Measures of Central Tendency Central tendency of a random variable is the concentration of its values in the central part of the distroution. E(X) = [x . f(x) dx, where Ry is the range space of X. A measure of central tendency is the value of the random variable which is RX representative of the entire distribution of the variable. (ii) Median : If X is a random variable, discrete or continuous, the value of The measures of central tendency, also known as measures of location or X = x for which the cumulative distribution function F(x) = 1/2 is called the averages, that are in common use are the following. median of X or the median of the distribution of X. (i) Arithmetic mean or simply mean In case of a discrete RV X, if there exist no x such that F(x) = 1/2, then the median M is defined as (ii) Median (iii) Mode M = = (xx+ Xx+1) (iv) Geometric Mean where, F(xx) 1/2, where Xx and Xx+1 are two consecutive values of X. (v) Harmonic Mean In case of a continuous RV X, the median M is defined by Definitions M (i) Arithmetic Mean : If X is a discrete random variable with probability [ foodx - [ f( ) dx-? where f(x) is the density function of X. mass function - co P(X = x ) = pi; i= 1,2,3, ... (iii) Mode : If X is a discrete RV, then that value of X for which the probability is maximum is called the mode of the distribution of X (or modal value of X). then the mathematical expectation of X or the arithmetic mean of X, denoted by E(X), is defined by, If X is a continuous RV with density function f(x), then the value of X = x for which f(x) is maximum [i. e., for which f'(x) = 0 and f"(x) X . Pi If X is a continuous random variable with probability density function f(x), then E(X) is defined by