Formula for Population Variance (Table of Contents)
What is Population Variance Formula?
Population variance is a measure of the spread of population data. Hence, population variance can be defined as the average of the distances from each data point in a particular population to the mean, squared and it indicates how data points are spread out in the population. Population variance is an important measure of dispersion used in statistics. Statisticians calculate variance to determine how individual numbers in a data set relate to each other.
While calculating population variance, the dispersion is calculated with reference to the population mean. Hence, we have to find out the population mean in order to calculate population variance. One of the post popular notifications of the population variance is σ2. This is pronounced as sigma squared.
Population variance can be calculated by using the following formula:
where
- σ2 is population variance,
- x1, x2, x3,…..xn are the observations
- N is the number of observations,
- µ is the mean of the data set
Explanation of the Population Variance Formula
The formula for population variance can be calculated by using the following five simple steps:
Step 1: Calculate the mean (µ) of the given data. In order to calculate the mean, add all the observations and then divide that by the number of observations (N).
Step 2: Make a table. Please note that constructing a table is not compulsory, but presenting it in a tabular format would make the calculations easier. In the first column, write each observation (x1, x2, x3,…..xn).
Step 3: In the second column, write the deviation of each observation from the mean (xi – µ).
Step 4: In the third column, write the square of each observation from the mean (xi – µ)2. In other words, square each of the numbers obtained in column 2.
Step 5: Subsequently we need to add the numbers obtained in the third column. Find the sum of the squared deviations and divide the sum so obtained by the number of observations (N). This will help us to obtain which is the population variance.
Examples of Population Variance Formula (with Excel Template)
Let’s see some simple to advanced examples of population variance equation to understand it better.
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Example #1
Calculate the population variance from the following 5 observations: 50, 55, 45, 60, 40.
Solution:
Use the following data for the calculation of population variance.
There are a total of 5 observations. Hence, N=5.
µ=(50+55+45+60+40)/5 =250/5 =50
So, the Calculation of population variance σ2 can be done as follows-
σ2 = 250/5
Population Variance σ2 will be-
Population Variance (σ2 ) = 50
The population variance is 50.
Example #2
XYZ Ltd. is a small firm and consists of only 6 employees. The CEO believes that there should not be high dispersion in the salaries of these employees. For this purpose, he wants you to calculate the variance of these salaries. The salaries of these employees are as under. Calculate the population variance of the salaries for the CEO.
Solution:
Use the following data for the calculation of population variance.
There are a total of 6 observations. Hence, N=6.
=(30+27+20+40+32+31)/6 =180/6 =$ 30
So, the Calculation of population variance σ2 can be done as follows-
σ2 = 214/6
Population Variance σ2 will be-
Population Variance (σ2 ) = 35.67
The population variance of the salaries is 35.67.
Example #3
Sweet Juice Ltd manufactures different flavors of juice. The Management Department purchases 7 big containers for storing this juice in the factory. The Quality Control Department has decided that it will reject the containers if the variance of the containers is above 10. Given are the weights of 7 containers in kg: 105, 100, 102, 95, 100, 98 and 107. Please advise the Quality Control Department on whether it should reject the containers.
Solution:
Use the following data for the calculation of population variance.
There are a total of 7 observations. Hence, N=7
=(105+100+102+95+100+98+107)/7 =707/7 =101
So, the Calculation of population variance σ2 can be done as follows-
σ2 = 100/7
Population Variance σ2 will be-
Population Variance (σ2 )= 14.29
Since the variance (14.29) is more than the limit of 10 decided by the Quality Control Department, the containers should be rejected.
Example #4
The management team of a hospital named Sagar Healthcare recorded that 8 babies had been born in the first week of March 2019. The doctor wanted to evaluate the health of the babies as well as the variance of the heights. The heights of these babies are as follows: 48 cm, 47 cm, 50 cm, 53 cm, 50 cm, 52 cm, 51 cm, 60 cm. Calculate the variance of the heights of these 8 babies.
Solution:
Use the following data for the calculation of population variance.
So, the Calculation of population variance σ2 can be done as follows-
In Excel, there is an inbuilt formula for population variance that can be used to calculate the population variance of a group of numbers. Select a blank cell and type this formula =VAR.P(B2:B9). Here, B2:B9 is the range of cells you want to calculate the population variance from.
Population Variance σ2 will be-
Population Variance (σ2 ) = 13.98
Relevance and Use
Population variance is used as a measure of dispersion. Let us consider two population sets with the same mean and number of observations. Data set 1 consists of 5 numbers – 55, 50, 45, 50 and 50. Data set 2 consists of 10, 50, 85, 90 and 15. Both the data sets have the same mean, which is 50. But, in data set 1, the values are close to each other while data set 2 has dispersed values. The variance gives a scientific measure of this closeness/dispersion. Data set 1 has a variance of only 10 while data set 2 has a huge variance of 1130. Thus, a large variance indicates that the numbers are far from the mean and from each other. A small variance indicates that the numbers are close to each other.
Variance is used in the field of portfolio management while carrying out asset allocation. Investors calculate the variance of asset returns to determine optimal portfolios by optimizing the two major parameters – return and volatility. Volatility measured by variance is a measure of the risk of a particular financial security.
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