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:

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

Example:

Calculate variance from the following distribution of marks:

Solution:

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