Summary: This variance calculator computes sample and population variance with live steps, formulas and a chart. It accepts labeled numeric inputs, works offline through file:// and includes source-backed explanations for students, analysts and researchers.
Variance calculator for sample s squared and population sigma squared with coefficient of variation, formula, chart and steps. The calculator works offline, updates instantly and includes a worked example, plain-text formula, MathML, references and structured data.
Default values are loaded. Click any field and edit it; results and chart update automatically.
| Input or setting | Result or interpretation | Use this when |
|---|---|---|
| 2,4,4,4,5,5,7,9 | sample variance about 4.571 | squared spread |
| 10,10,10 | variance = 0 | squared spread |
| 1,2,3 | sample variance = 1 | squared spread |
sample variance: s^2 = sum((x_i - mean)^2) / (n - 1); population variance: sigma^2 = sum((x_i - mu)^2) / NVariance is the average squared distance from the mean, using n minus 1 for samples and N for populations.
2,4,4,4,5,5,7,9
sample variance about 4.571; population variance = 4
This page uses the shared statistics core for distribution functions, quantiles and exact integer counting where needed. The formula is shown in plain text so screen readers and search engines can parse it reliably.
Use Load example in the calculator to reproduce this reference result.
{
"tool": "Variance Calculator",
"input": "2,4,4,4,5,5,7,9",
"output": "sample variance about 4.571; population variance = 4",
"formula": "sample variance: s^2 = sum((x_i - mean)^2) / (n - 1); population variance: sigma^2 = sum((x_i - mu)^2) / N"
}| Calculator | Example input | Expected output |
|---|---|---|
| Variance Calculator | 2,4,4,4,5,5,7,9 | sample variance about 4.571; population variance = 4 |
What does p = 0.03 mean? If the null hypothesis and model assumptions were true, a result at least this extreme would occur about 3% of the time. The American Statistical Association cautions that a p-value alone does not measure effect size, practical importance or the probability that Hâ‚€ is true.3
For most classroom and professional reports, pair the calculator result with the question you are answering. A mean or median summarizes location, but spread explains consistency. A confidence interval estimates plausible values, while a hypothesis test evaluates compatibility with a null model. Regression and correlation describe association, so they should be reported with a chart and residual or outlier review. When a result is statistically significant, still ask whether the effect is large enough to matter in the real setting.
| Statistic | Small | Medium | Large | Use |
|---|---|---|---|---|
| Cohen's d | 0.2 | 0.5 | 0.8 | t-test effect size |
| Cramér's V | 0.1 | 0.3 | 0.5 | chi-square association |
| |r| | 0.10 | 0.30 | 0.50 | correlation strength |
| R² | 0.01 | 0.09 | 0.25 | variance explained |
Five steps, almost identical to SD. (1) Find the mean. (2) Subtract the mean from each value to get deviations. (3) Square those deviations. (4) Sum the squared deviations. (5) Divide by n for population variance, or by n − 1 for sample variance. Quick example: {3, 5, 7, 9}, mean = 6, squared deviations = {9, 1, 1, 9}, sum = 20. Sample variance = 20/3 ≈ 6.67. Population variance = 20/4 = 5. Skip step 5's square root — that's where SD comes from, not variance.
Population variance, written σ², uses every member of the population and divides the sum of squared deviations by N. Sample variance, written s², uses a subset and divides by n − 1 instead. The minus-one is Bessel's correction — it offsets the underestimation that comes from using the sample mean rather than the true population mean. For most real-world studies, you have a sample, so s² is the working choice. With huge n, σ² and s² are basically the same; with small n, the gap matters.
Because dividing by n underestimates the true population variance when you're using the sample mean. The sample mean is, by construction, the value closest to your data, so deviations from it are systematically smaller than deviations from the unknown population mean would be. Dividing by n − 1 inflates the result enough to correct that bias. The formal name is Bessel's correction. With n = 5, dividing by 4 instead of 5 gives a 25% larger estimate; with n = 1000, the difference is barely noticeable.
Square it. Variance = SD². If SD = 7, then variance = 49. Going the other direction: SD = √variance, so a variance of 100 corresponds to an SD of 10. They're two views of the same spread. SD lives in the original units (kg, cm, rupees), which makes it easier to interpret day-to-day. Variance lives in squared units (kg², cm², rupees²), which is mathematically cleaner for derivations and statistical theory but harder to interpret intuitively. Both come from identical computation up to the final square root.
Most calculators take a list of numbers separated by commas, spaces, or new lines. Paste your data and select sample or population variance — that choice matters. Sample variance divides by n − 1 (use this when your data is a subset of a larger group). Population variance divides by n (use only when your data covers the entire population). The output is in squared units of your original data. To return to the original scale, take the square root and you've got the standard deviation.
A high variance signals that your data points are spread widely around the mean — values are scattered, not clustered. Low variance means most observations sit close to the mean. The interpretation depends on context: if exam scores have a high variance, the class shows mixed performance; if manufacturing measurements have a high variance, your process may be unreliable. Variance is in squared units, which can feel awkward, so analysts often report standard deviation instead. Comparing variances across datasets is meaningful only when the units and scales are similar.
Variance is extremely sensitive to outliers because every deviation gets squared before being summed. Squaring magnifies extreme values disproportionately — a single far-out point can inflate variance dramatically. Take {2, 4, 6, 8, 10}: variance ≈ 8. Swap 10 for 100, and variance balloons to roughly 1500. That's why variance (and the SD derived from it) can mislead in the presence of outliers. For skewed or outlier-heavy data, use the IQR or median absolute deviation, both of which are robust to extreme values.
Two main functions. VAR.S(range) gives sample variance, dividing by n − 1 — use this when your data is a sample. VAR.P(range) gives population variance, dividing by n — use this when your data covers the full population. Example: =VAR.S(A1:A20) computes the sample variance of cells A1 through A20. Older spreadsheets used VAR (same as VAR.S) and VARP (same as VAR.P), and those still function but are deprecated. The .S/.P pattern matches STDEV functions, making it consistent across Excel's statistical toolkit.