{"id":67,"date":"2018-10-29T10:27:09","date_gmt":"2018-10-29T10:27:09","guid":{"rendered":"http:\/\/mgmtp15.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=67"},"modified":"2018-10-29T10:44:51","modified_gmt":"2018-10-29T10:44:51","slug":"measures-of-dispersion-mean-absolute-deviation-standard-deviation-variance-coefficient-of-variation","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/mgmtp15\/chapter\/measures-of-dispersion-mean-absolute-deviation-standard-deviation-variance-coefficient-of-variation\/","title":{"rendered":"Measures of Dispersion: Mean Absolute Deviation, Standard Deviation, Variance, Coefficient of Variation"},"content":{"raw":"
\r\n\r\n \r\n\r\nMeasures of Dispersion: Mean Absolute Deviation, Standard Deviation, Variance, Coefficient of\u00a0<\/strong>Variation<\/strong>\r\n\r\n \r\n\r\nLearning Objectives:<\/strong>\r\n\r\n \r\n\r\nAfter the completion of this module the student will understand:\r\n\r\n \r\n\r\n\u00d8\u00a0 Range\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 Mean Absolute Deviation\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 Computation of Mean Deviation\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 Characteristics of mean deviation\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 Uses of mean deviation\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 Standard Deviation\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 Computation of Standard Deviation\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 Characteristics of Standard Deviation\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 Uses of Standard Deviation\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 Quartile deviation or Semi Inter Quartile range\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 Variance\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 Relative measures of dispersion\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 Coefficient of dispersion\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 Coefficient of variation\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 Standard error\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 Expression for the standard error of mean\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 Probable error\r\n\r\n \r\n\r\n \r\n\r\n1.<\/strong>\u00a0\u00a0\u00a0 Introduction<\/strong>\r\n\r\n \r\n

Any measure of central tendency or average has its own limitations and gives us an idea only about that central value of the set of observations around which all the observations have a tendency to lie, but it fails to give any idea about the way in which they are distributed. There can be a number of series each of which has the same mean but differs from others in respect of the pattern in which the observations are distributed. To follow this point Consider the following series.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Series A<\/td>\r\n9<\/td>\r\n9<\/td>\r\n9<\/td>\r\n9<\/td>\r\n9<\/td>\r\n9<\/td>\r\n9<\/td>\r\n<\/tr>\r\n
Series B<\/td>\r\n6<\/td>\r\n7<\/td>\r\n8<\/td>\r\n9<\/td>\r\n10<\/td>\r\n11<\/td>\r\n12<\/td>\r\n<\/tr>\r\n
Series C<\/td>\r\n1<\/td>\r\n2<\/td>\r\n4<\/td>\r\n5<\/td>\r\n11<\/td>\r\n13<\/td>\r\n27<\/td>\r\n<\/tr>\r\n
Series D<\/td>\r\n3<\/td>\r\n15<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n

In the above series, we observe that arithmetic mean of every series is 9, but the pattern in which the observations are distributed is different in different series. In series A the mean is 9 and all the observations are same. In series B also, the mean is 9 and the observations are scattered ranging from 6 to 12 but not very much scattered. In series C,\u00a0the mean is the same value 9 but the observations are too much scattered ranging from 1 to 27. In series D there are only two observations the mean of which is 9.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

From the above example it is quite obvious that for studying a series, a study of the extent of scattering of the observations of dispersion is also essential along with the study of the central tendency in order to throw more light on the nature of the series. The following are the different measures of dispersion which are in common use.<\/p>\r\n \r\n\r\n2.\u00a0\u00a0 Range<\/strong>\r\n\r\n \r\n\r\n2.1. Definition:<\/strong>\r\n\r\n \r\n

The range is the simplest measure of dispersion. It is the difference between the highest and lowest terms of a series of observations.<\/p>\r\n \r\n\r\n2.2.Computation of Range:<\/strong>\r\n\r\n \r\n\r\nWhere, XH = Highest variate value\r\n\r\n \r\n\r\nand XL = Lowest variate value\r\n\r\n \r\n\r\n2.3.Merits and demerits of range<\/strong>\r\n\r\n \r\n\r\n(i)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Its value usually increases with the increase in the size of the sample.\r\n\r\n \r\n\r\n(ii)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 It is usually unstable in repeated sampling experiments of the same size and large ones.\r\n\r\n \r\n\r\n(iii)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 It is a very rough measure of dispersion and is entirely unsuitable for precise and accurate studies.\r\n\r\n \r\n\r\n(iv)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 The only merit possessed by \u2018Range\u2019 are that it is (i) simple (ii) easy to understand and (iii) quickly calculated. It is often used in certain industrial work.\r\n\r\n \r\n\r\n3.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Mean Deviation<\/strong>\r\n\r\n\u00a0<\/strong>\r\n\r\n3.1.\u00a0 <\/strong>Definition:<\/strong>\r\n\r\n\u00a0<\/strong>\r\n

If the deviations of all the observations from their mean are calculated, their algebraic sum will be zero. When this sum is always zero, it is impossible to get the average of these deviations. In order to overcome this difficulty, these deviations are added irrespective of plus or minus sign and then the average is calculated. The deviations without any plus or minus sign are known as absolute deviations. The mean of these absolute deviations is called the mean deviation. If the deviations are calculated from the mean, the measure of dispersion is called mean deviation about the mean. As a matter of fact mean deviation can be calculated from any average, and for that, the absolute deviations from that average will be calculated.<\/p>\r\n\r\n<\/div>\r\n \r\n\r\n3.2.Computation of mean deviation:<\/strong>\r\n

\"\"Mean deviation about other averages:<\/span><\/div>\r\n
M.D. about A= 1 \u2211 \u20d3 (\u00a0\u00a0 \u2212\u00a0\u00a0 )\u20d3<\/span><\/div>\r\n
M.D. about Md.=1 \u2211 \u20d3 (\u00a0\u00a0 \u2212\u00a0\u00a0 . )\u20d3<\/span><\/div>\r\n
\r\n\r\nM.D. about Mo. = 1 \u2211 \u20d3 (\u00a0\u00a0 \u2212\u00a0\u00a0 . )\u20d3<\/span>\r\n\r\n3.3.Characteristics of mean deviation:<\/strong>\r\n\r\n \r\n\r\n(i)\u00a0 A notable characteristic of mean deviation is that it is the least when calculated about the median.\r\n\r\n \r\n

(ii) Standard deviation is not less than the mean deviation in a discrete series i.e. it is either equal to or greater than the M.D. about mean.<\/p>\r\n \r\n

