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

$$100$$


$$400$$


$$400$$

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