Variance Calculator

Population variance (divide by N): your numbers are the entire group you care about — every student in one class, every unit produced today.

Sample variance (divide by n − 1): your numbers are a subset standing in for a larger group — 30 survey responses out of thousands of customers.

The choice only changes the divisor, but on small data sets that difference is substantial, so make the call before quoting a result.

Variance is the average of the squared distances between each value and the mean. Squaring does two jobs at once: it removes the minus signs (so deviations below the mean cannot cancel deviations above it) and it weights large deviations more heavily than small ones, making variance especially sensitive to values far from center.

Because every deviation is squared, variance carries squared units — a variance of 6.5 cm² for plant heights measured in centimeters. Taking the square root converts back to the original units, which is exactly what the standard deviation calculator reports as its headline number.

A sample almost never captures the full spread of the population it came from, because the extreme values of a population are exactly the ones least likely to be drawn. On top of that, the sample mean is computed from the same data, so it sits closer to the sample values than the true population mean would. Both effects make the raw average of squared deviations an underestimate.

Dividing by n − 1 instead of n — a correction named after Friedrich Bessel — enlarges the result just enough to cancel that bias on average. The quantity n − 1 is the number of degrees of freedom: once the sample mean is fixed, only n − 1 deviations can vary freely, because they must sum to zero. With large samples the two divisors barely differ (dividing by 999 versus 1000), but with five values the sample variance is 25% larger than the population version, which is far from a rounding detail.

Although the standard deviation is easier to quote, variance itself is the quantity that statistical theory is built on.

• Finance: portfolio risk models add and decompose variances, because variances of independent returns combine by simple addition while standard deviations do not.
• Experiment design: ANOVA — literally the analysis of variance — splits total variance into between-group and within-group parts to test whether treatments differ.
• Machine learning: the bias-variance trade-off describes how a model that fits training data too flexibly produces predictions with high variance on new data.
• Manufacturing: process engineers track the variance of critical dimensions, since a shift in spread often signals tool wear before the mean moves at all.

If you only need a quick sense of center before studying spread, the mean, median, mode calculator handles that side of the analysis, and the descriptive statistics calculator combines both perspectives in a single report.

A biology student measures five seedlings, in centimeters: 10, 12, 9, 15, 14. Treating them as a sample from a larger tray, the calculator works through:

1. Mean: 10 + 12 + 9 + 15 + 14 = 60, so the mean is 60 ÷ 5 = 12.
2. Deviations from the mean: −2, 0, −3, 3, 2.
3. Squared deviations: 4, 0, 9, 9, 4.
4. Sum of squares: 4 + 0 + 9 + 9 + 4 = 26.
5. Divide by n − 1 = 4: sample variance s² = 26 ÷ 4 = 6.5, giving a sample standard deviation of √6.5 ≈ 2.5495.

Had those five seedlings been the entire population of interest — say, the only plants in the experiment — the divisor would be N = 5 instead, giving a population variance of 26 ÷ 5 = 5.2 and a standard deviation of √5.2 ≈ 2.2804. Same data, different question, different divisor: that is the entire population-versus-sample distinction in one example.