(iii) When an average other than the A.M. is calculated as a measure of central tendency, M.D. about that average is the only suitable measure of dispersion.<\/p>\r\n \r\n\r\n4.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Standard Deviation<\/strong>\r\n\r\n\u00a0<\/strong>\r\n\r\n4.1.Definition:<\/strong>\r\n\r\n\u00a0<\/strong>\r\n

Calculation of standard deviation is also based on the deviations from the arithmetic mean. In thecase of mean deviation the difficulty, that the sum of the deviations from the arithmetic mean is always zero, is solved by taking these deviations irrespective of plus or minus signs. But here, that the difficulty is solved by squaring them and taking the square root of their average. It is thus defined by thefollowingexpression.<\/p>\r\n \r\n

Standard Deviation (S.D.) = \u221a\u2211(\u00a0 \u2212\u00a0 )2 \u2026\u2026\u2026\u2026. (1)<\/span><\/p>\r\n

Where,\u00a0\u00a0 X= An observation or variate value<\/span><\/p>\r\n

\u00b5\u00a0 = Arithmetic mean of the population N= Number of given observations.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

According to the expression given in (1), thepopulationmean\u00b5 is required for finding the standard deviation (S.D.) of a given set of observations. Generally, \u00b5 is not known. Therefore it is replaced by X\u035e, which is the mean of the given set of observations, and then the S.D. of the given data is given by<\/p>\r\nStandard Deviation (S.D.) = \u221a\u2211(\u00a0 \u2212\u00a0 )2\u2026\u2026\u2026\u2026\u2026(2)\r\n\r\n \r\n\r\n(X \u2212 X\u035e)2 = deviation from mean\u00a0s-10<\/strong>\r\n\r\n \r\n

Here, it should be noted that formula (2) gives the S.D. of the given set of data which itself is assumed to be the population with \u00b5=X\u035e. Therefore we shall this S.D. as the \u2018population S.D.\u2019 Thus,<\/p>\r\n\r\n<\/div>\r\n

<\/div>\r\n
population S.D. (=\u03c3) = \u221a \u2211( \u2212 )2 \u2026\u2026\u2026\u2026\u2026 (3)<\/div>\r\n
<\/div>\r\n
\r\n\r\nIn case of frequency distribution-\r\n\r\nPopulation S.D. (=\u03c3) = \u221a \u2211 ( \u2212 )2 \u2026\u2026\u2026\u2026\u2026 (4)\r\n\r\nSample S.D.: <\/strong>In case, when the given set of data is not a population but is a sample drawn from\r\n\r\n\u035e\r\n

a large population, the population mean \u00b5 is not known. Therefore, in its place, we use X which is the estimate of\u00b5 obtained from the sample observations. The result is that we cannot calculate the population S.D. (\u03c3), but, in its place, we calculate its estimate (S). We represent the estimates of population parameters, \u00b5 and \u03c3, in the following way:<\/p>\r\n \r\n\r\nX\u035e= Estimate of (\u00b5)\r\n\r\n \r\n\r\nS= Estimate of (\u03c3)\r\n\r\n<\/div>\r\nPThe best estimates (S) of the population S.D. (\u03c3) is given by\r\n\r\nS (sample S.D.) = \u221a \u2211 ( \u2212 )2 \u2026\u2026\u2026\u2026\u2026 (5)\r\n

\r\n\r\ns-11<\/strong>\r\n\r\n \r\n\r\n4.2.Computation of Standard Deviation<\/strong>\r\n\r\n \r\n

For computing S.D., in every case, we have to calculate the arithmetic mean, which increases the labor of calculation work. Therefore, to avoid it, theshort cut method should be used in which any value of the variate is chosen as the arbitrary mean and then the standard deviation is calculated by the following process:<\/p>\r\n \r\n\r\nSuppose, A is the arbitrary mean and d is the deviation of the variate value from A. i.e. d = X-A\r\n\r\nwe have, \u2211\u00a0 (\u00a0\u00a0 \u2212\u00a0\u00a0 )2 =\u2211\u00a0 \u00a02 \u2212 (\u2211 \u00a0\u00a0)2\r\n

Therefore, for this, we require the columns of d, fd, and fd2. In the column of d we shall find a factor equal to the width of the class interval \u201ci\u201d common to all the figures in that column. After taking out this factor as common, the columns now will be of d\/I, fd\/I and fd2\/i2. With the help of these symbols, the values of \u2211\u00a0 (\u00a0\u00a0 \u2212 X\u035e)2 and S.D. will be calculated as given bellow.<\/p>\r\n\u2211\u00a0 \u00a0 \u00a0=\u00a0 \u00d7 \u2211\r\n\r\n \r\n\r\n\u2211 2<\/sup> = 2 \u00d7 \u2211 2<\/sup>\r\n\u2211 ( \u2212 )2<\/sup> = 2 \u00d7 {\u2211 2<\/sup> \u2212 (\u2211 )2<\/sup>} . . = \u00d7 [\u221a 1 {\u2211 2<\/sup> \u2212 (\u2211 )2<\/sup> }]\r\n\r\n \r\n\r\n \r\n\r\ns-12<\/strong>\r\n\r\n \r\n\r\nIf we use the symbol D for d\/I, the above expressions will be written as\r\n\r\n \r\n\r\n\u2211\u00a0\u00a0\u00a0\u00a0\u00a0 =\u00a0 \u00d7 \u2211<\/span>\r\n\r\n \r\n\r\n\u2211 <\/span>2<\/sup> = 2 \u00d7 \u2211<\/span> 2<\/sup>\r\n\r\n \r\n\r\n\u2211 ( \u2212 )<\/span>2<\/sup> = 2 \u00d7 {\u2211<\/span> 2<\/sup> \u2212 (\u2211 )<\/span>2<\/sup>} . . = \u00d7 [\u221a 1 {\u2211 <\/span>2<\/sup> \u2212 (\u2211 )<\/span>2<\/sup> }]<\/span>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\ns-13<\/strong>\r\n\r\n \r\n\r\nExample:<\/strong>\r\n\r\n \r\n\r\nCalculation of S.D.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Class<\/td>\r\nFrequency\u00a0f<\/td>\r\nMid values\u00a0X<\/td>\r\nd= X-A<\/td>\r\nd\/I or D<\/td>\r\nfD<\/td>\r\nfD2<\/td>\r\n<\/tr>\r\n
0-10<\/td>\r\n1<\/td>\r\n5<\/td>\r\n-20<\/td>\r\n-2<\/td>\r\n-2<\/td>\r\n4<\/td>\r\n<\/tr>\r\n
10-20<\/td>\r\n3<\/td>\r\n15<\/td>\r\n-10<\/td>\r\n-1<\/td>\r\n-3<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
20-30<\/td>\r\n5<\/td>\r\n25<\/td>\r\n0<\/td>\r\n0<\/td>\r\n0<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
30-40<\/td>\r\n4<\/td>\r\n35<\/td>\r\n+10<\/td>\r\n+1<\/td>\r\n+4<\/td>\r\n4<\/td>\r\n<\/tr>\r\n
40-50<\/td>\r\n2<\/td>\r\n45<\/td>\r\n+20<\/td>\r\n+2<\/td>\r\n+4<\/td>\r\n8<\/td>\r\n<\/tr>\r\n
Total<\/td>\r\n15<\/td>\r\n-<\/td>\r\n-<\/td>\r\n-<\/td>\r\n+3<\/td>\r\n19<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\n<\/div>\r\n
\r\n\r\n \r\n\r\n\u2211 ( \u2212 )2<\/sup> = 2 \u00d7 {\u2211 2<\/sup> \u2212 (\u2211 )2<\/sup> }\r\n\r\n \r\n\r\n= 102 \u00d7 [9 \u2212 32 ] 15\r\n\r\n= 100 \u00d7 [ 276 ] 15\r\n\r\n= 1840\r\n\r\n\u2211 ( \u2212 )2 \u035e Population S.D. =\u221a , (Here, \u00b5= X)\r\n\r\n \r\n\r\n\u03c3 = \u221a184015 = 11.07\r\n\r\n \r\n\r\nSample S.D. = \u221a\u2211 ( \u2212 )2 = \u221a1840\r\n\r\n\u22121 14\r\n\r\nOr S= 11.46\r\n\r\n \r\n\r\n4.3.Characteristics of Standard Deviation<\/strong>\r\n\r\n\u00d8\u00a0\u00a0 It is rigidly defined.\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 Its computation is based on all the observations.\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 If all the variate values are the same, S.D.=0\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 S.D. is least affected by fluctuations of sampling.\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 It is affected by the change of scale, but not affected by the change of origin.\r\n\r\n \r\n\r\n4.4.Uses of Standard Deviation<\/strong>\r\n\r\n \r\n

