107 LearnersLast updated on June 12th, 2025
Population Variance
Population variance is a mathematical measure that describes how data points in an entire population are distributed relative to the population mean. The population variance reflects the overall variability within the dataset. Let us now see more about population variance and how it is calculated.
What is Population Variance?
Population variance is a measure that shows how distributed the values are in a group. It's determined by calculating the average of the squared differences between the mean and each value. A high variance indicates that the values are well distributed, while a low variance means that they are close to the mean. We use the formula given below to calculate the population variance:
<formula>
Where,
- is the population variance
- xi is each individual data point
- is the population mean
- N is the total number of data points in the entire population
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Difference between Population Variance and Sample Variance
There are a lot of differences between population variance and sample variance. Some of them are given below:
| Population Variance | Sample Variance |
|
The population variance measures the spread of all the data points in an entire population |
The sample variance measures the spread of data points in a sample from the population |
|
The population variance uses all the data points from the population |
The sample variance uses only a subset or a selected sample from the population |
| The formula used to calculate population variance is: <formula> |
The formula to calculate the sample variance is: <formula> |
|
We use population variance when the data for the entire population is available |
We use sample variance to analyze a sample to estimate population characteristics |
Common mistakes and How to Avoid them in Population Variance
When working on problems of population variance, students tend to make mistakes. Here are some common mistakes and how to avoid them
Mistake 1
Mixing up the Formulas
Students must practice writing the formulas of population variance and sample variance. They should memorize the formulas to avoid confusion between population variance and sample variance.
Mistake 2
Not Squaring the Deviations
Students must always remember to square each deviation before summing up the values.
Mistake 3
Errors in Calculating the Mean
Students must always use the right mean (average) and not miss any number while calculating the mean. They must also remember not to round the mean too soon, as it can cause incorrect variance results.
Mistake 4
Misinterpreting the Meaning of Variance
Students should remember that variance is expressed in squared units and standard deviation is in the original units of the data.
Mistake 5
Using Incorrect Data Units
Students must remember to convert all the units to the same unit before calculating the answer.
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Real life applications of Population Variance
There are a lot of ways in which we apply population variance in our day-to-day life. Let us now see in what fields and applications we use population variance:
- Healthcare:
We use population variance in healthcare to analyze the variance in the recovery time for patients undergoing the same treatment. It is also used to measure variance in blood pressure, and track infection rate variance to identify hotspots.
- Finance and Investments:
Population variance is used to measure the variance of stock returns, assess investment volatility, diversify assets, and minimize overall risks. It is also used to calculate variance in claim amounts to set premium rates accurately.
- Education:
We use population variance in education to assess the variance in student scores to identify consistent or varied performance, it is also used to compare variance in student outcomes across different teaching methods and to measure variance in graduation rates across districts to assess educational inequality.
Solved examples on Population Variance
Problem 1
Population A: {5, 7, 9} Population B: {3, 5, 7} Find the difference between their means.
The difference between their means is 2
Explanation
Mean of A:
μA = = 7
Mean of B:
μB = = 5
Difference:
Δμ = 7 – 5 = 2
Problem 2
Population A: {10, 20, 30, 40} Population B: {15, 25, 35, 45} Determine the difference between their means.
The difference between their means is -5
Explanation
Mean of A:
μA = = 25
Mean of B:
μB = = 30
Difference:
Δμ = 25 - 30 = -5
Problem 3
Population A: {2, 4, 6, 8, 10} Population B: {1, 3, 5, 7, 9} Determine the difference between their means.
The difference between their means is 1
Explanation
Mean of A:
μA = (2 + 4 + 6 + 8 + 10)/5 = 30/5 = 6
Mean of B:
μB = (1 + 3 + 5 + 7 + 9)/5 = 255 = 5
Difference:
Δμ = 6 - 5 = 1
Problem 4
Population A: {100, 110, 120} Population B: {90, 100, 110} Determine the difference between their means.
The difference between their means is 10
Explanation
Mean of A:
μA = (100 + 110 + 120)/3 = 330/3 = 110
Mean of B:
μB = (90 + 100 + 110)/3 = 300/3 = 100
Difference:
Δμ = 110 - 100 = 10
Problem 5
Population A: {5, 10, 15, 20} Population B: {2, 4, 6, 8} Determine the difference between their means.
The difference between their means is 7.5
Explanation
Mean of A:
μA = (5 + 10 + 15 + 20)/4 = 50/4 = 12.5
Mean of B:
μB = (2 + 4 + 6 + 8)/4 = 20/4 = 5
Difference:
Δμ = 12.5 - 5 = 7.5
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FAQs on Population Variance
1.What is Population Variance?
2.What is the formula of population variance?
3.What does population variance tell us about a dataset?
4.Can the population variance be zero?
5.Is it possible for population variance to be negative?
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Jaipreet Kour Wazir
About the Author
Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref
Fun Fact
: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!
