Measure of Dispersion
The dispersion is the measure of variations in the values of the variable. It measures the degree of scatteredness of the observation in a distribution around the central value.
The measure of dispersion which is easiest to understand and easiest to calculate is the range.
Range is defined as the difference between two extreme observation of the distribution.
Range of distribution = Largest observation – Smallest observation.
Mean deviation for ungrouped data
For n observations x1, x2, x3,…, xn, the mean deviation about their mean is given by
Mean deviation about their median M is given by
Mean deviation for discrete frequency distribution
Let the given data consist of discrete observations x1, x2, x3,……., xn occurring with frequencies f1, f2, f3,……., fn respectively in case
Mean deviation about their Median M is given by
Mean deviation for continuous frequency distribution
where xi are the mid-points of the classes, and M are respectively, the mean and median of the distribution.
Variance is the arithmetic mean of the square of the deviation about mean .
Let x1, x2, ……xn be n observations with as the mean, then the variance denoted by σ2, is given by
If σ2 is the variance, then σ is called the standard deviation is given by
Standard deviation of a discrete frequency distribution is given by
Standard deviation of a continuous frequency distribution is given by
Coefficient of Variation
In order to compare two or more frequency distributions, we compare their coefficient of variations. The coefficient of variation is defined as
Note: The distribution having a greater coefficient of variation has more variability around the central value, then the distribution having a smaller value of the coefficient 0f variation.