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.

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

$V$ = $\frac{\sum (X-\mu )^{2}}{N}$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

$S$ = $\frac{\sum (X - \mu )^{2}}{N - 1}$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

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 $3.4969$.