#### 3.1.2 Measures of Dispersion

Measures of dispersion are the quantities that characterize the ‘spread’ of the data, such as range, inter–quartile range, or standard deviation.

##### The range

The range R is the difference between the maximum and minimum value of Xi. This measure is unstable since it depends upon the two extreme values of the data and not the entire set. The extreme values can result from exceptional observations, but the range is useful to show the extent (or limits) of the data.

In order to reduce the influence of the extreme values, inter–quartile range (IQR) or inter–decile range is often used as indicators of the dispersion of the data.

Inter–quartile range = q3q1

Inter–decile range = q9 q1

##### Standard deviation

Standard deviation (denoted by s or s.d.) is the root mean square of the deviations from the arithmetic mean. The standard deviation indicates the average distance of the observations from the mean of the data set.

To get a better estimate of the standard deviation of the population (denoted by s), standard deviation is often computed with n–1 instead of n in the denominator. However, for large values of n (n ³ 30) there is practically no difference between the two definitions. Variance is the square of the standard deviation.

The standard deviation is an absolute measure of deviation that expresses variation in the same units as the original data. Coefficient of Variation (C V) is a relative measure that indicates the magnitude of variation relative to the magnitude of the mean. Coefficient of variation is computed as follows:

CV= s/m