Ungrouped Data : If the number of observations is odd, then the median is the middle value of the rearranged form of the observations either in the ascending or in the descending order. Thus for the observations 5, 10, 25, 15, 50, 40, 35, the arranged order is 5, 10, 15, 25, 35, 40, 50 and the median is the middle value i.e. 25 (3 observations are lower than 25 i.e. 5, 10, 15 and other 3 observations are higher than the median i.e. 35, 40 and 50) If the ungrouped data contains even number of observations, then the median is calculated as the arithmetic mean of the two middle numbers, provided the data is arranged either in ascending or descending order. Thus in a list of observations 5, 10, 25, 15, 50, 40, 35, 17 the rearranged data in the descending order becomes 50, 40, 35, 25, 17, 15, 10 and 5, and the two middle values are 25 and 17. 25  17 By taking the average of the two i.e. 2 = 21, we have obtained Median as 21. Grouped Data or Frequency Distribution : In case of frequency distribution, the values of the variables are given as x1, x2, x3, etc. and their related freqsuencies as f1, f2, f3 etc. Where f = N, i.e. the sum total of frequencies. Then the median is calculated in the following manner : 1. Rearrange the data in terms of ‘less than’ cumulative frequency distribution 2. Find N/2. CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - I 55 3. Check where the cumulative frequency just exceeds the value of N/2 4. The value of the variable corresponding to the cumulative frequency in step (3) is the Median. Continuous Frequency Distribution : In a continuous frequency distribution data, we obtain the value of the median in the same manner as above, but here the value of the variable will be a class interval and not a specified value. This is called the Median class. For obtaining median, we use the following formula. F Ih N HG JKMedian = l + f c 2 Where l = lower limit of the median class h = magnitude or the width of the median class f = frequency of the median class c = cumulative frequency of the class preceeding the median class. and N = f i.e. the total cumulative frequency . 3.5 Calculation of Mode Mode is the value, which occurs most frequently in a set of observations. In simple language “Mode is the value which has the greatest frequency density in its immediate neighbourhood”. Computation of Mode As per the simple definition given above, the mode is the value of the variable corresponding to the maximum frequency in a frequency distribution. Hence in a table of frequency distribution given below, the mode is the variable value for the maximum frequency. CU IDOL SELF LEARNING MATERIAL (SLM)

56 Quantitative Techniques for Managers Thus taking a case of Data array arranged in the ascending order DataArray in Ascending Order 0 2 4 8 18 2 24 52 12 19 2 5 15 19 3 0 2 6 16 19 1 32 16 20 2 13 3 72 16 5 20 1 7 16 16 We observe that value 16 has occured maximum number of times (5 times) i.e. the maximum frequency is that for the value 16, the Mode, therefore for this data is 16. Using the same data and arranging it as frequency distribution, we get DataArranged asFrequency Distribution Class Frequency 000–2 9 003–5 5 006–8 4 09–11 0 12–14 1 15–17 6 18–20 6 Here the maximum frequency occurs for class-interval 0-2 (9 times). Thus 0-2 is the modal class. This is called the Grouping Method. CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - I 57 Multimodal Distributions : We can encounter cases, where there can be more than one observation indicating modal value or in a frequency distribution pattern, more than one class as modal class. This happens when the frequency of the value or a class interval is the same. These cases are called multimodal cases or multimodal distributions. Taking a case of values given in the table. There are two modal values 4 and 7. This is called Bimodal case. DataArranged inAscending Order 1 469 2 44 7 9 23 4 72 94 24 89 32 5 2 8 10 3 Since values 2 are occuring 3 times which are higher than the neighboring values, this is also a modal value, though not of same frequency. But values 4 and 9 are multimodal values. Mode of continuous frequency distribution : In case of continuous frequency distribution, the class pertaining to the maximum frequency is called the modal class. The mode interpolation formula is given below : b gh f1  f0 b g b gMode = l + f1  f0  f2  f1 Where = l + h( f1  f0) 2 f1  f0  f2 l = lower limit of the class h = magnitude or width of the modal class f0 = frequency of the class preceeding the modal class f1 = frequency of the modal class CU IDOL SELF LEARNING MATERIAL (SLM)

