Measures of dispersion

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MEASURES OF DISPERSION The measures of central tendency serve to locate the center of the distribution,but they do not reveal how the items are spread on either side of the center.This characteristics of a freq distn is referred to as dispersion.small dispersion indicates high uniformity of the items.For example:A marketing manager may be interested to know the avg sales of each product & the amount of fluctuations in the average sales because he wants to know the avg level of performance & also the variability of such performance. OBJECTS OF MEASURING DISPERSION To test the reliability of an average : If the variation is small ,that is, extent of dispersion is less ,then it indicates high uniformity & the avg is more reliable. To compare the variability of two series : Higher the degree of variability,lesser is the consistency or uniformity in the values of the series . To identify the causes of variability& to control it :Measures of dispersion help to identify the causes of variation & take remedial measures .In industrial production efficient operation requires control of quality variation,the causes of which are sought through inspection & quality control programmes To serve as a basis for further statistical analysis : Measures of variation.are necessary for studying correlation ,regression hypothesis testing ,forecasting ,quality control ,and so on.

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It was developed by karl pearson & is most commonly used measure of relative dispersion .It is used to compare the consistency variability or uniformity of two or more groups.The one for which C.V is less is said to be more consistent , more uniform ,less variable,more stable and more homogeneous. It is given by----- Inter- relationship between measures of dispersion 3Q.D=2S.D 5M.D=6Q.D 5M.D=4S.D 4S.D=5M,D=6Q.D PROBLEMS Calculate Range ,Q.D , M.D & .S.D. Also their relative measures X :48 ,52 , 60 ,45 ,65 , 72 , 80 , 50 X : 4 , 6 , 8 , 10 , 12 , 14 , 16 freq :2 , 1 , 3 , 6 , 4 , 3 , 1

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c) C.I 0-10 10-20 20-30 30-40 40-50 50-60 60-70 F 8 12 17 14 9 7 4 You are given the data pertaining to kilowatt hours of electricity consumed by 100 persons in a city Consumption 0-10 0-20 20-30 30-40 40-50 No. of users 6 25 36 20 13 Calculate the range within which the middle 50% of the consumers fall . organization A organization B No. of workers 550 650 Avg monthly wages 5000 4500 Variance 900 1600 Which org pays larger wages &in which org is greatervariability.

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ABSOLUTE & RELATIVE MEASURES OF DISPERSION An absolute measure of dispersion is one which is expressed in terms of statistical units that are used in the data. For ex : kg , Rs ,pounds . Relative measure of dispersion is one which is independent of the units of measurements .It is also called coefficient of dispersion .They are used for making comparisons between two or more series &are usually expressed in percentage form . Different measures of dispersion are : Range . Quartile deviation . Mean deviation . Standard deviation . RANGE : It is defined as the difference between the largest & the smallest value in the distribution .It is given by --------

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Its relative measure is called coefficient of range given by---------- QUARTILE DEVIATION It is defined as half of inter quartile range. It is given by-------.Its relative measure is called coefficient of quartile deviation given by --------= MEAN DEVIATION :A.M of the absolute deviations taken from an average .I t is given by ----------. Its relative measure is called coefficient of M.D given by-----------. STANDARD DEVIATION : It is defined as root mean square deviation taken from A.M.It is most important & widely used measure of dispersion It satisfies most of the properties of a good measure of dispersion . It is root of variance. It is given by--------. Greater the S.D greater is the variability of the values from A.M COEFFICIENT OF VARIATION : The corresponding relative measure of S.D is coefficient of variation .

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A work -standards expert observes the amount of time(mins) required to prepare a sample of 10 business letters in the office with observations in ascending order :5 ,5 , 5 , 7 ,9 14 ,15 ,15 ,16 ,18 a)Determine the Range & middle 70% range for the sample b) If the sample mean of the data is10.9, then calculate the mean absolute deviation & variance . The following is the % of sales met by each three persons X : 88 68 89 92 103 Y : 76 88 90 86 79 Z :104 88 118 88 123.which salesman is more consistent . Public transportation & the automobiles are the two options an employee can use to get to work each day.samples of time(in mins) recorded for each option are : Public transportation :29 29 32 37 33 25 29 32 41 34 Automobile :29 31 33 32 34 30 31 32 35 33

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a)compute sample mean & sample S.D for each option. b)which method of transportation should be preferred ? In two factories A & B ,the avg.weekly wages & S.D are as follows: Factory Avg. S.D. No. of workers A 460 50 100 B 490 40 80 Which factory pays higher amount as weekly wages ? Which factory shows greater variability in the distribution of wages ? What is the mean & S.D. of all the workers in two factories taken together ?

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The mean & S.D.of a set of 100 observations were found to be 40 &5 respectively .But by mistake a value 50 was taken in place of 40.Re-calculate the correct mean & S.D. A machine co.has a contract with one of his customers to supply machined pump gears.One requirement is that the diameter of its gears be within specific limits.The following data is of diameter (in inches) of a sample of 20 gears: 4.01 4.00 4.02 4.03 4.00 3.98 3.99 3.99 4.01 4.02 3.99 3.98 3.97 4.00 4.02 4.01 4.02 4.00 4.01 3.99 What can be said about the 95% of the gears.

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Thanks Tinku Kumar MBA