\u00d8\u00a0 It is used in computing different statistical quantities like, regression coefficient, correlation coefficient,<\/p>\r\netc. and in connection with business cycle analysis.\r\n\r\n \r\n\r\n\u00d8\u00a0\u00a0 It is also used in testing the reliability of certain statistical measures.\r\n\r\n \r\n\r\ns-15<\/strong>\r\n\r\n \r\n\r\n4.5.Quartile Deviation or Semi inter quartile range<\/strong>\r\n\r\n \r\n

The measure of dispersion is expressed in terms of quartiles and known as quartile deviation or semi inter quartile range. \u22122<\/p>\r\n \r\n\r\nWhere,Q1 = Lower quartile\r\n\r\n \r\n\r\nQ3= Upper Quartile\r\n\r\n \r\n

It is not a measure of the deviation from any particular average. For symmetrical and moderately skew distributions the quartile deviation is usually two-third of the standard deviation.<\/p>\r\nQ.D.=23 \u00d7 ( .\u00a0\u00a0 . )\r\n\r\n \r\n\r\n5.\u00a0<\/strong>Variance<\/strong>\r\n\r\n\u00a0<\/strong>\r\n

Variance is the square of the standard deviation.<\/p>\r\n\u00a0<\/strong>\r\n\r\nVariance = (S.D.)2\r\n\r\n \r\n

The variance of a population is generally represented by the symbol \u03c32and its unbiased estimate calculated from the sample, by the symbol S2.<\/p>\r\n \r\n

6.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Relative Measures of Dispersion<\/strong><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\u00a0<\/strong>\r\n

The measures of dispersion, which we studied so far, are the absolute measures of dispersion, and are represented it\u2019s the same units in which the observations are represented, e.g., gms., cm., meters, hectares, etc. When we have to compare the dispersions of two or more distributions, it will not be proper to compare their absolute measures of dispersions, because, the distributions or the data may differ from one another.<\/p>\r\n\u00a0<\/strong>\r\n\r\n(i)\u00a0\u00a0\u00a0\u00a0 With respect to their averages\r\n\r\n \r\n\r\n(ii)\u00a0\u00a0 With respect to their dispersions\r\n\r\n \r\n\r\n(iii)With respect to their averages and dispersions both (iv) With respect to their units\r\n\r\n \r\n

Therefore, they will not be comparable. Under such circumstances, their comparison is possible with the help of relative measures of dispersion.<\/p>\r\n \r\n\r\n6.1.Coefficient of Dispersion<\/strong>\r\n\r\n \r\n\r\nIt is computed by the following expression:\r\n\r\n \r\n\r\nCoefficient of Dispersion =\r\n\r\n \r\n\r\ns-17<\/strong>\r\n\r\n \r\n\r\n6.2.Coefficient of Variation (C.V.)<\/strong>\r\n\r\n \r\n\r\nThis is also a relative measure of dispersion. It is especially important on account of the widely used measures of central tendency and dispersion i.e., Arithmetic mean, and Standard Deviation. It is given by\r\n

C.V. = .. .. \u00d7 100<\/p>\r\n

It is expressed in percentage and used to compare the variability in the two or more series. Lesser value of thecoefficient of variation indicates more consistency.<\/p>\r\n \r\n\r\n7.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Standard Error 7.1.Definition<\/strong>\r\n\r\n\u00a0<\/strong>\r\n

The Standard deviation of the sampling distribution of a statistic (estimate) is known as the standard error of that statistic (estimate).<\/p>\r\n\u00a0<\/strong>\r\n

If we take all possible samples from the population of the same size and get a sampling distribution of means, it can be proved that the mean of this sampling distribution of means is the population mean and its standard deviation, the standard error of the mean.<\/p>\r\n\u00a0<\/strong>\r\n

