Population Variance
Population variance is a type of variance that is used to determine the variability of the population data with respect to the mean. The square root of the population variance will give the population standard deviation. It is an absolute measure of dispersion that indicates the spread of data in a data set.
There are two types of data available, namely, ungrouped and grouped data. Thus, there are two formulas to calculate the population variance. In this article, we will learn more about population variance, its formulas, and various associated examples.
1.  What is Population Variance? 
2.  Population Variance Formula 
3.  Population Variance and Sample Variance 
4.  Population Variance and Standard Deviation 
5.  FAQs on Population Variance 
What is Population Variance?
Population variance is a measure of dispersion that determines how far each data point is from the population mean. Population variance can be defined as the average of the square of the deviations from the data's mean value. Population refers to each and every observation in a finite group. The population variance is calculated on the population. However, when the number of observations increases then a few data points are selected that can represent the entire population. These specific data points form a sample and the variance calculated on this data is called the sample variance. The sample variance can be used to estimate the population variance.
Population Variance Example
Suppose a data set is given as {3, 7, 11}. The mean is 7. Add the square of the distances of each data point from the mean to get 32. This value is divided by the total number of observations (3) to get 10.67. This is the population variance.
Population Variance Formula
In statistics, two kinds of data can be encountered. The first is ungrouped data. When the data has not been organized and remains in its raw form it is known as ungrouped data. The second is grouped data. When data is classified into categories and is presented in a methodical form, it is called grouped data. The population variance formulas for both types of data are given below:
 Ungrouped Data: \(\sigma ^{2}\) = \(\frac{\sum_{i=1}^{n}(x_{i}\mu)^{2}}{n}\)
 Grouped data: \(\sigma ^{2}\) = \( \frac{\sum_{i=1}^{n}f\left ( m_{i}\overline{x} \right )^{2}}{N}\)
n = total number of observations.
N = \(\sum_{i=1}^{n} f_{i}\)
f = the frequency of occurrence of an observation for grouped data
Mean for grouped data, \(\overline{x}\) = \(\frac{\sum_{i=1}^{n} m_{i}f_{i}}{\sum_{i=1}^{n} f_{i}}\)
Mean for ungrouped data, \(\mu = \frac{\sum_{i=1}^{n}x_{i}}{n}\)
\(m_{i}\) = Midpoint of the i^{th} interval.
Population Variance and Sample Variance
Population variance and sample variance are both measures of absolute deviation. The value of the population variance is lower than the sample variance. Furthermore, the sample variance, on average, should be almost equal to the population variance. The table gives the differences between the population variance and sample variance.
Population Variance  Sample Variance 
Population variance is calculated on the population data.  Sample data is used to calculate the sample variance. 
The value is not dependent on the research methods used, as it is a parameter of the population.  The value depends on the sampling practices and research techniques used. 
The formulas to calculate the population variance are \(\sigma ^{2}\) = \(\frac{\sum_{i=1}^{n}(x_{i}\mu)^{2}}{n}\) and \(\sigma ^{2}\) = \( \frac{\sum_{i=1}^{n}f\left ( m_{i}\overline{x} \right )^{2}}{N}\) 
The formulas to calculate sample variance are s^{2} = \(\frac{\sum_{i=1}^{n}(x_{i}\mu)^{2}}{n1}\) and s^{2} = \( \frac{\sum_{i=1}^{n}f\left ( m_{i}\overline{x} \right )^{2}}{N1}\) 
Population Variance and Standard Deviation
Standard deviation can be defined as the square root of the variance. Thus, if the square root of the population variance is taken it results in the standard deviation. The standard deviation represents the average distance from the population mean. The population variance does not yield information that can be used. Thus, the standard deviation is a better measure to check the spread of data points and interpret results. For example, suppose weights of a certain number of people are given. The mean is 150 kgs and the variance is 10000. The significance of the value of population variance cannot be understood easily. However, the population standard deviation is 100. This tells us that the weights of the people lie between 50 kgs (mean  100) to 250 kgs (mean + 100).
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Important Notes on Population Variance
 Population variance is computed on the population data and is used to measure the deviation of the data points from the mean of the population.
