# Variance

Variance is another absolute measure of dispersion. It is defined as the average of the squared difference between each of the observations in a set of data and the mean. For sample data the variance is denoted by ${S^2}$ and the population variance is denoted by ${\sigma ^2}$(sigma square).

The sample variance ${S^2}$ has the formula:
${S^2} = \frac{{\sum {{\left( {X – \overline X } \right)}^2}}}{n}$

Here $\overline X$ sample is the mean andb$n$ is the number of observations in the sample.

The population variance ${\sigma ^2}$ is defined as:
${\sigma ^2} = \frac{{\sum {{\left( {X – \mu } \right)}^2}}}{N}$

Here$\mu$ is the mean of the population and $N$ is the number of observations in the data. It may be remembered that the population variance ${\sigma ^2}$ is usually not calculated. The sample variance ${S^2}$ is calculated and if need be, this ${S^2}$ is used to make inferences about the population variance.

The term $\sum {\left( {X – \overline X } \right)^2}$ is positive; therefore ${S^2}$ is always positive. If the original observations are in centimeters, the value of the variance will be (centimeter)2. Thus the unit of ${S^2}$ is the square of the units of the original measurement.

For a frequency distribution the sample variance ${S^2}$ is defined as:
${S^2} = \frac{{\sum f{{\left( {X – \overline X } \right)}^2}}}{{\sum f}}$

For a frequency distribution the population variance ${\sigma ^2}$ is defined as:
${\sigma ^2} = \frac{{\sum f{{\left( {X – \mu } \right)}^2}}}{{\sum f}}$

In simple terms we can say that variance is the square of the standard deviation.
${\text{Variance = (Standrad Deviation}}{{\text{)}}^{\text{2}}}$

Example:

Calculate the variance for the following sample data: 2, 4, 8, 6, 10, and 12.

Solution:

 $X$ ${\left( {X – \overline X } \right)^2}$ $2$ ${(2 – 7)^2} = 25$ $4$ ${(4 – 7)^2} = 9$ $8$ ${(8 – 7)^2} = 1$ $6$ ${(6 – 7)^2} = 1$ $10$ ${(10 – 7)^2} = 9$ $12$ ${(12 – 7)^2} = 25$ $\sum X = 42$ $\sum {\left( {X – \overline X } \right)^2} = 70$

$\overline X = \frac{{\sum X}}{n} = \frac{{42}}{6} = 7$
${S^2} = \frac{{\sum {{\left( {X – \overline X } \right)}^2}}}{n}$
${S^2} = \frac{{70}}{6} = \frac{{35}}{3} = 11.67$
${\text{Variance = }}{S^2} = 11.67$

Example:

Calculate variance from the following distribution of marks:

 Marks No. of Students $1 – 3$ $40$ $3 – 5$ $30$ $5 – 7$ $20$ $7 – 9$ $10$

Solution:

 Marks $f$ $X$ $fX$ ${\left( {X – \overline X } \right)^2}$ $f{\left( {X – \overline X } \right)^2}$ $1 – 3$ $40$ $2$ $80$ $4$ $160$ $3 – 5$ $30$ $4$ $120$ $0$ $0$ $5 – 7$ $20$ $6$ $120$ $4$ $80$ $7 – 9$ $10$ $8$ $80$ $16$ $160$ Total $100$ $400$ $400$

$\overline X = \frac{{\sum fX}}{{\sum f}} = \frac{{400}}{{100}} = 4$
${S^2} = \frac{{\sum f{{\left( {X – \overline X } \right)}^2}}}{{\sum f}} = \frac{{400}}{{100}} = 4$
${\text{Variance = }}{S^2} = 4$