📊 Statistics & Probability
Variance Calculator
Calculate sample and population variance from any dataset. Free variance calculator with mean and SD output.
What This Calculator Tells You
Variance measures how spread out a set of numbers is — how far, on average, each value sits from the mean. A small variance means the numbers cluster tightly; a large one means they swing widely.
Paste in your numbers and the calculator returns both the sample and population variance, plus the mean. Take the square root of variance to get the standard deviation, which is in the same units as your data.
What to Enter
- Numbers — your data values, separated by commas, spaces or new lines. You can paste a column straight from a spreadsheet.
Which result should you use? Sample variance (divides by n − 1) when your numbers are a sample from a bigger group — for example, 7 days of sales used to judge a typical week. Population variance (divides by n) when you have every value in the group you care about.
Worked Examples
1. How steady are daily sales?
A store sold 42, 38, 51, 45, 39, 47 and 44 units over seven days.
- Result → sample variance 20.57, population variance 17.63, mean 43.71.
- Standard deviation = √20.57 ≈ 4.5 units, so a typical day lands within about 4–5 units of 44.
2. Two suppliers with the same average
Supplier A delivers in 30, 31, 29, 30 and 30 days; Supplier B in 30, 45, 15, 38 and 22 days. Both average 30 days.
- Supplier A → sample variance 0.5. Supplier B → sample variance 144.5.
- Same mean, very different reliability — variance is what tells them apart.
3. Sample vs population on a textbook data set
The values 2, 4, 4, 4, 5, 5, 7, 9:
- Result → population variance 4.00 (standard deviation 2), sample variance 4.57.
- Dividing by n − 1 makes the sample figure slightly larger, correcting for the fact that a sample tends to underestimate spread.
How It’s Calculated
- s: Sample standard deviation
- Σ: Summation
- x: Observed value
- n: Number of periods or sample size
- σ: Population standard deviation
- μ: Population mean
- N: Population / total count
How to Read Your Result
- Variance is in squared units. If your data is in dollars, variance is in "dollars squared" — hard to picture. Report the standard deviation for everyday communication.
- Compare spread relative to the mean. A variance of 100 is large for values around 10 and tiny for values around 10,000. The coefficient of variation (standard deviation ÷ mean) puts different data sets on the same scale.
- Zero variance means every value is identical.
- For the difference between the two measures, read standard deviation vs variance.
Common Mistakes to Avoid
- Using population variance for a sample. Most real data is a sample; dividing by n understates the spread.
- Reporting variance to non-technical readers. Squared units confuse people — use standard deviation.
- Ignoring outliers. One extreme value can inflate variance dramatically. Check whether it is an error before drawing conclusions.
- Mixing units. Make sure every value is in the same unit (all dollars, all days).
When to Use This Calculator
- Compare the consistency of suppliers, processes or sales channels.
- Check how volatile monthly revenue or costs are.
- Prepare inputs for confidence intervals, z-scores and other statistics.
- Check homework or exam answers step by step.
Frequently Asked Questions
What is the formula for variance?
Population variance σ² = Σ(x − μ)² ÷ N. Sample variance s² = Σ(x − x̄)² ÷ (n − 1). In both, you subtract the mean from each value, square the result, add them up and divide.
Why does sample variance divide by n − 1?
A sample's own mean is always closer to its values than the true population mean is, which makes the raw spread look smaller than it really is. Dividing by n − 1 (Bessel's correction) removes that bias.
Can variance be negative?
No. It is built from squared differences, so the smallest possible value is zero.
How do I get standard deviation from variance?
Take the square root. A variance of 20.57 is a standard deviation of about 4.54.
Do you store the values I enter?
No. All calculations happen in your browser. Your data never leaves your device.
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