# 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:

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

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

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