Variance Formula

In probability theory and statistics, variance measures how far a set of numbers are spread out. It is a numerical value used to indicate how widely individuals in a group vary. If individual observations vary considerably from the group mean, the variance is big; and vice versa.

A variance of zero indicates that all the values are identical. Variance is always non-negative: a small variance indicates that the data points tend to be very close to the mean and hence to each other while a high variance indicates that the data points are very spread out around the mean and from each other.

The variance of a population is denoted by $\sigma^{2}$ and the variance of a sample by $s^{2}$ .

The variance of a population is defined by the following formula:

$\huge \sigma^{2}=\frac{\sum\left(x-\overline{x}\right)^{2}}{n}$

The variance of a sample is defined by slightly different formula:

$\huge S^{2}=\frac{\sum\left(x-\overline{x}\right)^{2}}{n-1}$

Where,
$\sigma^{2}$ = Variance
x = Item given in the data
$\overline{x}$ = Mean of the data
n = Total number of items.
$s^{2}$ = Sample variance

Solved examples of Variance

Question: Find the variance for the following set of data representing trees heights in feet: 3, 21, 98, 203, 17, 9

Solution:

Step 1: Add up the numbers in your given data set.

3 + 21 + 98 + 203 + 17 + 9 = 351

Step 2: Square your answer:

351 × 351 = 123201

…and divide by the number of items. We have 6 items in our example so:

$\large \frac{123201}{6}$ =20533.5

Step 3: Take your set of original numbers from Step 1, and square them individually this time:

3 × 3 + 21 × 21 + 98 × 98 + 203 × 203 + 17 × 17 + 9 × 9

Add the squares together:

9 + 441 + 9604 + 41209 + 289 + 81 = 51,633

Step 4: Subtract the amount in Step 2 from the amount in Step 3.

51633 – 20533.5 = 31,099.5

Set this number aside for a moment.

Step 5: Subtract 1 from the number of items in your data set. For our example:

6 – 1 = 5

Step 6: Divide the number in Step 4 by the number in Step 5. This gives you the variance:

$\large \frac{31099.5}{5}$ = 6219.9

Step 7: Take the square root of your answer from Step 6. This gives you the standard deviation:

$\large \sqrt{6219.9}$ = 78.86634

The answer is 78.86

More topics in Variance Formula
Covariance Formula Coefficient of Variation Formula

Practise This Question

Which the following are inhibitory hormones from the hypothalamus?

1. GnRH

2. LH

3. PIH

4. Somatostatin