As it is not possible to draw and study all possible samples, we have to get and we get the estimate of the standard error from a single sample. If S be the standard deviation of the sample of size N, the estimate of the standard error of mean is given by \u221a .<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\nS-18<\/strong>\r\n\r\n \r\n\r\n7.2.Expression for the standard error of mean<\/strong>\r\n\r\n \r\n\r\nLet there be a sample of N observations, X1, X2, X3\u2026\u2026\u2026XN which have been drawn at\r\n\r\n \r\n\r\nrandom from a population, the variance of which is \u03c32.\r\n\r\n\u035e\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1\r\n\r\nNow, Mean X\u00a0\u00a0 =\u00a0\u00a0\u00a0\u00a0\u00a0 (\u00a0 1 +\u00a0 2 +\u00a0 3 + \u22ef + Variance of mean\r\n\r\nV(X\u035e)= 12 {\u00a0 (\u00a0 1 +\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 2 +\u00a0 3 + \u22ef +\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 )}\r\n\r\n=\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 12 [\u00a0 (\u00a0 1) +\u00a0\u00a0 (\u00a0 2) + \u22ef +\u00a0\u00a0 (\u00a0\u00a0 )]\r\n\r\n \r\n\r\nSince, V(X1) =V(X2) = V(X3)=\u2026\u2026\u2026\u2026\u2026\u2026\u2026.= V(XN)= \u03c32\r\n\r\n\u035e\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 2\r\n\r\nV(X)=\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 2 =\r\n\r\nS.E. of X\u035e= \u221a\r\n\r\n \r\n\r\nBut since, in practice, \u03c3 is not known, it is replaced by its unbiased estimate S.\r\n\r\nS.E. of X\u035e= \u221a\r\n\r\n \r\n\r\n8.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Probable Error<\/strong>\r\n\r\n\u00a0<\/strong>\r\n\r\nThe quartile deviation of the sampling distribution of means is known as aProbable error\r\n\r\n \r\n\r\nand is 0.67449times the standard error.\r\n\r\n \r\n\r\nP.E. = 0.67449 (S.E.)\r\n\r\n \r\n\r\nThree times the probable error is roughly twice the standard error. This measure of dispersion has no particular advantage and moreover involves a troublesome factor 0.67449. This is why it has gone out of use and has given place to standard error.\r\n\r\n \r\n\r\n9.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Summary<\/strong>\r\n\r\n\u00a0<\/strong>\r\n

This module provides an overview to students to understand the techniques that are used to measure the extent of variation or the deviation (also called thedegree of variation) of each value in the dataset from a measure of central tendency, usually the mean or median. Such statistical techniques are called measures of dispersion (or variation). A small dispersion among values in the data set indicates that data are clustered that data are clustered closely around the mean. The mean is therefore considered representative of the data, i.e. mean is reliable average. Conversely, a large dispersion among values in the data set indicates that the mean is not reliable, i.e. it is not representative of data. The symmetrical distribution of values in two or more sets of data may have same variation but differ greatly in terms of A.M. On the other hand, two or more sets of data may have the same A.M. values but differ in variation.<\/p>\r\n\r\n<\/div>\r\n

    \r\n \t
  1. Self-check exercise with solution<\/strong><\/li>\r\n<\/ol>\r\n \r\n\r\nQ.1.<\/strong>Thefollowing data give the number of passengers traveling by airplane from one city to another in one week.\r\n\r\n \r\n\r\n115, 122, 129, 113, 119, 124, 132, 120, 110, 116<\/strong>\r\n\r\n \r\n

    Calculate the mean and standard deviation and determine the percentage of class that lie between (i) \u00b5\u00b1\u03c3 (ii) \u00b5\u00b12\u03c3 and (iii) \u00b5\u00b13\u03c3. What percentage of cases lie outside these limits.<\/p>\r\n \r\n\r\nCalculation of Mean and Standard Deviation\r\n\r\n \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    X<\/td>\r\nX- X\u035e<\/td>\r\n(X- X\u035e)\/2<\/td>\r\n<\/tr>\r\n
    115<\/td>\r\n-5<\/td>\r\n25<\/td>\r\n<\/tr>\r\n
    122<\/td>\r\n2<\/td>\r\n4<\/td>\r\n<\/tr>\r\n
    129<\/td>\r\n9<\/td>\r\n81<\/td>\r\n<\/tr>\r\n
    113<\/td>\r\n-7<\/td>\r\n49<\/td>\r\n<\/tr>\r\n
    119<\/td>\r\n-1<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
    124<\/td>\r\n4<\/td>\r\n16<\/td>\r\n<\/tr>\r\n
    132<\/td>\r\n12<\/td>\r\n144<\/td>\r\n<\/tr>\r\n
    120<\/td>\r\n0<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
    110<\/td>\r\n-10<\/td>\r\n100<\/td>\r\n<\/tr>\r\n
    116<\/td>\r\n-4<\/td>\r\n16<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nSolution:\r\n
      \r\n \t
    • \u00b5 = \u2211 = 1200= 120 and \u03c32= \u2211(\u00a0 \u2212X\u035e)2 = 436 =43.6<\/li>\r\n<\/ul>\r\n \r\n\r\n1010\r\n\r\nThe percentage of cases that lie between a given limit are as follows:\r\n\r\n \r\n\r\n\r\n\r\n\r\n\r\n
      Interval<\/td>\r\nValues within interval<\/td>\r\n<\/td>\r\nPercentage of<\/td>\r\nPercentage<\/td>\r\n<\/tr>\r\n
      \u00b5\u00b1\u03c3 = 120\u00b16.60\r\n\r\n= 113.4 and 126.6<\/td>\r\n113,\u00a0 115,\u00a0 116,\u00a0 119,120, 122, 124<\/td>\r\n70%<\/td>\r\n70%<\/td>\r\n30%<\/td>\r\n<\/tr>\r\n
      \u00b5\u00b12\u03c3 = 120 \u00b1 2\r\n\r\n= 106.80 and 133.20<\/td>\r\n110,\u00a0 113,\u00a0 115,\u00a0 116,\u00a0119,\u00a0 120,\u00a0 122,\u00a0 124,\u00a0129, 132<\/td>\r\n100%<\/td>\r\n100%<\/td>\r\nnil<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nQ.2. What do you understand by dispersion?\r\n\r\n \r\n\r\nA.2. A measure of dispersion is designed to state numerically the extent to which individual observations vary on the average.\r\n\r\n \r\n\r\nQ.3. What are the different measures of dispersion?\r\n\r\n \r\n\r\nA.3. (1) Absolute measures: (i) Mean Deviation (ii) Standard Deviation (ii) Quartile Deviation, Range.\r\n\r\n \r\n\r\n(2) Relative measures: (i) Coefficient of variation (ii) Coefficient of Mean deviation\r\n\r\n \r\n

      Learn More:<\/strong><\/p>\r\n\r\n

        \r\n \t
      1. Sharma, J K (2014). In: Business Statistics, II eds., S Chand & Company, N Delhi.<\/li>\r\n \t
      2. Chandel, S.R.S. (2006). In: A Handbook of Agricultural Statistics, Anchal Prakashan mandir, Kanpur<\/li>\r\n \t
      3. Pal, N. and Sarkar, S. (2005). In: Statistics Concepts and Applications, Prentice Hall of India, New Delhi.<\/li>\r\n \t
      4. Gupta, K.R. (2012). Practical Statistics, Atlantic Publishers & Distributors (P) Ltd., New Delhi.<\/li>\r\n<\/ol>\r\n ","rendered":"
        \n