58 Quantitative Techniques for Managers f2 = frequency of the class succeeding the modal class. An alternate formula can be used for calculation of mode as under. NML OPQf2 Mode = l + f1  f2 × h where the notations have the same meaning as given above. Mode by method of grouping (distribution irregular) : Calculation of mode can be done as above if the frequency table is regular. If it is not, as is the case below, the method of grouping is to be applied. Let us take the frequency distribution as follows x 10 11 12 13 14 15 16 17 18 19 y 5698725321 Here frequency is first increasing upto 9, then decreasing to 2, increasing again to 5 and then decreasing to 1, thus irregular. In such cases, various frequencies can be grouped together, initially in a group of 2 and then moving leaving the first and so on. These groupings can be done for 3 frequencies and so on. At the end of this computation, we work out the maximum number of times a particular frequency occurs. It will then be called the modal class and mode can be worked out accordingly as given in problem 8.28. 3.6 Empirical Relation among Mean, Mode and Median The following empirical relationship has been developed by Prof. Karl Pearson to connect Mean, Mode and Median. Mode = Mean – 3 (Mean – Median) (i) 1 (ii) or Mean – Median = (Mean – Mode) (iii) 3 or Mode = 3 Median – 2 Mean CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - I 59 3.7 Percentiles As the name suggests, these values divide the set of observations into 100 equal parts and hence are 99 in number i.e. P1, P2, P3 . . . . P99 such that P1 < P2 < P3 < . . . . < P99. The calculation of various percentile can be done in the following manner F Ii  N G JPi = lp + h H K100 – C, i = 1, 2, 3, = 99 where lp = lower limit of the class containing Pi h = magnitude of the class containing Pi f = frequency of the class containing Pi and C = cumulative frequency of the class proceeding the class containing Pi. Graphical Method of Locating Partition Values :These values can be best located with the help of the curves called “ogives” or “Cumulative frequency Curves” [less than ogives and more than ogives] described under paragraph 7.5 in the previous chapter. The procedure followed is as under 1. Draw the “less than ogive” for the data given. 2. Obtain the desired partition values from the curve by scaling out the type of partition value and obtain its numerical values from the corresponding point on the curve. 3.8 Quartiles These are so called because these values divide the given data set into four equal parts. There are three such points, namely the first, second and third quartile, devoted by Q1, Q2 and Q3 respectively such as Q1 < Q2 < Q3. These values indicate that Q1 has 25 per cent of the items of the distribution below it and 75 percent of the items above it. Q2, the second quatile divides the values into two equal parts, 50 per cent lower and 50 percent higher than Q2. Thus it coincides with Median, described above. Q3, also known as third quartile or upper quartile would have 75 per cent observations below it and only 25 per cent one of. CU IDOL SELF LEARNING MATERIAL (SLM)

60 Quantitative Techniques for Managers The calculation of quartiles can be done as follows – Q1 = lq1 + GFGHG N  C IJJJK × h = lq1 + h HGF N – CJKI 4 f f 4 GFHGG 3N  C JKIJJ h GHF 3N  CKIJ 4 f 4 and Q3 = lq3 + × h = lq3 + f Where N = Total cumulative frequency C = Cumulative frequency upto the lower limit of quartile l = Lower limit of the quartile class f = Frequency of the quartile class h = Width of the class interval Deciles These are the values, which divide the set of observations into ten equal parts (hence the name). These values are denoted as D1, D2, D3 . . . . etc., such that D1 < D2 < D3 < . . . . < D9. In such case, D5 will coincide with the Median. For computing the values of Deciles. We can use the formula F Ih HG KJDi = ld + f iN –C 10 (where i = 1, 2, 3, . . . . 9) where ld = lower limit of the class containing Di f = frequency of the class containing Di h = magnitude of the class containing Di and C = cumulative frequency of the class preceeding the class containing Di. 3.9 Solved Problems Problem 1 Calculate the average value of age for a class of 10 students with their ages as under : 11, 12, 13, 13, 10, 13, 12, 11, 10, 12 years. CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - I 61 Solution : Average age = x = 11  12  13  13  10  13  12  11  10  12 10 117 = 10 = 11.7 years Problem 2 The following table indicates the marks obtained by students in a class test. Calculate the average level of marks of the class. Marks obtained :0 2 3 4 5 6 7 89 Number of students : 11 10 9 21 12 17 8 22 15 Solution :  f = 11 +10 + 9 + 21 + 12 + 17 + 8 + 22 + 15 = 125  fx = 11 × 0 + 10 × 2 + 9 × 3 + 21 × 4 + 12 × 5 + 17 × 6 +8 × 7 + 22 × 8 + 15 × 9 = 660 fx = f 660 = 125 = 5.3 marks The same data can be tabulated as follows and then used to calculate the average of the observations. Number of marks (x) Number of students (f) Product (f.x) 0 11 0 2 10 20 3 9 27 CU IDOL SELF LEARNING MATERIAL (SLM)

62 Quantitative Techniques for Managers 4 21 84 5 12 60 6 17 102 7 8 56 8 22 176 9 15 135 N = f = 125 (fx) = 660 660  Mean marks or class average = x = 125 = 5.3 marks. Problem 3 Computing Arithmetic mean using mid-values of class intervals, given below is the table for service time on a work station; calculate the average service time for the place. Class Interval (minutes) : 0-10 10-20 20-30 30-40 40-50 50-60 Frequency (No. of Cars) : 6 5 8 12 5 15 Solution : The data can be put into the tabulated form as follows: Class intervals Mid. values (x) Frequency (f) fx 0-10 5 6 30 10-20 15 5 75 20-30 25 8 200 30-40 35 12 420 40-50 45 5 225 50-60 55 15 825 N = f = 51fx = 1775  fx Hence average Service time =  f CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - I 63 or, 1775 Problem 4 = 51 = 34.8 minutes The data in respect of daily sales of refrigerators for a company are given below (in thousands). Calculate the average sales per day for the company. 14, 23, 8, 26, 5, 27, 46, 45, 32, 30, 42, 7, 6, 32, 15, 28, 12, 16, 22, 36, 26, 15, 40, 31, 29 Solution : The raw data can first be tabulated in the following manner : Daily Sales (in thousands) Mid-values (x) Frequency (f) f. (x) 1-10 5.5 4 22.0 11-20 15.5 5 72.5 21-30 25.5 7 178.5 35.5 6 213.0 31 – 40 45.5 3 136.5 41 – 50 f = 25 fx = 622.5 Hence Average Daily Sales = x  fx = f 622.5 Say 25 thousand refrigerators. = = 24.9 25 If we calculate the average daily sales by using raw data or ungrouped data, we get x x = f CU IDOL SELF LEARNING MATERIAL (SLM)

