Enter one data set and choose whether it is a sample or the complete population. The calculation keeps the squared deviations and denominator visible.
Result
Sample variance: 4
s2=3−18=4
Conditions
Every observation must be a finite real number measured on a common quantitative scale.
Sample variance requires at least two observations and divides by n minus 1.
Population variance divides by N only when the entered values are the complete population of interest.
Variance has squared measurement units; its nonnegative square root is the corresponding standard deviation.
Steps
Calculate the sample mean The three values total 12.xˉ=12/3=4
Find and square each deviation The deviations are negative 2, 0, and 2.(−2)2+02+22=8
Use the sample denominator Because the data are identified as a sample, divide by n minus 1.s2=3−18=4
Cross-check with standard deviation The positive square root returns spread in the original units.s=4=2
Deviations from the sample mean for the default data
Value
Deviation
Squared deviation
2
-2
4
4
0
0
6
2
4
✓
Independent check
Multiplying the sample variance 4 by n minus 1, which is 2, recovers the squared-deviation total 8. Squaring the standard deviation 2 returns the same variance.
Scope
What this variance calculator covers
For 2, 4, and 6, the mean is 4 and the squared deviations total 8. Population variance is 8/3, while sample variance is 8/2, or 4.
Sample variance
Use deviations from the sample mean and the n minus 1 denominator.
Examples: 2, 4, 6, 10, 12
Population variance
Use deviations from the population mean and the N denominator for a complete population.
Examples: 2, 4, 6, -3, 0, 3
Deviation and sum-of-squares table
Keep every observation, signed deviation, and squared deviation visible before division.
Examples: x_i-\bar{x}, (x_i-\bar{x})^2
Standard-deviation cross-check
Take the nonnegative square root of variance and square it back to verify the pair.
Examples: s=\sqrt{s^2}, \sigma=\sqrt{\sigma^2}
How to use it
Enter enough information for one clear task
1
Enter the observations
Use the raw finite values and include repetitions. Summary statistics alone are not a substitute for the data list.
2
Choose sample or population
Base the choice on the role of the data, not on list length or the result you expect.
3
Inspect the denominator
Confirm n minus 1 for a sample or N for a complete population before interpreting the result.
4
Keep the units straight
Variance uses squared units. Use standard deviation when a spread measure in the original units is easier to interpret.
Worked inputs
Examples to try
Use these examples to recognize the method, compare equivalent forms, and check your own work.
Three-value sample
The squared deviations total 8; divide by 2.
2,4,6
Expected result
s2=4,s=2
The same values as a population
Divide the same squared-deviation total by N = 3.
2,4,6
Expected result
σ2=38,σ=38
Constant population
Every deviation from the mean is zero.
5,5,5,5
Expected result
σ2=0,σ=0
Two-value sample
The mean is 11 and the squared deviations total 2; divide by 1.
10,12
Expected result
s2=2,s=2
Population centered at zero
The squared deviations total 18; divide by 3.
−3,0,3
Expected result
σ2=6,σ=6
Decimal population
The mean is 1.5 and the squared deviations total 0.5.
1.0,1.5,2.0
Expected result
σ2=61≈0.1667
Complete example
Calculate sample and population variance from the same data
The squared-deviation numerator is identical for both roles. The interpretation of the observations determines which denominator answers the intended question.
2,4,6
1
Find the center
The total 12 divided by 3 gives 4.
xˉ=μ=4
2
Calculate the common numerator
Square the deviations negative 2, 0, and 2.
SS=4+0+4=8
3
Apply each denominator
Use 2 for a sample and 3 for a population.
s2=8/2=4,σ2=8/3
4
Return to original units
Take the nonnegative square roots for standard deviation.
s=2,σ=8/3
s2=4 if sampled,σ2=38 if complete population
Verification: Both denominators return the same numerator when multiplied back: 4 times 2 equals 8, and 8/3 times 3 equals 8.
Avoidable errors
Common mistakes and how to fix them
Choosing the denominator by list length
Problem: Using the population formula because the list looks large enough.
Why it matters: Sample or population is a statement about the data source and target group, not the number of entries.
Better approach: Decide whether the entered values are the entire population of interest before calculating.
Forgetting to square deviations
Problem: Adding signed deviations and dividing.
Why it matters: Deviations from their arithmetic mean sum to zero, which would erase spread.
Better approach: Square each deviation before adding.
Confusing variance with standard deviation
Problem: Reporting 4 as both variance and standard deviation.
Why it matters: Standard deviation is the nonnegative square root of variance.
Better approach: Keep squared-unit variance and original-unit standard deviation under separate labels.
Rounding the mean too early
Problem: Rounding the center before calculating deviations.
Why it matters: Every squared deviation then inherits the rounding error.
Better approach: Keep full precision through the variance calculation and round only the displayed result.
Trust the result for the right reasons
Checks, assumptions, and limits
How results are checked
The observation count, sum, and mean are recomputed from the full input.
Signed deviations are checked to sum to zero within numeric tolerance.
The displayed denominator is multiplied by variance to recover the sum of squared deviations.
The reported standard deviation is squared to recover variance within displayed rounding.
When to stop and revise the input
A one-value sample has no sample variance because n minus 1 is zero.
Variance is sensitive to extreme observations because deviations are squared.
The numerical calculation cannot determine whether a sample is representative or observations are independent.
Values whose squared deviations exceed finite browser arithmetic are rejected instead of displaying infinity.
Common questions
Calculate variance with the correct denominator FAQ
What is the difference between sample and population variance?
Population variance describes the complete population and divides by N. Sample variance is the usual estimator from sampled observations and divides by n minus 1.
Can variance be negative?
No. Variance is a sum or average of squared deviations, so it is always zero or positive.
Why are variance units squared?
Each deviation has the original measurement unit and is squared before averaging. Taking the square root produces standard deviation in the original unit.
Why does one extreme value change variance so much?
Variance squares distances from the mean. A value twice as far from the mean contributes four times the squared deviation.
Sources and curriculum alignment
This page follows standard introductory mathematics notation and learning sequences. Use these references to continue with a complete course treatment.
