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Population Variance Formula
Before learning the population variance formula, let us recall what is population variance. Population variance (σ^{2}) is the squared variation of all values (X_{i}) of a random variable (X) from the population mean (μ) over the whole population. This formula lets us measure the spread of random variables from the population mean. Let us learn the population variance formula in the section below.
What Is Population Variance Formula?
The population variance formula is stated below:
Population Variance (σ) = \( \dfrac{1}{N} \sum^N_{i=1} (x_i  \mu)^2 \)
where
 N is the population size
 x_{i} is the i^{th} value
 μ is the population mean.
Let us see the applications of the population variance formula in the following section.
Solved Examples Using the Population Variance Formula

Example 1. Given the following population data, find its population variance.
X 21 42 37 16 31 28 33 41 12 Solution.
Population Mean = \( \frac{21+42+37+16+31+28+33+41+12}{9} \) = 261/9 = 29 units
Using the population variance formula,
Population Variance = \( \frac{(21  29)^2+(42  29)^2+(37  29)^2+(16  29)^2+(31  29)^2+(28  29)^2+(33  29)^2+(41  29)^2+(12  29)^2}{9} \) = 102.22 units^{2}
Answer: Population variance of the given dataset is 102.22 units^{2}

Example 2: Find the population variance of the age of people having ages: 20, 32, 25, 44, 19
Solution:
Number of people = 5
The mean of the population is (20 + 32 + 25 + 44 + 19)/5 = 140/5 = 28 years
Using the population variance formula,
Population Variance = ((20  28)^{2} + (32  28)^{2} + (25  28)^{2} + (44  28)^{2} + (19  28)^{2})/5 = 85.2 years^{2}
Answer: The population variance of the given data is 85.2 years^{2}
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