         <\/p>\n

        Measures of Dispersion: Mean Absolute Deviation, Standard Deviation, Variance, Coefficient of\u00a0<\/strong>Variation<\/strong><\/p>\n

         <\/p>\n

        Learning Objectives:<\/strong><\/p>\n

         <\/p>\n

        After the completion of this module the student will understand:<\/p>\n

         <\/p>\n

        \u00d8\u00a0 Range<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 Mean Absolute Deviation<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 Computation of Mean Deviation<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 Characteristics of mean deviation<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 Uses of mean deviation<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 Standard Deviation<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 Computation of Standard Deviation<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 Characteristics of Standard Deviation<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 Uses of Standard Deviation<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 Quartile deviation or Semi Inter Quartile range<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 Variance<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 Relative measures of dispersion<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 Coefficient of dispersion<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 Coefficient of variation<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 Standard error<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 Expression for the standard error of mean<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 Probable error<\/p>\n

         <\/p>\n

         <\/p>\n

        1.<\/strong>\u00a0\u00a0\u00a0 Introduction<\/strong><\/p>\n

         <\/p>\n

        Any measure of central tendency or average has its own limitations and gives us an idea only about that central value of the set of observations around which all the observations have a tendency to lie, but it fails to give any idea about the way in which they are distributed. There can be a number of series each of which has the same mean but differs from others in respect of the pattern in which the observations are distributed. To follow this point Consider the following series.<\/p>\n\n\n\n\n\n\n
        Series A<\/td>\n9<\/td>\n9<\/td>\n9<\/td>\n9<\/td>\n9<\/td>\n9<\/td>\n9<\/td>\n<\/tr>\n
        Series B<\/td>\n6<\/td>\n7<\/td>\n8<\/td>\n9<\/td>\n10<\/td>\n11<\/td>\n12<\/td>\n<\/tr>\n
        Series C<\/td>\n1<\/td>\n2<\/td>\n4<\/td>\n5<\/td>\n11<\/td>\n13<\/td>\n27<\/td>\n<\/tr>\n
        Series D<\/td>\n3<\/td>\n15<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

         <\/p>\n

        In the above series, we observe that arithmetic mean of every series is 9, but the pattern in which the observations are distributed is different in different series. In series A the mean is 9 and all the observations are same. In series B also, the mean is 9 and the observations are scattered ranging from 6 to 12 but not very much scattered. In series C,\u00a0the mean is the same value 9 but the observations are too much scattered ranging from 1 to 27. In series D there are only two observations the mean of which is 9.<\/span><\/p>\n<\/div>\n

        \n

         <\/p>\n

        From the above example it is quite obvious that for studying a series, a study of the extent of scattering of the observations of dispersion is also essential along with the study of the central tendency in order to throw more light on the nature of the series. The following are the different measures of dispersion which are in common use.<\/p>\n

         <\/p>\n

        2.\u00a0\u00a0 Range<\/strong><\/p>\n

         <\/p>\n

        2.1. Definition:<\/strong><\/p>\n

         <\/p>\n

        The range is the simplest measure of dispersion. It is the difference between the highest and lowest terms of a series of observations.<\/p>\n

         <\/p>\n

        2.2.Computation of Range:<\/strong><\/p>\n

         <\/p>\n

        Where, XH = Highest variate value<\/p>\n

         <\/p>\n

        and XL = Lowest variate value<\/p>\n

         <\/p>\n

        2.3.Merits and demerits of range<\/strong><\/p>\n

         <\/p>\n

        (i)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Its value usually increases with the increase in the size of the sample.<\/p>\n

         <\/p>\n

        (ii)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 It is usually unstable in repeated sampling experiments of the same size and large ones.<\/p>\n

         <\/p>\n

        (iii)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 It is a very rough measure of dispersion and is entirely unsuitable for precise and accurate studies.<\/p>\n

         <\/p>\n

        (iv)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 The only merit possessed by \u2018Range\u2019 are that it is (i) simple (ii) easy to understand and (iii) quickly calculated. It is often used in certain industrial work.<\/p>\n

         <\/p>\n

        3.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Mean Deviation<\/strong><\/p>\n

        \u00a0<\/strong><\/p>\n

        3.1.\u00a0 <\/strong>Definition:<\/strong><\/p>\n

        \u00a0<\/strong><\/p>\n

        If the deviations of all the observations from their mean are calculated, their algebraic sum will be zero. When this sum is always zero, it is impossible to get the average of these deviations. In order to overcome this difficulty, these deviations are added irrespective of plus or minus sign and then the average is calculated. The deviations without any plus or minus sign are known as absolute deviations. The mean of these absolute deviations is called the mean deviation. If the deviations are calculated from the mean, the measure of dispersion is called mean deviation about the mean. As a matter of fact mean deviation can be calculated from any average, and for that, the absolute deviations from that average will be calculated.<\/p>\n<\/div>\n

         <\/p>\n

        3.2.Computation of mean deviation:<\/strong><\/p>\n

        \"\"Mean deviation about other averages:<\/span><\/div>\n
        M.D. about A= 1 \u2211 \u20d3 (\u00a0\u00a0 \u2212\u00a0\u00a0 )\u20d3<\/span><\/div>\n
        M.D. about Md.=1 \u2211 \u20d3 (\u00a0\u00a0 \u2212\u00a0\u00a0 . )\u20d3<\/span><\/div>\n
        \n

        M.D. about Mo. = 1 \u2211 \u20d3 (\u00a0\u00a0 \u2212\u00a0\u00a0 . )\u20d3<\/span><\/p>\n

        3.3.Characteristics of mean deviation:<\/strong><\/p>\n

         <\/p>\n

        (i)\u00a0 A notable characteristic of mean deviation is that it is the least when calculated about the median.<\/p>\n

         <\/p>\n

        (ii) Standard deviation is not less than the mean deviation in a discrete series i.e. it is either equal to or greater than the M.D. about mean.<\/p>\n

         <\/p>\n

        (iii) When an average other than the A.M. is calculated as a measure of central tendency, M.D. about that average is the only suitable measure of dispersion.<\/p>\n

         <\/p>\n

        4.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Standard Deviation<\/strong><\/p>\n

        \u00a0<\/strong><\/p>\n

        4.1.Definition:<\/strong><\/p>\n

        \u00a0<\/strong><\/p>\n

        Calculation of standard deviation is also based on the deviations from the arithmetic mean. In thecase of mean deviation the difficulty, that the sum of the deviations from the arithmetic mean is always zero, is solved by taking these deviations irrespective of plus or minus signs. But here, that the difficulty is solved by squaring them and taking the square root of their average. It is thus defined by thefollowingexpression.<\/p>\n

