Measures of Dispersion Though a measure of central tendency may be taken as a representative value In the case of discrete random variable X, the quartiles are defined in a manner of the RV, it is important to know how the values are clustered around or similar to the definition of median i. e., scattered away from the measure of central tendency while studying the distribution of the RV. The property of the RV (or its distribution) by which its The value of X = x for which the cumulative distribution function F(x) = 1/4 values are clustered around or scattering away from the central value is called is called the first (lower) quartile Q1. If there exist no x such that F(x) = 1/4, dispersion or spread or scatter or variability. then the first quartile Q, is defined as Following are the measures of dispersion which are commonly used. 21 = = ( xx + Xx+1) (i) Quartile Deviation where, F(xx) 1/4, where xx and Xx+1 are two consecutive (ii) Mean Deviation values of X. (iii) Standard Deviation The value of X = x for which the cumulative distribution function F(x) = 1/2 is called the second (median) quartile Q2. If there exist no x such that Definitions F(x) = 1/2, then the second quartile Q2 is defined as (i) Quartile Deviation : The rth quartile of a continuous random variable X (or its distribution) with density function f(x), denoted by Q,, is defined by Q2 = 5 (Xx + * *+1) where, F(xx) 1/2, where xx and Xx+1 are two consecutive [ f ( x) dx - 7 ; r = 1,2,3 values of X. -0c The value of X = x for which the cumulative distribution function F(x) = 3/4 Here Q1 and Q3 are called the first (lower) and the third (upper) quartiles is called the third (upper) quartile Q3. If there exist no x such that F(x) = 3/4, respectively. Q2 is obviously the median (middle or second quartile). then the third quartile Q3 is defined as Now 23 - ", that is half of the difference between the upper and lower 23 = = ( xx + XX+1) 2 . quartiles, is called the semi - interquartile range or the quartile deviation and denoted by Q. D. i. e., where, F(xx) 3/4, where xx and Xx+1 are two consecutive values of X. Q. D. = 23 - Q1 2 Quartile deviation, Q. D. = 23- Q1 2