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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
xi | (xi -x̄) | (xi – x̄)2 |
24 | -26 | 676 |
53 | 3 | 9 |
53 | 3 | 9 |
36 | -14 | 196 |
21 | -29 | 841 |
84 | 34 | 1156 |
64 | 14 | 196 |
34 | -16 | 256 |
77 | 27 | 729 |
54 | 4 | 16 |
μx = Σxi/10=500/10=50 units |
| σx = Σ(xi −x̄)2/10=4084/10= 408.4 units2 |
Therefore, the variance of the particular data is 408.4 units2,
Example 2: Find the population variance of the given given data set
X | 21 | 42 | 37 | 16 | 31 | 28 | 33 | 41 | 12 |
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