64 Quantitative Techniques for Managers 14  23  8  26  5  27  46  45  32  30  42  7  6  32  15  28  12  16  22  36  26  15  40  31  29 = 25 613 = 25 = 24.5, say 24,500 refrigerators Thus there is only a marginal difference in the calculations using the raw data and the grouped frequency distribution data. Problem 5 Calculate the Arithmetic Mean by Step Deviation Method for following data : Values : 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70 Frequency : 7 9 15 11 27 18 5 Solution : Class interval Mid point (x) Frequency (f) x – 35 f.d 0-10 5 7 d= – 21 10-20 15 9 – 18 20-30 25 15 10 – 15 30-40 35 11 40-50 45 27 –3 0 50-60 55 18 –2 27 –1 36 0 1 2 60-70 65 5 3 15 f = 92 = n fd = 24 Taking x – 35 A = 35, h = 10, d = 10 h  fd x =A+ n CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - I 65 10  24 = 35 + 92 = 35 + 2.33 = 37.33 Problem 6 Let the table of observations be as under : Subject Marks Obtained (%) Weightage for W.X (X) Admissions (W) 67 Hindi 67 1 85 English 85 1 174 Physics 87 2 158 Chemistry 79 2 285 3 Maths 95 Find the Weighted Average. Solution : x = Weighted Average =  WX = 769 w W 9 = 85.5 Problem 7 For the frequency table given below, calculate Median. x: 0 1 2 3 4 5 6 7 8 9 51 23 3 f : 3 5 12 45 61 22 29 CU IDOL SELF LEARNING MATERIAL (SLM)

66 Quantitative Techniques for Managers Solution : Variable (x) Frequency (f) Less than cf 0 3 3 1 5 8 2 12 20 3 45 65 4 61 126 5 22 148 6 29 177 7 51 228 8 23 251 9 3 254 After obtaining less than cf as given above, Let us calculate median stepwise. (i) Less than cf has been obtained 254 (ii) N/2 = = 127 2 (iii) The value of variable just above the figure of 127 is 5 (against 148 i.e. just higher than 127) (iv) Hence Median of the observations is 5. Problem 8 Considering the following table, obtain the median value : Pay Scales (`) Number of employees in that pay scale (Nos. in hundreds) Less than 2,000 14 Less than 3,000 19 Less than 4,000 26 Less than 5,000 35 5,000 and above 42 CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - I 67 Solution : We first prepare the continuous frequency distribution table as given below. Pay Scales (`) Number of employees in Less than cf that pay scale (Nos. in hundreds) 0,000-2,000 14 14 2,000-3,000 19 – 14 = 5 193 3,000-4,000 26 – 19 = 7 26 4,000-5,000 35 – 26 = 9 35 5,000 and above 42 – 35 = 7 42 Here N = f = 42 Hence N 42 = = 21 22 Cumulative frequency just higher than this number is for class ` 3,000 – 4,000. Thus the class 3,000 – 4,000 is the median class, where l = 3,000; h = 1,000 f = 7; C = 19 1,000 Hence Median = 3,000 + 7 (21 – 19) = 3,285.7 (Median pay scale) or ` 3,286 Problem 9 Given the following distribution, calculate ‘Mode’. Variable (x) : 123456 Frequency (f) : 5 15 25 15 20 6 CU IDOL SELF LEARNING MATERIAL (SLM)

68 Quantitative Techniques for Managers Solution : The miximum frequency in this distribution is 25 and the value of the variable corresponding to this frequency is 3. Thus 3 is the Mode for the given distribution. Problem 10 Given the following data, calculate Mode. Variable (x) 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70 Frequency (f) 5 6 8 12 15 5 3 Solution : In this case, the maximum frequency occurred is 15 and this pertains to the class 40 – 50 Hence class 40-50 is the model class. Hence h( f1  f0 ) Mode = l + 2 f1 – f0 – f2 10(15  12) = 40 + (2  15)  (12  5) = 42.3 An alternate formula also can be used for calculation of Mode as follows F If2 GH JKMode = l + f1  f2 × h F I5 GH KJ= 40 + 15  5 × 10 = 42.5 Since the second formula is only approximate, there is slight variation in the calculated value. CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - I 69 Problem 11 Calculate mode for the following data : Marks No. of Students Marks No. of Students Below 10 4 Below 60 86 Below 20 6 Below 70 96 Below 30 24 Below 80 99 Below 40 46 Below 90 100 Below 50 67 Solution : Working out the class frequencies and writing the frequency distribution table, we obtain Marks Frequency Marks Frequency 0-10 4 50 – 60 86 – 67 = 19 10-20 60 – 70 96 – 86 = 10 20-30 6 – 4 =12 70 – 80 99 – 96 = 13 30-40 24 – 6 = 18 80 – 90 100 – 99 = 1 46 – 24 = 22 40-50 67 – 46 = 21 It is observed that frequencies are increasing first and then decreasing, the distribution is irregular and hence Method of grouping is used to locate modal class. Marks Grouped Frequencies 00-10 1 2345 6 10-20 20-30 46 30-40 2 20 24 40-50 50-60 18 40 60-70 70-80 22 43 42 61 80-90 21 40 62 19 29 50 10 13 32 3 4 14 1 CU IDOL SELF LEARNING MATERIAL (SLM)