         <\/p>\n

        Standard Deviation (S.D.) = \u221a\u2211(\u00a0 \u2212\u00a0 )2 \u2026\u2026\u2026\u2026. (1)<\/span><\/p>\n

        Where,\u00a0\u00a0 X= An observation or variate value<\/span><\/p>\n

        \u00b5\u00a0 = Arithmetic mean of the population N= Number of given observations.<\/span><\/p>\n<\/div>\n

        \n

         <\/p>\n

        According to the expression given in (1), thepopulationmean\u00b5 is required for finding the standard deviation (S.D.) of a given set of observations. Generally, \u00b5 is not known. Therefore it is replaced by X\u035e, which is the mean of the given set of observations, and then the S.D. of the given data is given by<\/p>\n

        Standard Deviation (S.D.) = \u221a\u2211(\u00a0 \u2212\u00a0 )2\u2026\u2026\u2026\u2026\u2026(2)<\/p>\n

         <\/p>\n

        (X \u2212 X\u035e)2 = deviation from mean\u00a0s-10<\/strong><\/p>\n

         <\/p>\n

        Here, it should be noted that formula (2) gives the S.D. of the given set of data which itself is assumed to be the population with \u00b5=X\u035e. Therefore we shall this S.D. as the \u2018population S.D.\u2019 Thus,<\/p>\n<\/div>\n

        <\/div>\n
        population S.D. (=\u03c3) = \u221a \u2211( \u2212 )2 \u2026\u2026\u2026\u2026\u2026 (3)<\/div>\n
        <\/div>\n
        \n

        In case of frequency distribution-<\/p>\n

        Population S.D. (=\u03c3) = \u221a \u2211 ( \u2212 )2 \u2026\u2026\u2026\u2026\u2026 (4)<\/p>\n

        Sample S.D.: <\/strong>In case, when the given set of data is not a population but is a sample drawn from<\/p>\n

        \u035e<\/p>\n

        a large population, the population mean \u00b5 is not known. Therefore, in its place, we use X which is the estimate of\u00b5 obtained from the sample observations. The result is that we cannot calculate the population S.D. (\u03c3), but, in its place, we calculate its estimate (S). We represent the estimates of population parameters, \u00b5 and \u03c3, in the following way:<\/p>\n

         <\/p>\n

        X\u035e= Estimate of (\u00b5)<\/p>\n

         <\/p>\n

        S= Estimate of (\u03c3)<\/p>\n<\/div>\n

        PThe best estimates (S) of the population S.D. (\u03c3) is given by<\/p>\n

        S (sample S.D.) = \u221a \u2211 ( \u2212 )2 \u2026\u2026\u2026\u2026\u2026 (5)<\/p>\n

        \n

        s-11<\/strong><\/p>\n

         <\/p>\n

        4.2.Computation of Standard Deviation<\/strong><\/p>\n

         <\/p>\n

        For computing S.D., in every case, we have to calculate the arithmetic mean, which increases the labor of calculation work. Therefore, to avoid it, theshort cut method should be used in which any value of the variate is chosen as the arbitrary mean and then the standard deviation is calculated by the following process:<\/p>\n

         <\/p>\n

        Suppose, A is the arbitrary mean and d is the deviation of the variate value from A. i.e. d = X-A<\/p>\n

        we have, \u2211\u00a0 (\u00a0\u00a0 \u2212\u00a0\u00a0 )2 =\u2211\u00a0 \u00a02 \u2212 (\u2211 \u00a0\u00a0)2<\/p>\n

        Therefore, for this, we require the columns of d, fd, and fd2. In the column of d we shall find a factor equal to the width of the class interval \u201ci\u201d common to all the figures in that column. After taking out this factor as common, the columns now will be of d\/I, fd\/I and fd2\/i2. With the help of these symbols, the values of \u2211\u00a0 (\u00a0\u00a0 \u2212 X\u035e)2 and S.D. will be calculated as given bellow.<\/p>\n

        \u2211\u00a0 \u00a0 \u00a0=\u00a0 \u00d7 \u2211<\/p>\n

         <\/p>\n

        \u2211 2<\/sup> = 2 \u00d7 \u2211 2<\/sup>
        \n\u2211 ( \u2212 )2<\/sup> = 2 \u00d7 {\u2211 2<\/sup> \u2212 (\u2211 )2<\/sup>} . . = \u00d7 [\u221a 1 {\u2211 2<\/sup> \u2212 (\u2211 )2<\/sup> }]<\/p>\n

         <\/p>\n

         <\/p>\n

        s-12<\/strong><\/p>\n

         <\/p>\n

        If we use the symbol D for d\/I, the above expressions will be written as<\/p>\n

         <\/p>\n

        \u2211\u00a0\u00a0\u00a0\u00a0\u00a0 =\u00a0 \u00d7 \u2211<\/span><\/p>\n

         <\/p>\n

        \u2211 <\/span>2<\/sup> = 2 \u00d7 \u2211<\/span> 2<\/sup><\/p>\n

         <\/p>\n

        \u2211 ( \u2212 )<\/span>2<\/sup> = 2 \u00d7 {\u2211<\/span> 2<\/sup> \u2212 (\u2211 )<\/span>2<\/sup>} . . = \u00d7 [\u221a 1 {\u2211 <\/span>2<\/sup> \u2212 (\u2211 )<\/span>2<\/sup> }]<\/span><\/p>\n<\/div>\n

        \n

         <\/p>\n

        s-13<\/strong><\/p>\n

         <\/p>\n

        Example:<\/strong><\/p>\n

         <\/p>\n

        Calculation of S.D.<\/p>\n\n\n\n\n\n\n\n\n\n
        Class<\/td>\nFrequency\u00a0f<\/td>\nMid values\u00a0X<\/td>\nd= X-A<\/td>\nd\/I or D<\/td>\nfD<\/td>\nfD2<\/td>\n<\/tr>\n
        0-10<\/td>\n1<\/td>\n5<\/td>\n-20<\/td>\n-2<\/td>\n-2<\/td>\n4<\/td>\n<\/tr>\n
        10-20<\/td>\n3<\/td>\n15<\/td>\n-10<\/td>\n-1<\/td>\n-3<\/td>\n3<\/td>\n<\/tr>\n
        20-30<\/td>\n5<\/td>\n25<\/td>\n0<\/td>\n0<\/td>\n0<\/td>\n0<\/td>\n<\/tr>\n
        30-40<\/td>\n4<\/td>\n35<\/td>\n+10<\/td>\n+1<\/td>\n+4<\/td>\n4<\/td>\n<\/tr>\n
        40-50<\/td>\n2<\/td>\n45<\/td>\n+20<\/td>\n+2<\/td>\n+4<\/td>\n8<\/td>\n<\/tr>\n
        Total<\/td>\n15<\/td>\n–<\/td>\n–<\/td>\n–<\/td>\n+3<\/td>\n19<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

