Statistics Calculator (Mean, Median, Standard Deviation)

How this calculation works

The mean is the sum of all values divided by how many there are. The median is the middle value once the data is sorted — for an even count, the average of the two middle values.

Variance measures spread: the average of squared distances from the mean. The sample variance divides by n − 1 (Bessel's correction) because a sample underestimates the true spread; the population variance divides by n.

Standard deviation is the square root of variance, bringing spread back into the same units as the data.

Worked example

Sample vs population — which standard deviation?

Use the population standard deviation (÷ n) when your numbers are the entire group you care about — every student in one class, every order this month.

Use the sample standard deviation (÷ n − 1) when your numbers are a sample drawn from a bigger group and you want to estimate that group's spread — survey responses, quality-control samples, experiment measurements. Dividing by n − 1 corrects the bias that makes small samples look tighter than the population really is. With large n the two values converge.

Mean or median — which average to report?

The mean uses every value, which makes it efficient but sensitive to outliers: one billionaire in the room drags the mean income up dramatically. The median only cares about rank, so it resists outliers — that is why house prices and incomes are usually reported as medians.

A quick skew check: if the mean is well above the median, the data has a long right tail (a few large values); well below, a long left tail.

Reading standard deviation

For roughly bell-shaped data, about 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3 — the empirical rule. A test with mean 70 and SD 8 means most scores land between 62 and 78.

Standard deviation shares the data's units (points, kg, ₩), which makes it more interpretable than variance, whose units are squared.