Variance formula

The two variance formulas

Sample variance: your numbers are a subset of a larger group.

s² = Σ(x − x̄)² / (n − 1)

Population variance: your numbers are the whole group.

σ² = Σ(x − μ)² / N

Both do the same four things: find the mean, subtract it from every value, square each difference, and add the squares up. The only difference is the divisor, n − 1 for a sample and N for a population. Coursework and most real data are samples, so when in doubt use n − 1. In a spreadsheet the two are =VAR.S(A1:A100) and =VAR.P(A1:A100).

Every symbol in the formula

The symbols in the standard deviation formula are the same ones, with one square root added.

Variance formula example, step by step

Data: 3, 5, 7, 7, 8.

1. Sum = 30, count n = 5, so the mean is 6.
2. Deviations from the mean: −3, −1, 1, 1, 2.
3. Squared: 9, 1, 1, 1, 4. Their sum, Σ(x − x̄)², is 16.
4. As a sample: 16 ÷ (5 − 1) = s² = 4.
   As a population: 16 ÷ 5 = σ² = 3.2.

Paste the five values into the calculator above to check them, and switch between sample and population to see only the divisor change. The square roots, 2 and about 1.789, are the matching standard deviations.

Why n − 1 for a sample

The sample mean is calculated from the very values whose spread you are measuring, so it sits closer to them than the true population mean does. Squared distances around it are therefore too small on average, and dividing by n − 1 instead of n corrects that bias exactly. The effect is large for small samples (a quarter more variance at n = 5) and negligible for large ones. Sample vs population works through the reasoning in full.

The computational (shortcut) formula

s² = [ Σx² − (Σx)² / n ] / (n − 1)

It avoids computing each deviation. On the example data, Σx² = 196 and (Σx)²/n = 900/5 = 180, so s² = (196 − 180) / 4 = 4, the same answer. It is handy by hand but fragile on a computer: when the values are large and close together the subtraction loses most of its significant digits, which is why this site uses Welford's algorithm instead.

Variance of a random variable

For a random variable X with mean μ = E[X]:

Var(X) = E[(X − μ)²] = E[X²] − (E[X])²

• Discrete: Var(X) = Σ (x − μ)² · p(x), summed over every value x.
• Continuous: Var(X) = ∫ (x − μ)² f(x) dx, integrated over the whole range.

Example, one roll of a fair die: E[X] = 3.5 and E[X²] = (1 + 4 + 9 + 16 + 25 + 36) / 6 = 91/6, so Var(X) = 91/6 − 3.5² = 35/12 ≈ 2.917. The expected value calculator does this for any probability table, and the standard deviation of a random variable takes the square root.

Variance of common distributions

The die example fits the discrete uniform row: k = 6 gives (36 − 1) / 12 = 35/12.

Variance for grouped and frequency data

When each value x occurs f times, weight every squared deviation by its frequency:

s² = Σf(x − x̄)² / (Σf − 1)  where  x̄ = Σfx / Σf

Example: the value 1 occurs 2 times, 2 occurs 3 times and 3 occurs 5 times. Then Σf = 10, Σfx = 23 and x̄ = 2.3. The weighted squared deviations are 2 × 1.69 + 3 × 0.09 + 5 × 0.49 = 6.1, so s² = 6.1 / 9 ≈ 0.678 (or σ² = 6.1 / 10 = 0.61 for a population). For class intervals, use each class midpoint as x; the result is then an approximation. The grouped-data calculator and the frequency table calculator show every column.

Variance of a sum or linear combination

Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab·Cov(X, Y)

Constants come out squared, and adding a constant does not change the variance at all: Var(X + c) = Var(X). When X and Y are independent their covariance is zero, so variances simply add, which is the main reason statistics works with variances rather than standard deviations. The covariance calculator gives Cov(X, Y) from paired data, and the portfolio standard deviation calculator applies this formula to two assets.

From variance to standard deviation

The standard deviation is the square root of the variance, s = √s², which puts the spread back into the data's own units: a variance of 16 cm² is a standard deviation of 4 cm. The standard deviation formula is therefore this page's formula with one more step, and standard deviation vs variance covers when to report which.

Copy the formulas

Plain text, for Word, Google Docs or email:

• s² = Σ(x − x̄)² / (n − 1)
• σ² = Σ(x − μ)² / N
• Var(X) = E[X²] − (E[X])²

LaTeX:

• s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}
• \sigma^2 = \frac{\sum (x_i - \mu)^2}{N}
• \operatorname{Var}(X) = E[X^2] - (E[X])^2

In Google Docs choose Insert → Equation and type the LaTeX commands followed by a space; in Word press Alt+= and paste the LaTeX line with LaTeX selected in the Equation ribbon.