70 Quantitative Techniques for Managers In order to compute modal class, we prepare frequency of maximum occurance of a class interval Column No. Max Frequency Class interval involved 1 22 30 – 40 50 – 60 2 40 20 – 30 3 0 – 40 40 – 50 3 43 50 – 60 60 – 70 4 62 30 – 40 40 – 50 50 – 60 5 50 30 – 40 40 – 50 6 61 40 – 50 20 – 30 30 – 40 40 – 50 Frequency of classes 25 5 31 From the frequency of occurance of various class intervals, we observe that the class 30 – 40 as well as 40 – 50 are occuring 5 times each. Hence this method does not clearly define the modal class. If maximum frequency had occured only once i.e. either only class 30 – 40 or class 40 – 50 it could have been declared as the modal class. Now, we can use the relationship given in paragraph 11.15 for calculation of mode i.e. Mode = 3 Median – 2 Mean For calculation of mean, we can use step – deviations method by taking. x – 45 d= 10 h  fd Then, Mean = A + N 10(–28) = 45 + 100 = 42.2 N Calculating Median, we have 2 = 50. Since cummulative frequency more than 50 is 67, the median class will be class 40 – 50. CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - I 71 F Ih H KHence Median = 40 + f N  46 2 10 = 40 + (50 – 46) 21 = 41.9 Now mode = 3 Median – 2 Mean = 3 × 41.9 – 2 × 42.2 = 41.3 3.10 Summary The unit is summarised by some of its important points as below:  Bimodal Distribution: A distribution of 2 observations occurring more frequently than the others in a set of values.  Geometric Mean: A measure of central tendency for multiplicative effects of the set of observations.  Mean: A measure of central tendency representing the arithmetic average of the given set of observations.  Measure of Central Tendency: The measure of values of observations indicating their position between the highest and the lowest values of observation denoting the central part of the observations.  Median: Middle point of a set of observations dividing the set into two halves.  Median Class: A frequency distribution class interval denoting the median value of observations,  Mode: The value most often occurring or being repeated in a set of observations  Weighted Average: A mean or average value calculated to take into account the importance of each value to the overall total. CU IDOL SELF LEARNING MATERIAL (SLM)

72 Quantitative Techniques for Managers Important Relationships used x  m= N x  x= n  fx  x = f  fd  x =A+h f  wx  x = w  GM = n x1, x2 , x3....... xn 1  HM = ML PO1 1  1 .... 1 N Qn x1 x2 xn  G2 = A × H L On  1 NM PQ Median= 2 th item in a data array FGH – cJKI =l+ h N f 2  Mode= value of most repeated observations b gh f1  f0 = l + 2 f1  f0  f2  Mode L Of2 MN QP= l + h f1  f2 CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - I 73 3.11 Key Words/Abbreviations  Geometric Mean: A measure of central tendency for multiplicative effects of the set of observations.  Mean: A measure of central tendency representing the arithmetic average of the given set of observations.  Measure of Central Tendency: The measure of values of observations indicating their position between the highest and the lowest values of observation denoting the central part of the observations.  Median: Middle point of a set of observations dividing the set into two halves.  Median Class: A frequency distribution class interval denoting the median value of observations, 3.12 Learning Activity 1. Find the missing frequency from the following distribution of sales of shops, given that the median sale of shops is ` 2,400. Sales in hundred ` : 0–10 10–20 20–30 30–40 40–50 No. of shops : 5 25 ? 18 70 2. An incomplete frequency distribution is given as follows: Variable Frequency 10-20 12 20-30 30 30-40 ? 40-50 65 50-60 ? 60-70 25 70-80 19 230 Total CU IDOL SELF LEARNING MATERIAL (SLM)