 The two formulas to calculate population variance are \(\sigma ^{2}\) = \(\frac{\sum_{i=1}^{n}(x_{i}\mu)^{2}}{n}\) (for ungrouped data) and \(\sigma ^{2}\) = \( \frac{\sum_{i=1}^{n}f\left ( m_{i}\overline{x} \right )^{2}}{N}\) (for grouped data).
 The population standard deviation can be obtained by taking the square root of the population variance.
Examples on Population Variance

Example 1: Find the population variance of the data set {12, 13, 12, 14, 19}
Solution: n = 5, \(\mu\)= (12 + 13 + 12 + 14 + 19) / 5 = 14
Using the formula, \(\sigma ^{2}\) = \(\frac{\sum_{i=1}^{n}(x_{i}\mu)^{2}}{n}\)
= \(\frac{(12  14)^{2} + (13 14)^{2} + (12 14)^{2} + (14  14)^{2}+ (19 14)^{2}}{5}\) = 6.8
Answer: Population variance = 6.8 
Example 2: Find the population variance for the following
\(x_{i}\) f 10  20 5 20  30 3 30  40 8 40  50 1
Solution:\(x_{i}\) f \(m_{i}\) (\(m_{i}  \overline{x}\))^{2} f(\(m_{i}  \overline{x}\))^{2} 10  20 5 15 167.44 837.22 20  30 3 25 8.64 25.93 30  40 8 35 49.84 398.75 40  50 1 45 291.04 291.04 \(\sum_{i=1}^{n} f_{i}\) = 17 \(\sum_{i=1}^{n}f\left ( m_{i}\overline{x} \right )^{2}\) = 1552.94
\(\overline{x}\) = \(\frac{\sum_{i=1}^{n} m_{i}f_{i}}{\sum_{i=1}^{n} f_{i}}\) = 27.94
Population Variance = \( \frac{\sum_{i=1}^{n}f\left ( m_{i}\overline{x} \right )^{2}}{N }\) = 1552.94 / 17 = 91.35
Answer: Population Variance = 91.35 
Example 3: Find the population standard deviation of the given data {12.3, 15.6, 7.9, 8.31, 2.59, 9.5, 10}
Solution: n = 7, \(\mu\)= (12.3 + 15.6 + 7.9 + 8.31 + 2.59 + 9.5 + 10) / 7 = 9.46
Population Variance, \(\sigma ^{2}\) = \(\frac{\sum_{i=1}^{n}(x_{i}\mu)^{2}}{n}\)
\(\sigma ^{2}\) = \(\frac{(12.39.46)^{2} + (15.69.46)^{2} + (7.99.46)^{2} + (8.31 9.46)^{2}+ (2.59 9.46)^{2}+ (9.59.46)^{2} + (109.46)^{2}}{7}\) = 13.86
Population Standard Deviation, \(\sigma\) = \(\sqrt{13.86}\) = 3.723
Answer: Population Standard Deviation = 3.723
FAQs on Population Variance
What is Population Variance in Statistics?
Population variance can be defined as the average of the squared deviations from the mean. The population data is used to calculate population variance.
What is the Formula for Population Variance?
The formulas for population variance are given as follows:
 Ungrouped Data: \(\sigma ^{2}\) = \(\frac{\sum_{i=1}^{n}(x_{i}\mu)^{2}}{n}\)
 Grouped data: \(\sigma ^{2}\) = \( \frac{\sum_{i=1}^{n}f\left ( m_{i}\overline{x} \right )^{2}}{N}\)
What is the Symbol of Population Variance?
Population variance is represented as sigma squared (\(\sigma ^{2}\)) while sample variance is denoted as s^{2}.
How to Find the Population Variance?
The steps to find the population variance are as follows:
 Find the mean of the given data.
 Subtract the mean from each observation.
 Add the sum of squares of the values obtained in the previous step.
 Divide the value from step 3 by the total number of observations (n) to get the population variance.
Is Population Variance the Same as Standard Deviation?
The square root of the population variance will give the standard deviation. The unit of standard deviation will be the same as the data set. However, as the population variance is squared hence, its unit will be different with respect to the data set.
Is Population Variance the Same as Sample Variance?
Population variance is calculated on the population data while sample variance is computed on the sample data. The sample variance is used to estimate the value of the population variance when the given data set is extremely large.
Is Population Variance a Random Variable?
Population variance is not a random variable. Population variance can be defined as the expectation of the squared deviation of a random variable from the mean.
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