The 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 a sample data the variance is denoted 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}

Where \overline X sample is mean andn 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}

Where\mu is the mean of the population and Nis 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 inference 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 centimeter, 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 words we can say that variance is the square of 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