         <\/p>\n<\/div>\n

        \n

         <\/p>\n

        \u2211 ( \u2212 )2<\/sup> = 2 \u00d7 {\u2211 2<\/sup> \u2212 (\u2211 )2<\/sup> }<\/p>\n

         <\/p>\n

        = 102 \u00d7 [9 \u2212 32 ] 15<\/p>\n

        = 100 \u00d7 [ 276 ] 15<\/p>\n

        = 1840<\/p>\n

        \u2211 ( \u2212 )2 \u035e Population S.D. =\u221a , (Here, \u00b5= X)<\/p>\n

         <\/p>\n

        \u03c3 = \u221a184015 = 11.07<\/p>\n

         <\/p>\n

        Sample S.D. = \u221a\u2211 ( \u2212 )2 = \u221a1840<\/p>\n

        \u22121 14<\/p>\n

        Or S= 11.46<\/p>\n

         <\/p>\n

        4.3.Characteristics of Standard Deviation<\/strong><\/p>\n

        \u00d8\u00a0\u00a0 It is rigidly defined.<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 Its computation is based on all the observations.<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 If all the variate values are the same, S.D.=0<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 S.D. is least affected by fluctuations of sampling.<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 It is affected by the change of scale, but not affected by the change of origin.<\/p>\n

         <\/p>\n

        4.4.Uses of Standard Deviation<\/strong><\/p>\n

         <\/p>\n

        \u00d8\u00a0 It is used in computing different statistical quantities like, regression coefficient, correlation coefficient,<\/p>\n

        etc. and in connection with business cycle analysis.<\/p>\n

         <\/p>\n

        \u00d8\u00a0\u00a0 It is also used in testing the reliability of certain statistical measures.<\/p>\n

         <\/p>\n

        s-15<\/strong><\/p>\n

         <\/p>\n

        4.5.Quartile Deviation or Semi inter quartile range<\/strong><\/p>\n

         <\/p>\n

        The measure of dispersion is expressed in terms of quartiles and known as quartile deviation or semi inter quartile range. \u22122<\/p>\n

         <\/p>\n

        Where,Q1 = Lower quartile<\/p>\n

         <\/p>\n

        Q3= Upper Quartile<\/p>\n

         <\/p>\n

        It is not a measure of the deviation from any particular average. For symmetrical and moderately skew distributions the quartile deviation is usually two-third of the standard deviation.<\/p>\n

        Q.D.=23 \u00d7 ( .\u00a0\u00a0 . )<\/p>\n

         <\/p>\n

        5.\u00a0<\/strong>Variance<\/strong><\/p>\n

        \u00a0<\/strong><\/p>\n

        Variance is the square of the standard deviation.<\/p>\n

        \u00a0<\/strong><\/p>\n

        Variance = (S.D.)2<\/p>\n

         <\/p>\n

        The variance of a population is generally represented by the symbol \u03c32and its unbiased estimate calculated from the sample, by the symbol S2.<\/p>\n

         <\/p>\n

        6.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Relative Measures of Dispersion<\/strong><\/p>\n<\/div>\n

        \n

        \u00a0<\/strong><\/p>\n

        The measures of dispersion, which we studied so far, are the absolute measures of dispersion, and are represented it\u2019s the same units in which the observations are represented, e.g., gms., cm., meters, hectares, etc. When we have to compare the dispersions of two or more distributions, it will not be proper to compare their absolute measures of dispersions, because, the distributions or the data may differ from one another.<\/p>\n

        \u00a0<\/strong><\/p>\n

        (i)\u00a0\u00a0\u00a0\u00a0 With respect to their averages<\/p>\n

         <\/p>\n

        (ii)\u00a0\u00a0 With respect to their dispersions<\/p>\n

         <\/p>\n

        (iii)With respect to their averages and dispersions both (iv) With respect to their units<\/p>\n

         <\/p>\n

        Therefore, they will not be comparable. Under such circumstances, their comparison is possible with the help of relative measures of dispersion.<\/p>\n

         <\/p>\n

        6.1.Coefficient of Dispersion<\/strong><\/p>\n

         <\/p>\n

        It is computed by the following expression:<\/p>\n

         <\/p>\n

        Coefficient of Dispersion =<\/p>\n

         <\/p>\n

        s-17<\/strong><\/p>\n

         <\/p>\n

        6.2.Coefficient of Variation (C.V.)<\/strong><\/p>\n

         <\/p>\n

        This is also a relative measure of dispersion. It is especially important on account of the widely used measures of central tendency and dispersion i.e., Arithmetic mean, and Standard Deviation. It is given by<\/p>\n

        C.V. = .. .. \u00d7 100<\/p>\n

        It is expressed in percentage and used to compare the variability in the two or more series. Lesser value of thecoefficient of variation indicates more consistency.<\/p>\n

         <\/p>\n

        7.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Standard Error 7.1.Definition<\/strong><\/p>\n

        \u00a0<\/strong><\/p>\n

        The Standard deviation of the sampling distribution of a statistic (estimate) is known as the standard error of that statistic (estimate).<\/p>\n

        \u00a0<\/strong><\/p>\n

        If we take all possible samples from the population of the same size and get a sampling distribution of means, it can be proved that the mean of this sampling distribution of means is the population mean and its standard deviation, the standard error of the mean.<\/p>\n

        \u00a0<\/strong><\/p>\n

        As it is not possible to draw and study all possible samples, we have to get and we get the estimate of the standard error from a single sample. If S be the standard deviation of the sample of size N, the estimate of the standard error of mean is given by \u221a .<\/p>\n<\/div>\n

        \n

         <\/p>\n

        S-18<\/strong><\/p>\n

         <\/p>\n

        7.2.Expression for the standard error of mean<\/strong><\/p>\n

         <\/p>\n

        Let there be a sample of N observations, X1, X2, X3\u2026\u2026\u2026XN which have been drawn at<\/p>\n

         <\/p>\n

        random from a population, the variance of which is \u03c32.<\/p>\n

        \u035e\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 1<\/p>\n

        Now, Mean X\u00a0\u00a0 =\u00a0\u00a0\u00a0\u00a0\u00a0 (\u00a0 1 +\u00a0 2 +\u00a0 3 + \u22ef + Variance of mean<\/p>\n

