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Variance Calculator
Use this free variance calculator to measure how spread out a set of numbers is. Paste your list of numbers and it shows the mean, the population variance and sample variance, both standard deviations, and the count, sum, minimum, and maximum — all in one place.
| Measure | Value |
|---|---|
| Count (N) | 8 |
| Sum | 40 |
| Mean | 5 |
| Minimum | 2 |
| Maximum | 9 |
| Population variance (σ²) | 4 |
| Sample variance (s²) | 4.57 |
| Population SD (σ) | 2 |
| Sample SD (s) | 2.14 |
How it works
Variance answers a simple question: how far do the numbers in a set typically sit from their average? Two sets of numbers can have the same mean and look completely different. The scores 4, 5, and 6 average 5, and so do the scores 0, 5, and 10 — but the second set is far more spread out. The mean alone cannot tell them apart. Variance can: the first set has a small variance, the second a much larger one.
The calculation has three steps. First, find the mean by adding all the values and dividing by how many there are. Second, for each value, subtract the mean and square the result — squaring matters, because it stops positive and negative differences from cancelling each other out, and it punishes values that sit far from the mean more than values that sit close. Third, average those squared differences. That average is the variance.
The two formulas differ only in the last step. Population variance divides by N, the number of values, and it is the right choice when your data is the entire group you care about — every student in a class, every sale in a month. Sample variance divides by N − 1 instead, and it is the right choice when your data is a sample drawn from a larger group. A sample tends to underestimate the spread of the full group, because a small sample rarely includes the most extreme values, so dividing by the slightly smaller N − 1 nudges the estimate upward to correct for that. When N is large the two answers are nearly identical; when N is small the difference matters. This calculator shows both, so you can pick the one that fits your data.
Standard deviation is the square root of variance, and it is usually the number people actually quote. Variance is measured in squared units — squared dollars, squared inches — which are hard to picture. Taking the square root brings the spread back into the original units of your data, so a standard deviation of 2 on test scores means a typical score sits about 2 points from the mean. If your data has a standard deviation of 2 and a mean of 5, most values fall roughly between 3 and 7, and values far outside that range are unusual.
Variance is the backbone of statistics. It is how investors compare the risk of two stocks, how factories check whether a machine is producing consistent parts, how teachers judge whether a test separated strong and weak students, and how scientists decide whether an experiment's result is real or just noise. A variance of zero means every value is identical — there is no spread at all. The larger the variance, the less the mean alone tells you about any single value in the set.
To use this calculator, paste your numbers separated by commas, spaces, or line breaks — any mix works. Decimals and negative numbers are fine. You need at least two numbers for sample variance, since dividing by N − 1 with a single value would mean dividing by zero; with one number the calculator still shows the population variance, which is zero, and explains that a sample variance needs more data.
Worked example
Find the variance of 2, 4, 4, 4, 5, 5, 7, 9.
Step 1: Find the mean — (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) ÷ 8 = 40 ÷ 8 = 5.
Step 2: Subtract the mean from each value and square the result: (2 − 5)² = 9, (4 − 5)² = 1, (4 − 5)² = 1, (4 − 5)² = 1, (5 − 5)² = 0, (5 − 5)² = 0, (7 − 5)² = 4, (9 − 5)² = 16.
Step 3: Add the squared differences — 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32.
Step 4: Population variance = 32 ÷ 8 = 4. Sample variance = 32 ÷ 7 = 4.57.
So the mean is 5, the population variance is 4, and the sample variance is 4.57. The population standard deviation is the square root of 4, which is 2, and the sample standard deviation is the square root of 4.57, about 2.14. The numbers cluster fairly tightly around the mean of 5 — only the 2 and the 9 sit more than a couple of points away — which is exactly what a variance of 4 describes.
FAQ
What is variance in simple terms?
Variance measures how spread out a set of numbers is around their average. A small variance means the numbers cluster tightly around the mean; a large variance means they are scattered far from it. It is the average of the squared differences from the mean.
What is the difference between population variance and sample variance?
Population variance divides by N, the number of values, and is used when your data is the entire group you care about. Sample variance divides by N minus 1, which corrects for the fact that a sample tends to underestimate the spread of the full group. This calculator shows both.
How is variance related to standard deviation?
Standard deviation is the square root of variance. Variance is in squared units, which are hard to picture, so taking the square root brings the spread back into the original units of your data — dollars, inches, points, or whatever you measured.
Can variance be negative?
No. Variance is an average of squared differences, and squares are never negative, so variance is always zero or positive. A variance of zero means every value in the set is identical.