74 Quantitative Techniques for Managers You are given that Median value is 46. (a) Using median formula, fill up the missing frequencies. (b) Calculate the Arithmetic Mean of the complete table. 3.13 Unit End Questions (MCQ and Descriptive) A. Descriptive Types Questions 1. What are various means to identify the nature of data distribution? 2. Distinguish Median and Mode for a group of data. 3. How do you use the data for calculation of various central tendency measures? 4. Calculate the mean of the following frequency distribution relating to the marks secured by students in statistics. Marks No. of Students Marks No. of Students 0–5 1 40–45 20 5–10 6 45–50 25 10–15 8 50–55 12 15–20 7 55–60 7 20–25 11 60–65 6 25–30 10 65–70 5 30–35 10 70–75 4 35–40 17 75–80 1 5. Following are the marks (out of 100) obtained by students in a test. 70, 55, 51, 42, 57, 45, 60, 47, 63, 53, 33, 65, 39, 82, 55, 64, 50, 25, 65, 75, 30, 20, 58, 52, 36, 45, 42, 35, 40, 61,53, 59, 49, 41, 15, 52, 46, 42, 45, 39, 55, 65, 45, 63, 54, 48, 64, 35, 26, 18. (i) Make a frequency distribution taking a class-interval of 10 marks (Take the first interval as 0-10) (ii) Draw a histogram and a frequency polygon from the frequency distribution. (iii) Find out the average marks of the students. CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - I 75 6. The mean weight of a student in a group of students is 119 lbs. The individual weights of 5 of them are 115, 109, 129, 117 and 114 lbs. What is the weight of the sixth student? 7. In thefollowing grouped data. x are the mid-values of the class intervals and c is a constant. If the arithmetic mean of the original distribution is 35.84, find its class interval. x-c : –21 –14 –7 0 7 14 21 Total f: 2 12 19 29 20 13 5 100 8. From the following data of income distribution, calculate the arithmetic mean. It is given that (i) the total income of the persons in the highest group is ` 435 and (ii) none is earning less than ` 20. Income (`) : Below Below Below Below Below Below Over No. of persons : 30 40 50 60 70 80 80 16 36 61 76 87 96 5 9. For a certain frequency table, which has only been partly reproduced here, the mean was found to be 1.46. Calculate the missing frequencies. Number of accidents : 01 2345 Frequency (No. of days) : 46 ? ? 25 10 5 Total 200 10. The average monthly wage of all workers in a factory is ` 444. If the average wages paid to male and female workers are ` 480 and ` 360 respectively, find the percentage of male and female workers employed by the factory. 11. Twelve persons gambled on a certain night. Seven of them lost at an average rate of ` 10.50 while the remaining five gained at an average of ` 13.00. Is the information given above correct? If not why? 12. For two frequency distributions given below, the mean calculated from the first was 25.4 and that from the second was 32.5. Find the value of x and y. Class Distribution Frequency I Distribution Frequency II 10-20 20 4 20-30 15 8 30-40 10 4 CU IDOL SELF LEARNING MATERIAL (SLM)

76 Quantitative Techniques for Managers 40-50 x 2x 50-60 y y 13. The following data give the number of employees and the total wages paid in the three departments of a manufacturing unit. Departments No. of employees Total wages (`) A 432 1,08,864 B 517 1,62,855 C 51 25,704 If a bonus amounting to ` 63 is given to each employee, what is the average percentage increase per employee for each department and for the total? 14. Calculate simple and weighted arithmetic averages from the following data and comment on them. Designation Monthly Salary (`) Strength of the Cadre Class I officers 1,500 10 Class II officers 800 20 Subordinate Staff 500 70 Clerical Staff 250 100 Lower Staff 100 150 15. The frequency distribution of weight in grams of mangoes of a given variety is given below. Calculate the arithmetic mean and the median. Weight in grams : 410–419 420–429 430–439 440–449 450–459 460–469 470–479 Number of mangoes : 14 20 42 54 45 18 7 16. Find mean and median from the data given below : Marks obtained : 0-10 10-20 20-30 30-40 40-50 50-60 No. of students : 12 18 27 20 17 6 CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - I 77 17. Calculate the median annual income of a group of employees from the data given below: Annual Income in ` Number of Employees Under 2,000 15 2,000–2,999 32 3,000–3,999 65 4,000–4,999 79 5,000–5,999 90 6,000–6,999 57 7,000–7,999 36 8,000–8,999 14 18. Calculate the mode for the following data : Annual wages No. of workers Annual wages No. of workers 150 Upto ` 1,200 12 Upto ` 1,600 220 300 Upto ` 1,300 30 Upto ` 1,700 330 Upto ` 1,400 80 Upto ` 1,800 Upto ` 1,500 100 Upto ` 1,900 19. Find out the Mean, Median and Mode for the following series : Size (below) : 5 10 15 20 25 30 35 Frequency : 1 3 13 17 27 36 38 20. Obtain the Median and Mode for the following table No. of days absent No. of students No. of days absent No. of students Less than 5 29 Less than 25 634 Less than 10 239 Less than 30 644 Less than 15 469 Less than 35 650 Less than 20 584 Less than 40 655 CU IDOL SELF LEARNING MATERIAL (SLM)

