Measures of Dispersion

Central Tendency Measures describe the central values of the sample, whereas the Dispersion Measures describe the data spread in the sample. Both together contributes to the Descriptive Statistics.

Earlier we have seen Range, Quantiles, Percentiles, and Interquartiles Range measures. The mean absolute deviation and variance measures are described here. The rest of the spread measures are given below.

Standard Deviation (SD): Both variance and standard deviation provide the same information. One can always be obtained from the other. In other words, the process of computing a standard deviation always involves computing a variance. Since standard deviation is the square root of the variance, it is always expressed in the same units as the raw data:

For sufficiently large data sets, approximately 68% of the data are contained within one standard deviation of the mean, 95% contained within two standard deviations. 97.7% of the data are contained within three standard deviations (S) from the mean.

The Mean Square Error (MSE): MSE of an estimate is the variance of the estimate plus the square of its bias. Therefore, if an estimate is unbiased, then its MSE is equal to its variance. This is one of the measures usually used to assess the quality of forecasts made by a statistical or physical modelling system.

Coefficient of Variation (CV): Coefficient of Variation is the absolute relative deviation with respect to the mean, provided mean is not zero, expressed in percentage:

CV =100 |S/| %

CV is independent of the unit of measurement. In estimation of a parameter, when its CV is less than 10%, the estimate is assumed acceptable. The inverse of CV, namely, 1/CV is called the Signal-to-noise Ratio.

The CV is used to represent the relationship of the standard deviation to the mean, suggesting how representative the mean is of the numbers from which it came. It expresses the standard deviation as a percentage of the mean. That is it reflects the variation in a distribution relative to the mean.

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