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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 s^{2} and the standard deviation is denoted by s.

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.

s^{2 }= [(-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

s^{2 }= [(-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%

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.