Measures of central tendency and dispersion

    Quantitative phenomena can often be described by a measure of central tendency (the "average", or arithmetic mean), and a measure of dispersion, either the varianc
e or the standard deviation.

Mean
= sum of i individual values of variable X, divided by number of individuals N

     [read as, "X bar"]

The intuitive measure of dispersion is the average difference from the mean: however, the differences would be both above and below the means, and their sum would be zero.
     To express average dispersion in terms of magnitude without regard to sign, the difference from the mean is squared.

Variance = average squared deviation of N individuals from the mean. By definition,

  [read as, "sigma squared"]

        Calculation of the variance by this formula is cumbersome, and variance is more easily calculated as

    
        This traditional calculation can be remembered as  "mean of squares" minus "square of means" [MOSSOM].
                   With a hand calculator, this requires only two summations, of the individual values, and their squares.

Standard deviation = square root of the variance. This expresses dispersion in the same units as the mean.

          If all individuals of a population are included, the parametric standard deviation (s) is identical with the square root of the variance



         Typically, where measurements are made on a finite sample size N, the non-parametric standard deviation is


        When N is small, the sample s is larger than the parametric value, but as N increases, the correction term becomes negligible.

Covariance = average squared deviation of N individuals for two variables, X and Y :

covariance
Note that the covariance of two variables X and Y is identical to that for one variable X with itself.



All text material ©2024 by Steven M. Carr