Variance Calculator
Calculate variance and standard deviation for a set of numbers. Choose population variance (σ²) or sample variance (s²).
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How to use this calculator
Variance measures the average squared deviation from the mean. Population variance divides by N; sample variance divides by N−1 (Bessel's correction) to give an unbiased estimate.
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Enter your numbers separated by commas or new lines in the text area.
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Choose whether your data is a full population or a sample.
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Variance, standard deviation, mean, count, and range are calculated instantly.
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Sample variance uses N−1 in the denominator (Bessel's correction) for an unbiased estimate.
Frequently asked questions
How should I type the numbers in?
Separate values with commas, semicolons, spaces or new lines, in any mix, so "4.5; 5.1; 6.2" and one number per line both work. Use a decimal point. A decimal comma such as 3,5 is split into the two numbers 3 and 5. Always check the Count (N) row after a calculation to confirm it matches the number of values you meant to enter.
What happens to text or typos in the list?
Any token that is not a number is silently dropped instead of raising an error. The input 1,2,x,4 is processed as 1, 2, 4 with N = 3. The only warning is the Count row, so compare it with your expected count.
How many values does each variance type need?
Sample variance needs at least 2 values because it divides by N − 1, and shows an error otherwise. Population variance accepts a single value and returns 0, since one number has no spread. The Divisor row shows which of N or N − 1 was used.
Does the sample variance give an unbiased standard deviation too?
Only the variance. Dividing by N − 1 makes s² an unbiased estimate of the population variance, but its square root s slightly underestimates the population standard deviation on average. The effect is small beyond a handful of observations and is normally ignored.
Why is variance in odd-looking squared units?
Because each deviation is squared. Data in dollars give variance in dollars squared. Variance is still useful because variances of independent quantities add: Var(X + Y) = Var(X) + Var(Y). Standard deviations do not add this way. The calculator shows both so you can use whichever fits the task.
Is the calculation numerically reliable for large values?
Yes for typical use. It subtracts the mean first and then squares the differences, instead of using the sum-of-squares shortcut that can lose precision. The set 1000000, 1000001, 1000002 returns a sample variance of exactly 1. Outputs are rounded to six decimal places.
What can this calculator not do?
It does not accept frequencies or weights, grouped data, or a confidence interval for the variance, and it does not flag outliers. Because deviations are squared, one extreme value can dominate the answer, so look at Min, Max and Range alongside the variance.
Variance calculator: population or sample, from a pasted list
Choosing the divisor
The calculator finds the mean, squares every value’s distance from it, adds those squares and divides. You pick the divisor with the Variance type setting: N for a complete population, N − 1 for a sample. The same sum of squared deviations feeds both, and the result panel shows it so you can verify the arithmetic. It then reports the standard deviation, the count, the minimum, the maximum and the range.
| Data | Sum of squares | Population variance (÷N) | Sample variance (÷N−1) |
|---|---|---|---|
| 4.5, 5.1, 6.2, 7.8 | 6.3 | 1.575 | 2.1 |
| −5, 5, −5, 5 | 100 | 25 | 33.333333 |
Worked example: four measurements
Enter 4.5, 5.1, 6.2 and 7.8. The mean is 5.9, the squared deviations add up to 6.3, and N = 4. Sample variance is 6.3 ÷ 3 = 2.1 with a standard deviation of 1.449138. Choosing population instead gives 6.3 ÷ 4 = 1.575 and 1.25499. The range is 3.3, from the smallest value 4.5 to the largest 7.8.
Worked example: zero mean, large spread
The values −5, 5, −5, 5 have a mean of exactly 0, so each squared deviation is 25 and the total is 100. Population variance is 25 (σ = 5) and sample variance is 33.333333 (s = 5.773503). The example shows why you must pick the type deliberately: with only four values the sample figure is a third larger than the population one.
Why squaring and why N − 1
Squaring stops positive and negative deviations from cancelling and gives extra weight to values far from the mean. The N − 1 divisor corrects for the fact that deviations are measured from the sample’s own mean, which sits closer to the data than the true mean does. For two values, 3 and 5, the sample variance is 2 and the standard deviation is 1.414214, while the population variance would be 1. The gap between the two versions shrinks quickly as the sample grows.
Input handling and limits
The text box accepts commas, semicolons, spaces and line breaks, but it silently ignores anything that is not a number, so verify the Count row. Weighted data, grouped frequency tables and variance confidence intervals are not supported. Extreme values can dominate the squared sum, so check the minimum, maximum and range before relying on the figure.
Learn more from an authoritative source:
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Results are estimates for informational purposes only and do not constitute professional financial, medical, legal, or technical advice. Read full disclaimer →