78 Quantitative Techniques for Managers 21. Calculate the arithmetic mean and the median of the frequency distribution given below. Hence calculate the mode using the empirical relation among the three. Class limits Frequency Class limits Frequency 130–134 5 150–154 17 135–139 15 155–159 10 140–144 28 160–164 1 145–149 24 Total 100 22. Calculate Mode from the following data : Monthly wage No. of Monthly wage No. of (in `) workers (in `) workers 200-250 4 400–450 33 250-300 6 450–500 17 300-350 20 500–550 8 350-400 12 550–600 2 Calculate Mode by graphic method also. 23. From an ordinary frequency table from the following cumulative distribution of marks obtained by 22 students and calculate (i) Arithmetic Mean, (ii) Median and (iii) Mode. Marks Number of students Below 10 3 Below 20 8 Below 30 17 Below 40 20 Below 50 22 CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - I 79 24. Calculate the Mode, Median and Arithmetic average from the following data : Class f Class f 0–2 8 25–30 45 2–4 12 30–40 60 4–10 20 40–50 20 10–15 10 50–60 13 15–20 16 60–80 15 20–25 25 80–100 4 25. What are various means to identify the nature of data distribution? 26. Distinguish Median and Mode for a group of data. 27. How do you use the data for calculation of various central tendency measures? B. Multiple Choice/Objective Type Questions 1. Which measure of central tendency can be used for both numerical and categoriced variables? (a) Quartiles (b) Median (c) Mode (d) Mean 2. Which of the following statistics always corresponds to 75th percentile in a distribution? (a) Mean (b) Median (c) Mode (d) Third quartile 3. Which of the following statements about the median is not true? (a) It is measure of central tendency (b) It is equal to second quartile (c) The median is more affected by extreme value than the mean. (d) It is equal to the mode in bell-shaped, symmetrical distribution. CU IDOL SELF LEARNING MATERIAL (SLM)

80 Quantitative Techniques for Managers 4. The quartiles, median, percentiles and deciles are measures of central tendency classified as __________. (a) Paired average (b) Deviation average (c) Central average (d) Positioned averages 5. Change of origin and scale is used for calculation of the __________. (a) Arithmatic mean (b) Geometric mean (c) Weighted mean (d) Lower and upper quartile Answers 1. (c), 2. (d), 3. (c), 4. (d), 5. (a) 3.14 References References of this unit have been given at the end of the book. CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - II 81 UNIT 4 DESCRIPTIVE SUMMARY MEASURES - II Structure: 4.0 Learning Objectives 4.1 Introduction 4.2 Measures of Dispersion 4.3 Solved Problems 4.4 Summary 4.5 Key Words/Abbreviations 4.6 LearningActivity 4.7 Unit End Questions (MCQ and Descriptive) 4.8 References 4.0 Learning Objectives After studying this unit, you will be able to:  Define the concept of spread of date.  Discuss the various measures of dispersion such as Range and standard deviation.  Utilisation of above concepts through problem solving.  Capability assessment through self-assessment problems. 4.1 Introduction The measure of central tendency i.e. averages are indicative of the concentration of the observed data about the central part of the distribution system, but they do not reveal the spread of the data on either side of the central pivot. Hence inspite of having the same average value, the observations may CU IDOL SELF LEARNING MATERIAL (SLM)

82 Quantitative Techniques for Managers vary widely and no concrete idea comes out of averages. The concepts of averages have their own utility, but their limitations too. The idea of spread about the central value of the observation is called “Dispersion”. Whereas a small dispersion indicates high uniformity of the observations, the large dispersion would mean less uniformity. In any business function, the concept of dispersion is very useful so that variability of performance can be used to monitor and control this variability bringing in better product/service/ sales etc. To differentiate the concept from the averages, let us consider the following set of observations. Set A 20 20 25 25 30 30 35 35 Total 220 Mean 27.5 Set B 15 20 25 30 35 30 25 40 Total 220 Mean 27.5 Set C 3 6 18 36 72 36 24 25 Total 220 Mean 27.5 It can be seen that in all the sets, the total sum and the average value is same, but the variation in set A is from 20 to 35. in set B, from 15 to 40. Whereas in set C, it is from 3 to 72. If these are the figures for sales, you can well understand the plight of the sales/marketing manager and that of the production manager to cope up with the types of demand. This variation of observations is called “Dispersion or spread”. This information has been represented in Figure 4.1 : Fig. 4.1: Spreadof observations From the values of the observations given above for set A, B and C, it is observed that the data is closely centered around the mean value of 27.5, whereas for set B, the data is more spread out from the mean. For set C. the data is well spread out from 3 to 72. The concept of wide dispersion CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - II 83 can be made use of in high risk business situations, where spread of data indicates greater level of risk due to earnings varying widely. 4.2 Measures of Dispersion The following measures of dispersion are commonly used 1. Range 2. Quartile Deviation 3. Mean Derviation 4. Standard Deviation and Variance 1. Range The difference between the highest and the lowest observation or the value of the variable is called the “Range”.  Range (R) = Xmax – Xmin Where X is the variable. Thus taking the values for set A, B and C in para 12.1 the three ranges can be writen as follows. For set A ; R = Xmax – Xmin = 35 – 20 = 15 For set B ; R = Xmax – Xmin = 40 – 15 = 25 For set C ; R = Xmax – Xmin = 72 – 3 = 69 The limitation of this measure is that it takes into account only the extreme values of the variable and ignores all other data. As can be seen that there can be a wide variation of its magnitude from one set of observation to the other. Inspite of this great limitation, this measure of dispersion is widely used in Industrial Quality Control for construction of quality control charts. It is also widely used in risk analysis for an investment proposal. The expression for coefficient of range = Xmax  Xmin Xmax  Xmin CU IDOL SELF LEARNING MATERIAL (SLM)

