# Variance and Standard Deviation

Sub Topics

Variance and standard deviation of a data are closely associated. They measure how the items in a data is spread out, means the measure of variability in a data.

The variance in a data is the mean of the squares of the deviation of each item from the mean of the data. The standard deviation in a set is just the square root of the variance. The Greek letter s is used as a symbol for these. The variance is denoted as s2 and the standard deviation is denoted by s.

## Formula for Variance and Standard Deviation

Consider a data with a set of numbers 1, 2, 3, 4, 5.

The mean of the set is 3.

The deviation of items from the mean are (1 - 3), (2 - 3), (3 - 3), (4 - 3) and (5 - 3) or -2, -1, 0, 1 and 2. Suppose you want to compute, for the purpose of studying the average deviation of the whole set, you will end up with a misleading conclusion that on an average there is no deviation between the items and the mean of the set. Because, $\frac{[(-2) + (-1) + (0) + (1) + (2)]}{5}$ = $\frac{0}{5}$ = 0.

To give realistic picture, mathematicians brought out the concept of finding the mean of the squares of the deviation and the same is called variance.

Now let us calculate the variance in the same set instead of average variation.

s2 = [(-2)2 + (-1)2 + (0)2 + (1)2 + (2)2]/5 = $\frac{(4 + 1 + 0 + 1 + 4)}{5}$ = $\frac{10}{5}$ = 2

The standard deviation s is therefore +2, which is about 1.414. The numbers 2, 3, 4 are within one standard deviation formula from the mean which is about 60%

Now let us calculate the variance in the set of numbers 3, 4, 5, 6, 7, 8, 9

The mean is 6

s2 = [(-3)2 + (-2)2 + (-1)2 + (0)2 + (1)2 + (2)2 + (3)2]/7 = $\frac{(9 + 4 + 1 + 0 + 1 + 4 + 9)}{7}$ = 4

The standard deviation s is therefore +4, which is 2. The numbers 4, 5, 6, 7, 8 are within one standard deviation from the mean which is about 71%

## Mean Variance and Standard Deviation

The earlier section gives an idea how a standard deviation varies from the mean of the data. considering a normal distribution and conducting a number of sample experiments, mathematicians have arrived at the following conclusion.

In a normal distribution, about 68% of the scores are within one standard deviation of the mean and about 95% of the scores are within two standard deviations of the mean.

The above conclusion helps a lot in estimating some statistical figures. It is explained through an illustrated example.

## Variance and Standard Deviation Examples

Below you could see variance and standard deviation example