Variance is a measure of dispersion. Measures of dispersions are different from measures of central tendency. Measures of central tendency give an insight to one of the important characteristics of a distribution (mean, median or mode of a set of data). But they fail to give us any idea as to how the individual values differ from the central value, i.e. whether they are closely packed around the central value or widely scattered away from it. Two distributions may have the same mean and the same total frequency, yet they may differ in the extent to which the individual values may be spread about the average. The magnitude of such a variation is called dispersion. See picture below. All the three curves have the same mean, but the spread around the mean is different for each of them.
One such measure of dispersion is the measure of amount of variation.
Variance definition: The measure of amount of variation can be defined as the mean of the squares of the difference from the mean of each of the observations. For calculating variance we follow the following steps:
(1) First of all find the mean of the data set. Let the mean = x ¯. Also let each of the observations be = x_i, where i = 1,2,3... n for n observations.
(2) Then for each of the observation, do x ¯ - x_i. That is to say we find the difference between the mean and each of the observations one by one.
(3) Next we square each of the numbers that we got in step (2). So in other words we find (x ¯ - x_i)^2 for each of the observation.
(4) Now we add up all the numbers we got in step (3). That means we find ∑(x ¯ - x_i)^2
(5) Finally we find the average of the sum we found in step(3). Therefore we divide ∑(x ¯ - x_i)^2 by n (the total number of observations). The result thus obtained is called the for measure of amount of variation.
Variance formula: Based on the steps we stated above, let us now write the formula for measure of amount of variation. The measure of amount of variation can be denoted by V. Using the same notations that we used above the formula for measure of amount of variation can be stated as follows:
V = [ ∑ (x ¯ - x_i)^2]/n
Note here that whether we take x ¯ - x_i or x_i- x ¯, it would not make a difference since it is squared. So measure of amount of variation can also be:
V = [ ∑ (x_i- x ¯)^2]/n