Measures of Dispersion Though a measure of central tendency may be taken as a representative value Let X be a continuous RV and p(a) be the mean deviation about a. of the RV, it is important to know how the values are clustered around or scattered away from the measure of central tendency while studying the i. e., (a) = E{IX - al} distribution of the RV. The property of the RV (or its distribution) by which its values are clustered around or scattering away from the central value is called dispersion or spread or scatter or variability. p(a) = [ 1x - al . f(x) dx - [ (a - x) . f(x) ax+ / (x -a) . f(x) dx -00 -00 Following are the measures of dispersion which are commonly used. [ : in (-oo, a), x a] (i) Quartile Deviation (ii) Mean Deviation " ' (@) - [ fax+ (a-of(@da- [cuf() ax+ (a-a)f(a) a -00 (iii) Standard Deviation Definitions (ii) Mean Deviation : Mean Deviation of a random variable X (discrete or continuous) about the mean value u is defined by : M. D. = E(|X - /} $' (a) = [f()ax+ [ f()ax - [ f(x)dx- [ f(xx) dx Xbox - Hel . Pi, if X is discrete RV = F(a) - {1 - F(a)} i. e., M. D. = Now, again differentiating p'(a) w.r.t. a, we get |x - ul . f(x) dx, if X is continuous RV : $"(a) = f(a) + f(a) = 2f(a) > o [: f(x) is the density function] Mean deviations about the averages median and mode may also be required in : a is the minima for $(a) and value of a can be obtained by, statistical analysis. $'(a) = 0 = F(a) - {1 -F(a)} =0 = 2F(a) =1 = F(a) = 1/2 Property of Mean Deviation i. e., a is the median. Mean deviation about the median is the minimum Therefore, mean deviation about a is minimum, when a is the median