A measure of central tendency like mean, median or mode give a single value as the representative of the whole data set. A single central value is not sufficient to describe the data set clearly. Suppose we find the average annual income of two localities to be equal. From this, we cannot conclude that the individual incomes in the two localities are distributed in a similar manner around the mean. In one locality the disparities between the incomes of the families may be large and in another all the families may earn more or less the same amount. Hence to understand how the data values are spread around the central value we need a measure of dispersion or variation. Variance is one such measure which convey the spread of data set. Variance is calculated using all the observations in the data set and indeed the most meaningful measure of dispersion used both in descriptive and inferential statistics.

The variance is the arithmetic mean of the squares of distance or deviation of each value from the mean.
The symbol σ2 is used to denote the population variance while s2 is used to represent sample variance.The squared deviations are used rather than the actual differences, because the sum of actual deviations of all values from the mean is zero. Since squared deviations are always positive, the variance is a positive measure for all data distributions.
The formula used to compute population variance is

σ2 = $\frac{\sum (X-\mu )^{2}}{N}$Where X represents an individual value, μ the population mean and N the population size.

The formula used for computing the sample variance is

S2 = $\frac{\sum (X-\overline{X})^{2}}{n-1}$
Where X represents an individual value of the sample, X sample mean and n the sample size.

When the sample is considered as the population itself, then the formula given for population variance itself is used.

The variance of a random variable X is given by,

Var(X) = E(X2) - μ2.
Where E(X2) is the expectation of X2 and μ is the mean of random variable X which equal to E(X).
Standard deviation is the positive square root of the variance. The symbols σ and S are used correspondingly to represent population and sample standard deviations. The corresponding formulas are hence,

Population standard deviation σ = $\sqrt{\frac{\sum (X-\mu )^{2}}{N}}$ and

Sample standard deviation S = $\sqrt{\frac{\sum (X-\overline{X})^{2}}{n-1}}$
The variance var(X) of a random variable X has the following properties.
  1. Var(X + C) = Var(X), where C is a constant.
  2. Var(CX) = C2.Var(X), where C is a constant.
  3. Var(aX + b) = a2.Var(X), where a and b are constants.
  4. If X1, X2,...Xn are n independent random variables, then Var(X1 + X2 +......+ Xn) = Var(X1) + Var(X2) +........Var(Xn).
In order to apply the formula given for calculating the variance, a table is made to make the calculations easier.

Solved Example

Question: Find the variance of the numbers 3, 8, 6, 10, 12, 9, 11, 10, 12, 7.
Step 1:

Compute the mean of the 10 values given.

x = $\frac{3+8+6+10+12+9+11+10+12+7}{10}$ = $\frac{88}{10}$ = 8.8

Step 2: Make a table with three columns, one for the X values, the second for the deviations and the third for squared deviations.

  X - X 
  3   -5.8    33.64 
  8   -0.8    0.64
  6   -2.8    7.84
 10    1.2
 12    3.2   10.24
   0.2     0.04
 11    2.2
  10    1.2
  -1.8    3.24
 Total     0

Step 3:

As the data is not given as sample data we use the formula for population variance.

σ2 = $\frac{\sum (X-\bar X )^{2}}{N}$   (Here $\mu = \bar X$)

     = $\frac{73.6}{10}$ = 7.36