        V(X\u035e)= 12 {\u00a0 (\u00a0 1 +\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 2 +\u00a0 3 + \u22ef +\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 )}<\/p>\n

        =\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 12 [\u00a0 (\u00a0 1) +\u00a0\u00a0 (\u00a0 2) + \u22ef +\u00a0\u00a0 (\u00a0\u00a0 )]<\/p>\n

         <\/p>\n

        Since, V(X1) =V(X2) = V(X3)=\u2026\u2026\u2026\u2026\u2026\u2026\u2026.= V(XN)= \u03c32<\/p>\n

        \u035e\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 2<\/p>\n

        V(X)=\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 2 =<\/p>\n

        S.E. of X\u035e= \u221a<\/p>\n

         <\/p>\n

        But since, in practice, \u03c3 is not known, it is replaced by its unbiased estimate S.<\/p>\n

        S.E. of X\u035e= \u221a<\/p>\n

         <\/p>\n

        8.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Probable Error<\/strong><\/p>\n

        \u00a0<\/strong><\/p>\n

        The quartile deviation of the sampling distribution of means is known as aProbable error<\/p>\n

         <\/p>\n

        and is 0.67449times the standard error.<\/p>\n

         <\/p>\n

        P.E. = 0.67449 (S.E.)<\/p>\n

         <\/p>\n

        Three times the probable error is roughly twice the standard error. This measure of dispersion has no particular advantage and moreover involves a troublesome factor 0.67449. This is why it has gone out of use and has given place to standard error.<\/p>\n

         <\/p>\n

        9.\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>Summary<\/strong><\/p>\n

        \u00a0<\/strong><\/p>\n

        This module provides an overview to students to understand the techniques that are used to measure the extent of variation or the deviation (also called thedegree of variation) of each value in the dataset from a measure of central tendency, usually the mean or median. Such statistical techniques are called measures of dispersion (or variation). A small dispersion among values in the data set indicates that data are clustered that data are clustered closely around the mean. The mean is therefore considered representative of the data, i.e. mean is reliable average. Conversely, a large dispersion among values in the data set indicates that the mean is not reliable, i.e. it is not representative of data. The symmetrical distribution of values in two or more sets of data may have same variation but differ greatly in terms of A.M. On the other hand, two or more sets of data may have the same A.M. values but differ in variation.<\/p>\n<\/div>\n

          \n
        1. Self-check exercise with solution<\/strong><\/li>\n<\/ol>\n

           <\/p>\n

          Q.1.<\/strong>Thefollowing data give the number of passengers traveling by airplane from one city to another in one week.<\/p>\n

           <\/p>\n

          115, 122, 129, 113, 119, 124, 132, 120, 110, 116<\/strong><\/p>\n

           <\/p>\n

          Calculate the mean and standard deviation and determine the percentage of class that lie between (i) \u00b5\u00b1\u03c3 (ii) \u00b5\u00b12\u03c3 and (iii) \u00b5\u00b13\u03c3. What percentage of cases lie outside these limits.<\/p>\n

           <\/p>\n

          Calculation of Mean and Standard Deviation<\/p>\n

           <\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n
          X<\/td>\nX- X\u035e<\/td>\n(X- X\u035e)\/2<\/td>\n<\/tr>\n
          115<\/td>\n-5<\/td>\n25<\/td>\n<\/tr>\n
          122<\/td>\n2<\/td>\n4<\/td>\n<\/tr>\n
          129<\/td>\n9<\/td>\n81<\/td>\n<\/tr>\n
          113<\/td>\n-7<\/td>\n49<\/td>\n<\/tr>\n
          119<\/td>\n-1<\/td>\n1<\/td>\n<\/tr>\n
          124<\/td>\n4<\/td>\n16<\/td>\n<\/tr>\n
          132<\/td>\n12<\/td>\n144<\/td>\n<\/tr>\n
          120<\/td>\n0<\/td>\n0<\/td>\n<\/tr>\n
          110<\/td>\n-10<\/td>\n100<\/td>\n<\/tr>\n
          116<\/td>\n-4<\/td>\n16<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

           <\/p>\n

          Solution:<\/p>\n

            \n
          • \u00b5 = \u2211 = 1200= 120 and \u03c32= \u2211(\u00a0 \u2212X\u035e)2 = 436 =43.6<\/li>\n<\/ul>\n

             <\/p>\n

            1010<\/p>\n

            The percentage of cases that lie between a given limit are as follows:<\/p>\n

             <\/p>\n\n\n\n\n\n
            Interval<\/td>\nValues within interval<\/td>\n<\/td>\nPercentage of<\/td>\nPercentage<\/td>\n<\/tr>\n
            \u00b5\u00b1\u03c3 = 120\u00b16.60<\/p>\n

            = 113.4 and 126.6<\/td>\n

            		113,\u00a0 115,\u00a0 116,\u00a0 119,120, 122, 124<\/td>\n70%<\/td>\n70%<\/td>\n30%<\/td>\n<\/tr>\n
            \u00b5\u00b12\u03c3 = 120 \u00b1 2<\/p>\n

            = 106.80 and 133.20<\/td>\n

            		110,\u00a0 113,\u00a0 115,\u00a0 116,\u00a0119,\u00a0 120,\u00a0 122,\u00a0 124,\u00a0129, 132<\/td>\n100%<\/td>\n100%<\/td>\nnil<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

             <\/p>\n

            Q.2. What do you understand by dispersion?<\/p>\n

             <\/p>\n

            A.2. A measure of dispersion is designed to state numerically the extent to which individual observations vary on the average.<\/p>\n

             <\/p>\n

            Q.3. What are the different measures of dispersion?<\/p>\n

             <\/p>\n

            A.3. (1) Absolute measures: (i) Mean Deviation (ii) Standard Deviation (ii) Quartile Deviation, Range.<\/p>\n

             <\/p>\n

            (2) Relative measures: (i) Coefficient of variation (ii) Coefficient of Mean deviation<\/p>\n

             <\/p>\n

            Learn More:<\/strong><\/p>\n

              \n
            1. Sharma, J K (2014). In: Business Statistics, II eds., S Chand & Company, N Delhi.<\/li>\n
            2. Chandel, S.R.S. (2006). In: A Handbook of Agricultural Statistics, Anchal Prakashan mandir, Kanpur<\/li>\n
            3. Pal, N. and Sarkar, S. (2005). In: Statistics Concepts and Applications, Prentice Hall of India, New Delhi.<\/li>\n
            4. Gupta, K.R. (2012). Practical Statistics, Atlantic Publishers & Distributors (P) Ltd., New Delhi.<\/li>\n<\/ol>\n

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