Variance Formula

Variance Formula

In probability and statistics, variance is defined as the expected value of a random variable’s squared variation from its mean value.mInformally, variance calculates how far apart a set of data (random) is all from their own mean value.

Variance is a measure of how different data points are from the mean. A variance, according to Layman, is a way of measuring just how far a dataset (numbers) is spread out from its mean (average) value. The term “variance” refers to determining the expected difference in deviation from the actual value. As a result, variance is defined as the standard deviation of a given data set. The greater the variance, the greater the scatter from the mean, and the lower the variance, the less the scatter from the mean. As a result, it is known as a measure of data spread from the mean.

Variance Formula

S² = ∑(x₁ – x̄)²/(n-1)

Here, 

S² = Sample Variance,

x₁= value of one observation

x̄ = mean value of all observations

n = number of observations 

Solved Examples

Example 1: Find the variance of the following data set: 24, 54, 53, 36, 21, 84, 64, 34, 77, 53

Solution:

x̄ = (24+54+53+36+21+84+64+34+77+53)/10

   = 500/10

   = 50

Therefore, the variance of the particular data is 408.4 units²,

Example 2: Find the population variance of the given given data set

Solution:

Mean of the population = (21+42+37+16+31+28+33+41+12)/9 = 261/9 = 29

Using the formula of population variance,

[(21-29)²+(42-29)²+(37-29)²+(16-29)²+(31-29)²+(28-29)²+(33-29)²+(41-29)²+(12-29)²]/9

= 920/9 = 102.22 unit²

Therefore, the population variance of the data set is 102.22 unit²