84 Quantitative Techniques for Managers 2. Quartile Deviation The Quartile Deviation (QD) is the difference of the third and first quartile divided by two. Thus QD = Q3 – Q1 2 The calculation of quaritiles can be done as follows : F Ih H KQ1= l + f N F 4 FH KIand h 3N F Q3 = l + f 4 where N = Total cumulative frequency F = Cumulative frequency upto the lower limit of quartile l = Lower limit of the quartile class f = Frequency of the quartile class h = Width of the class interval If Q3 = 35, Q1 = 15 Then QD = 35 – 15 = 10 2 This indicates that the dispersion on either side of mean is 10, when measured in terms of the quartile deviation. It is a better measure of dispersion than the range. There are different names for these ranges, commonly known as interfractile range. If half the data is below or equal to this value, it is called 0.5 fractile or Median. A fractile is the cut off point for a certain fraction of a sample. If your distribution is known, then the fractile is just the cut-off point where the distribution reaches a certain probability. In a normal distribution, a fractile of 0 . 5 is equal to the mean,  CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - II 85 If 25 per cent of the data lies at or below the given value, it is called 0.25 fractile or first quartile. Thus these values are like percentages. Interfractile range is, therefore, a measure of the spread between two fracticles of the frequency distribution. This is also called Inter-quartile range, when it is spread between the first and the third quartile or 0.25 fractile and 0.75 fractile. Hence inter quartile range = Q3 – Q1. The interquartile range (IQR) in particular is used to describe the dispersion of the data. The interquartile range (IQR) is defined as the range between the first and the third quartile. Please note that the IQR contains exactly 50 % of the data within the distribution. Based on the concept of percentage, the fractiles are named as percentiles when the data is divided into 100 equal parts. Thus decile divides the data into 10 equal parts. CU IDOL SELF LEARNING MATERIAL (SLM)

86 Quantitative Techniques for Managers 3. Mean Deviation The Mean deviation is defined as the arithmetic average of the deviations, when the deviations are taken from the average (mean, median or mode), taking all deviations as positive. Thus, when mean is used as the average, it is called Mean deviation about the mean; while mean median is used as the average, it is termed as the Mean Deviation about the Median. o t xi – x Thus Mean Deviation (MD) = n Where x is the mean of the deviation. In this case, we take the deviation of all observation values xi’s and the deviation of the mean is calculated from all the observations and totalled up as (xi – x) ( xi – x) MD =  f Whereas in a grouped frequency distribution, the frequency of occurence of each value of (xi – x ) is available and to obtain the mean of this deviation, we have to multiply by the frequency fi also and then summed up over the entire range for mean calculation. If we consider a grouped frequency distribution.  fi[xi – x] MD = fi Since the Mean Deviation depends on all the values of the observations, it is treated as a better measure of the dispersion comparing the Range and Quartile Deviation. 4. Standard Deviation and Variance In case of Mean Deviation, value of (xi – x ) can be negative. To avoid this situation, we use standard deviation as the measure. The best way is to square the expression (xi – x–). This will always lead to a positive value. The square of the standard deviation (i.e. the deviation about the mean) denoted by s2 (s; small sigma, a Greek alphabet, first suggested by Karl Pearson) is called the Variance. Let us take the following data x : 5, 10, 14, 20, 26 CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - II 87  xi The average of this data = x = n 5  10  14  20  26 = 5 75 = 5 = 15 Taking the deviations about this average or mean i.e., 15, we get deviation as (5-15), (10-15), (14-15), (20-15) and (26-15) i.e., –10, –5, –1, 5, 11 The square of these deviations are 100, 25, 1, 25, 121. The variance is defined as the average of the square of the deviations. Variance = 2 100  25  1  25  121 =5 272 = = 54.4 5 b gFor the ungrouped data 2 = x1  x 2 n and for the grouped data  fi (xi – x)2 2 = fi Thus, we can write the formula for variance (2, sigma squared) as follows 2 = (x – )2 N Where 2 = population variance CU IDOL SELF LEARNING MATERIAL (SLM)

88 Quantitative Techniques for Managers x = value of the observations  = population mean N = total number of observations or items in the population  = sum of all the squared values. This can also be written as follows  x2 2 = N – 2 We normally use x for the mean of a sample out of the given population and  for the population mean. Similarly n is used as the size of the sample and N as the size of the population. Hence standard deviation is now defined as the positive square root of the variance or the square root of the means of squared deviations of the given observations from thier arithmetic means. (x – x)2 Standard Deviation = n Where x = FHG x JIK n In terms of population mean, the population standard deviation can be written as  = 2 (x – )2 = N =  x2 – 2 N Standard Score The standard deviation is found useful in specifying the deviation of the individual item in a distribution from the mean of the distribution, Standard score, is a measure of the standard deviations for a particular observation, Standard score, therefore, is the number of standard deviations a particular observation lies below or above the mean and can be written as Population standard score = x –   CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - II 89 Use of Standard Deviation (Chebyshev’s Theorem) A Russian Mathematician P.L. Chebyshev devised a theorem as “No matter what the shape of the distribution, at least 75 per cent of the value will fall within ±2 standard deviations from the mean of the distribution and at least 89 per cent of the values will be within ±3 standard deviations from the mean, It can be explained from a symmetrical bell-shaped curve as follows. Fig. 4.2. Observations around the mean Standard deviation for samples The relationship of standard deviation for a sample undergoes a modification as follows: (x – x)2 s2 = n –1  x2 nx 2 =– (n – 1) (n – 1) and (x – x)2 Where s= n –1 s = sample standard deviation s2 = sample variance x = value of each of the observations CU IDOL SELF LEARNING MATERIAL (SLM)

