Variance calculator

What variance measures

Variance is the average of the squared distances between each value and the mean. It answers the same question as the standard deviation — how spread out is this data? — but stops one step earlier, before the square root that would return the answer to the original units.

Sample: s² = Σ(x − x̄)² / (n − 1) Population: σ² = Σ(x − μ)² / N

Because it is built from squares, variance is always zero or positive, and it grows with the square of the scale of the data: double every value and the variance quadruples, while the standard deviation merely doubles.

Variance or standard deviation — which should you report?

For describing data to a reader, use the standard deviation. Its units match the data, so it can be compared directly against the mean and against individual values.

For doing further mathematics with the spread, use the variance. Its defining advantage is that variances of independent quantities add:

Var(X + Y) = Var(X) + Var(Y) (for independent X and Y)

No such rule holds for standard deviations. This additivity is what makes analysis of variance, portfolio risk decomposition and the whole machinery of linear models work, and it is the reason variance survives despite being harder to interpret.

A worked example

Data: 4, 8, 6, 5, 3. Sum = 26, n = 5, mean = 5.2.

Note that the middle column sums to exactly zero — that always happens, and it is precisely why the deviations must be squared before averaging. Dividing 14.8 by n − 1 = 4 gives a sample variance of 3.7; dividing by N = 5 gives a population variance of 2.96. The corresponding standard deviations are 1.9235 and 1.7205.

Converting between variance and standard deviation

If somebody has already given you one, you do not need the raw data to get the other — a single operation each way:

s = √(s²)    s² = s × s

The conversion carries no information about sample versus population — that choice was made when the variance was computed, and square-rooting cannot undo or change it. A sample variance square-roots to a sample standard deviation, and nothing else.

One consequence catches people out: because squaring is not linear, the standard deviation is not proportional to the variance. Doubling a variance multiplies the standard deviation by only √2 ≈ 1.414. So a group with four times the variance of another has twice its standard deviation, not four times. The full comparison is here.

If you need both numbers from raw data rather than one from the other, the variance and standard deviation calculator gives the sample and population versions of each side by side, and the mean, variance and SD table lays out every deviation and square in one table, the way a textbook shows the working.

Variance of a probability distribution

Everything above computes the variance of data you already have. The variance of a random variable — a probability distribution rather than a list of observations — uses a different formula, because each outcome carries a probability instead of counting once:

Var(X) = Σ (x − μ)² P(x)  where  μ = Σ x P(x)

There is no n and no n − 1 anywhere in it. The probabilities already sum to 1, so they do the averaging themselves, and there is no sample to correct for. A fair six-sided die has μ = 3.5 and Var(X) = 35/12 ≈ 2.9167, giving σ ≈ 1.7078. The standard deviation of a random variable guide works through this and a non-uniform example line by line.

For the standard named distributions the sum has already been done for you:

The Poisson row is worth a second look: its variance equals its mean. Count data whose variance runs well above its mean is overdispersed and is not Poisson, which is the standard diagnostic for that family.

Variance in Excel and Google Sheets

The older VAR() and STDEV() functions still work and are the sample versions. VARA() and STDEVA() differ in that they count text as 0 and booleans as 0 or 1, which is rarely what you want. The full Excel guide is here.