Page 1 : Chapter 4, MEASURES OF DISPERSION, , An average is a measure of central tendency. It does, ot show as how the variates are scattered about the, central value. In a distribution it is possible that all, glues May be spread over a narrow range. In another, qistribution with the same average, it is quite possible, at the values may be scattered over a wide range., ence, an average is not an adequate statistical constant, cate the second aspect namely how the individual, differ or vary from the central value. A study of, such a variation of the values in a distribution is, fundamentally important in a statistical investigation, and it can be well appreciated from the following example., , Series A : 10; 15, 20, 25, 30, Series B : 2, 8, 40, 35, 15, Series C : 20, 20, 20, 20, 20, , Since all the three series have the same mean one, may conclude that the three series are almost similar., But actually the three series differ widely from one, another. In series C none of the observations deviates, fom the AM and hence there is no dispersion. In series, A the variations of observations from the AM is very, small compared to series B. The measurement of the, scattering of observations or items in a distribution about, the average is called a measure of variation or dispersion., Measure of dispersion informs about the extent of, tepresentativeness of the average of a series. When, ue is significant, it implies that the average is, is tom being a representative figure. If however,, ree is not very significant, then the average, , baste to be a highly representative figure of the series., isthe of dispersion also enable comparison of two, , utions with regard to their variability., , to indi, yalues, , , , , , , , , , , , , , , Scanned with CamScanner

Page 5 : 1, , 4x, 2 10439,, , 3, , 10. 7944.67 = 74.67, 0, , 50.20 _ 24.47, , , , QD, Quartile coefficient ofdispersion, , 4.67, = sO. eel 2, , = 12.93, , 5-0 _ 74:87 80.20 _ 24.47, , sso, , z QQ, 7467-9020 124.87 = 0.195, , Merits and Demerits, , Merits, 1. Itis rigidly defined., , 2. Itiseasy to understand and simple to calculate,, 3, Itisnot unduly affected by extreme values,, 4. Itcanbe calculated for open-end distributions,, , Demerits, 1. {tis not based on all observations, , 2, Itisnot capable of further algebraic treatment, 3. Itis much affected by fluctuations of sampling., , 3. Mean Deviation, , Here the deviations of the individual values from some, origin are considered. The absolute values (numerical, values without considering the sign) of these deviations, are added. When this sum is divided by the number of, observations the result obtained is known as Mean, deviation. The mean deviation is thus defined as’ the, arithmetic mean of the absolute values of the deviations, of observations from some origin, say mean or median, , or mode., Thus for a raw data, , M.D about Mean =, , , , L|x-x|, n, , “7 Stay jon to Statistics, on to Statis, Ney yao!, , , , 139, x - X| stands for the absolute di, , oft BI * deviation of x x, wad as modulus of (x-X) or mod (x-x) somite, : 5, , is f taking deviation from, , stead Oo! mean, If we are ug, , est the mean deviation about median, eosin, w, , ». MD. about Median = =— Ml], n, Fora frequency data, MD about Meanis given by, , cmys = LEAF, yy, , MD about Median (Mp) = =£ K =M|, , Note, whenever nothing is mentioned about the measure of Central, tendency from which deviations are to be considered, deviations, , areto be taken from the mean and the required MD is MD about, mean., , (i) Coefficient of MD (about mean) = MD about mean., Mean, MD about median, (i) Coefficient of MD (about median) = Mea a, ‘Theorem:, Fora frequency distribution, the mean deviation from an average Ais, , minimum when Ais the median (thatis, the mean deviation is least when., measured from the median), , Proof, let S = Lf\x-Al ani), , where A is an average (mean, median or mode). Then, the, mean deviation from A is minimum when S is minimum., , Let x, be the item or the mid-point of the class just less than, A,so that, , XS XQ << Xe SAS Xe SSX ,- Then, , Scanned with CamScanner, , —_