90 Quantitative Techniques for Managers x = mean of the sample n = number of observations in the sample or sample size Coefficient of Variation The Standard Deviation is the measure of the dispersion of observations about their arithmetic mean and the unit of measurement of the standard deviation is the same as that of the arithmatic mean or of the variance itself. Hence the dispersion actually depends upon the measurement of the variable. In order to have the comparison of the dispersion about the arithmetic mean, we define it as the “Coefficient of Variation”, which is the ratio of standard deviation to the arithmetic mean. It is calculated as Coefficient of Variation =   100%   = standard deviation  = mean 4.3 Solved Problems Problem 1 Calculate the range and coefficient of range for the following data. Months Sales (in 000 units) Months Sales (in 000 units) April 145 October 180 May 151 November 180 June 161 December 182 July 164 January 184 August 165 February 185 September 170 March 190 Solution: Largest sales (Xmax) = 1,90,000 units Smallest sales (Xmin) = 1,45,000 units CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - II 91  Range = Xmax – Xmin = 1,90,000 – 1,45,000 = 45,000 units Coefficient of range = X max – X min X max  X min 1,90,000 – 1,45,000 = 1,90,000  1,45,000 45,000 = = 0.134 3,35,000 Problem 2 Evaluate an appropriate measure of dispersion for the following data Income (in `) : less than 5050-70 70-90 90-110 110-130 130-150 above 150 Number of persons : 54 100 140 300 230 125 51 Solution : The table below indicates the cf Income (in `) Number of persons ( f ) Less than (cf ) Less than 50 54 54 50-70 100 154 70-90 140 294 90-110 300 594 110-130 230 824 130-150 125 949 more than 150 51 1,000 Here N =  f = 1,000 N 1,000  4 = 4 = 250; CU IDOL SELF LEARNING MATERIAL (SLM)

92 Quantitative Techniques for Managers 3N 3  1,000 = = 750 44 Class interval for cf just greater than 250 i.e., 824 is 110-130 20  Q1 = 70 + 140 (250 – 154) = 83.714 20 Q3 = 110 + 230 (750 – 594) = 123.565  QD = Q3 – Q1 123.565 – 83.714 = = 9.925 22 Problem 3 Find the average or mean deviation from the median for the following distribution. Mark less than : 80 70 60 50 40 30 20 10 No. of students : 100 90 80 60 32 20 13 5 Solution : Converting the above data into ordinary frequency distribution. Marks cf Frequency Mid value |x–Md| f |x–Md| (f ) of class (x) 207.15 251.44 00-10 05 05 05 41.43 150.01 137.16 10-20 13 08 15 31.43 040.04 20-30 13 07 25 21.43 071.4 135.7 30-40 32 12 35 11.43 235.7 = 1228.6 40-50 60 28 45 01.43 50-60 80 20 55 03.57 60-70 90 10 65 13.57 70-80 100 10 75 23.57 F IhN –c = 40 + 10 (50 – 32) GH KJMedian = l + f2 28 CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - II 93 = 46.43 1  Mean deviation about median = N f |x – Md| 1228.6 = 100 = 12.29 Problem 4 Calculate the mean and standard deviation for the following data : Value : 90-99 80-89 70-79 60-69 50-59 40-49 30-39 Frequency : 2 12 22 20 14 4 1 Solution : Mid Value (x) Frequency (f) x – 64.5 fd fd2 d = 10 Class 94.5 2 6 18 84.5 12 3 24 48 90-99 74.5 22 2 22 22 80-89 64.5 20 1 0 0 70-79 54.5 14 0 –14 14 60-69 44.5 4 –1 –8 16 50-59 34.5 1 –2 –3 9 40-49 –3  fd = 27  fd2 = 127 30-39  f = N = 75 h  fd Mean = A + N 10  27 = 64.5 + 75 = 68.1 FG JISD = h  fd 2 –  fd 2 H KN N CU IDOL SELF LEARNING MATERIAL (SLM)

94 Quantitative Techniques for Managers GF JI= 10 × 127 – 27 2 = 12.5 H K75 75 Problem 5 The arithmetic mean and the standard deviation of a set of 9 items are 43 and 5 respectively. If an item of a value 63 is added to the set, find the mean and standard deviation of 10 items given. Solution : Given here n = 9, x = 43, s = 5 Since x Also x =n x = n x = 9 × 43 = 387 F I x2 x 2 G J2 = H Kn n – d i x2 2 = n –x x2 = n(s2 + x 2) = 9(25 + 432) = 16866 If new item 63 is added, n becomes 10 New x = x9 + x10 = 387 + 63 = 450  x 450 New mean = n = 10 = 45 New x2 = x2 + 632 CU IDOL SELF LEARNING MATERIAL (SLM)