For a given set of data, mean is said to be the average of all the data. But the mean will not give any information about how evenly or unevenly the data is spread. The variance of a given data gives the information about the variability of individual data in a group. Standard deviation is the square root of variance.

Suppose we have a data set, $28, 10, 6, 2, 8$. The average of this data will be $10.8$ which will be the mean of the data. Here we can notice that $4$ members of the set are closer to the mean but $28$ varies a lot from the mean. This difference of data sets is obtained using variance.

Formula

For a given data set, let the mean be $\mu$ and total number of members be $N$. Then the variance can be given by the formula for each element $X$,
$V$ = $\frac{\sum (X-\mu )^{2}}{N}$If a data set is given to be $10, 6, 24, 12$, then find the variance.

Solution:

Mean of the data set, $\mu$ = $13$

Variance, $V$ = $\frac{(13-10 )^{2}+(13-6 )^{2}+(13-24 )^{2}+(13-12 )^{2}}{4}$
               
                  = $\frac{3^2 + 7^2 + 11^2 + 1^2}{4}$ = $\frac{180}{4}$
                 
                  = $45$

Sample Variance

Sample variance is the variance of a part of the total population known as sample. The formula to find sample variance is little different from the formula of getting the variance. The formula for the sample variance is given by:
$S$ = $\frac{\sum (X - \mu )^{2}}{N - 1}$In the denominator, one is being subtracted from the total number of members as it is the sample data instead of the total population.

Variance and Standard Deviation

Both variance and standard deviation are measure of the scatter of data and they are independent on the data size or the mean of data. Standard deviation is the square root of the variance of a given data set.
Standard deviation, $\sigma$ = $\sqrt{V}$For example, If standard deviation of a data set is given to be $1.87$, then what will be its variance?

Since, variance is the square of the standard deviation, the variance will be $(1.87)^2$ which will be $3.4969$.