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Sample Variance (s²)
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Mean (x̄)
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Std. Deviation
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Count (n)
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Sum (Σx)
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Minimum
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Maximum
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Range
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Sum of Squares
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Coeff. of Variation
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What Is a Population Variance Tool?
A population variance calculator computes σ² — the exact measure of spread when you have every single data point from the entire population you are studying. Unlike sample variance, which divides by n − 1 to correct for estimation bias, population variance divides by N because there is no estimation involved — you have the complete data set. This calculator is used when data covers an entire class of students, every product in a batch, a full census, or any other scenario where no value is missing from the group of interest.
Formula
Population variance uses the full count N as the divisor because no correction is needed.
Population Variance
σ² = Σ(xᵢ − μ)² / N
σ²Population variance
xᵢEach data value in the population
μPopulation mean (true mean of all values)
NTotal count of values in the population
How to Calculate Population Variance Step by Step
Confirm you have the entire population.Population variance is only correct when your data set includes every member of the group you are studying.
Find the population mean (μ).Add all values and divide by N.
Compute deviations.Subtract μ from each value: (xᵢ − μ).
Square each deviation.Calculate (xᵢ − μ)².
Sum the squared deviations.Total all squared values: Σ(xᵢ − μ)².
Divide by N.σ² = Σ(xᵢ − μ)² / N. This is the population variance.
Practical Population Variance Examples
1All Students in a Class
Data:78, 82, 91, 67, 85, 73, 88, 79
Solution:Mean μ = 80.375. Sum of squared deviations = 444.875.
Calculate the exact variance of your complete data set in seconds:
Enter all values from your population into the input field.
The calculator returns population variance (σ²), population standard deviation (σ), and the mean.
Check the step-by-step breakdown to see individual deviations and squared deviations.
Use the population variance result directly — no Bessel's correction needed.
Benefits of Using a Population Variance Calculator
Gives the exact variance without estimation — accurate when you have complete data.
Prevents the mistake of applying Bessel's correction when it is not appropriate.
Shows the population standard deviation (σ) alongside variance for dual interpretation.
Handles large data sets instantly — no need for spreadsheet formulas.
Step-by-step output is ideal for academic assignments and quality reports.
Where Population Variance Is Commonly Used
Education
A teacher measuring score spread for an entire class uses population variance.
Census Data
Government statisticians working with complete national census records.
Batch Manufacturing
When every unit in a production lot is measured, population variance gives the exact spread.
Small Organizations
A startup analysing performance metrics for all 10 employees has the full population.
Complete Surveys
When every member of a defined group has responded, population formulas apply.
Population vs Census Data: Important Distinctions
Many users assume that any large data set counts as a "population." Understanding when to use σ² versus s² avoids systematic errors:
A population is defined by your research question
If you study "all students in Section A," then Section A's grades are the population. If you study "all students at the university," Section A is just a sample.
Census ≠ population for every question
A national census is a population for demographic questions but a sample if you use it to infer trends about future years.
Time-series data is usually a sample
Last year's monthly sales are a sample from the theoretical distribution of all possible monthly sales. Population variance would underestimate the true spread.
When in doubt, use sample variance
Bessel's correction (n − 1) only slightly increases the result but protects against underestimating variance. It is the safer default.
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Frequently Asked Questions — Population Variance Calculator
Everything you need to know about variance.
What is population variance?
Population variance (σ²) measures the exact spread of data points from the population mean (μ). The formula is σ² = Σ(xᵢ − μ)² / N, where N is the total number of data points in the population. Unlike sample variance, it divides by N (not N−1) because you have all the data — no estimation or correction is needed.
When should I use population variance?
Use population variance when your data set includes every member of the population you're studying. Examples: all students in a class, every product in a batch, complete census data, all transactions in a time period. If you have a sample (subset) from a larger population, use sample variance instead. The key question is: does your data represent the complete group?
What is the population variance of 2, 4, 4, 4, 5, 5, 7, 9?
For 2, 4, 4, 4, 5, 5, 7, 9: Mean (μ) = 40/8 = 5. Deviations: −3, −1, −1, −1, 0, 0, 2, 4. Squared: 9, 1, 1, 1, 0, 0, 4, 16. Sum of squares = 32. Population variance (σ²) = 32/8 = 4. Population standard deviation (σ) = √4 = 2.
What is the difference between σ² and s²?
σ² (population variance) divides by N and uses the population mean μ. s² (sample variance) divides by n−1 and uses the sample mean x̄. The sample variance is always slightly larger than the population variance for the same data set. This difference (Bessel's correction) matters most for small data sets and becomes negligible for large N.
Can population variance be zero?
Yes, population variance equals zero when all values are identical. If every data point in the population has the same value (e.g., {7, 7, 7, 7}), then every deviation from the mean is zero, making σ² = 0. This indicates zero variability — no spread at all. Any difference among values produces a positive variance.
How to find population variance in Excel?
In Excel, use =VAR.P(range) for population variance. Example: =VAR.P(A1:A20). For population standard deviation, use =STDEV.P(range). In Google Sheets, the same functions work. In Python: np.var(data) (default is population variance in NumPy). In R: var(x) * (n-1)/n because R's var() defaults to sample variance.