What is the difference between sample and population variance?

Population variance divides the sum of squared deviations by n, because the data represents the entire group. Sample variance divides by n − 1 — an adjustment called Bessel's correction that removes the bias that appears when you estimate spread from a sample rather than the whole population.

How do you calculate variance step by step?

Find the mean, subtract it from each value and square the result, add those squared deviations together, then divide the total by n for population variance or by n − 1 for sample variance.

What is the relationship between variance and standard deviation?

Standard deviation is the square root of the variance, so variance is the standard deviation squared. Variance is measured in squared units, while taking the square root returns the spread to the original units of the data.

Why does sample variance divide by n − 1 instead of n?

Dividing by n underestimates the true spread because the sample mean is, by construction, the value that minimizes the squared deviations within the sample. Using n − 1 — the degrees of freedom — inflates the estimate just enough to make it unbiased. This is Bessel's correction.

Can variance be negative?

No. Variance is an average of squared deviations, and squares are never negative, so variance is always zero or positive. A variance of exactly zero means every value in the data set is identical.

What are the units of variance?

Variance is expressed in the square of the data's units. If your data is in centimetres, the variance is in square centimetres. Because squared units are hard to interpret, people often report the standard deviation, which is back in the original units.

Should I use sample or population variance for my data?

Use population variance only when your numbers cover the entire group you care about. If your data is a sample drawn from a larger population — which is the usual case in statistics — use sample variance with the n − 1 divisor.

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n = 8

Method

Numbers

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Sample variance (s²)

4.5714

s² = Σ(x − x̄)² ÷ n − 1 = 32 ÷ 7

Std. dev (√)

2.1381

Mean (x̄)

Count (n)

Σ(x − x̄)²

Sum (Σx)

Range

Deviation breakdown

x	x − x̄	(x − x̄)²
2	−3	9
4	−1	1
4	−1	1
4	−1	1
5	0	0
5	0	0
7	2	4
9	4	16
Σ(x − x̄)²		32

Variance is in squared units; its square root (the standard deviation) returns to the original units.

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The basics

What is variance?

Variance is the single number that answers “how spread out is this data?” It takes the distance of every value from the mean, squares those distances so they cannot cancel out, and averages them. The squaring is the key idea — it punishes far-flung values heavily and keeps the result positive.

σ² = Σ(x − x̄)² ÷ n

Population variance

The average squared deviation when your numbers represent the entire population. You divide the sum of squared deviations by the count n.

Use n when the data is the whole group.

s² = Σ(x − x̄)² ÷ (n − 1)

Sample variance

The unbiased estimate when your numbers are a sample drawn from a larger population. Dividing by n − 1 corrects for the bias of estimating from limited data.

Use n − 1 when the data is a sample.

Method

How to calculate variance by hand

Four steps. The only choice is the divisor in the last one.

1
Find the mean
Add up all the values and divide by how many there are. This average x̄ is the centre every deviation is measured from.
x̄ = Σx ÷ n
2
Subtract the mean and square each result
For each value, compute x − x̄ and square it. Squaring removes negative signs so deviations on both sides count.
(x − x̄)²
3
Add the squared deviations
Sum every squared deviation into a single total — the sum of squares.
Σ(x − x̄)²
4
Divide by n or n − 1
Divide the sum of squares by n for a population, or by n − 1 for a sample. The quotient is the variance.
σ² = Σ(x − x̄)² ÷ n · s² = Σ(x − x̄)² ÷ (n − 1)

Worked examples

See it worked out

The same data set, computed as a sample and as a population, plus the smallest valid sample — every figure is calculated live, not hardcoded.

Given 4, 8, 6, 5, 3, 7 (Divide by n − 1 = 5)

Mean: x̄ = 33 ÷ 6 = 5.5

Squared deviations: 2.25 + 6.25 + 0.25 + 0.25 + 6.25 + 2.25

Sum of squares: Σ(x − x̄)² = 17.5

Divide: 17.5 ÷ 5

s² = 3.5√ = 1.8708 (std. dev)

Where it shows up

Why variance matters

Variance is the raw material of risk, quality, and uncertainty across many fields.

σ²ₚ

Finance

Portfolio risk is variance of returns — the more an asset's returns swing, the higher its variance and the riskier it is judged to be.

MSE

Machine learning

Mean squared error, the workhorse loss function, is the variance of a model's prediction errors around zero.

Manufacturing

Process capability indices compare the allowed tolerance to the variance of a process, flagging when output drifts out of spec.

Reference

Sample vs population, side by side

Both start identically — find the mean, then sum the squared deviations. Only the divisor differs.

	Population	Sample
Symbol	σ² (sigma squared)	s²
Divisor	n	n − 1
When to use	Data is the entire group	Data is a sample of a larger group
Bias	Exact for the population	Unbiased estimate (Bessel's correction)
Square root	σ — population std. dev	s — sample std. dev

Quick chart

The n vs n − 1 gap by sample size

For a fixed sum of squares, how much do the two divisors change the answer? The gap is huge for tiny samples and vanishes for large ones.

Sum of squares Σ(x − x̄)²

See how sample (÷ n − 1) and population (÷ n) variance compare as the sample grows.

n	Population σ²	Sample s²	Difference
2	20	40	+100%
3	13.33	20	+50%
4	10	13.33	+33.33%
5	8	10	+25%
8	5	5.71	+14.29%
10	4	4.44	+11.11%
20	2	2.11	+5.26%
30	1.33	1.38	+3.45%
50	0.8	0.82	+2.04%
100	0.4	0.4	+1.01%

With a small sample the choice of divisor matters a lot — at n = 2 the sample estimate is double the population one. By n = 100 the gap is about 1%, which is why the distinction fades for large data sets.

Worth knowing

Properties and pitfalls

A handful of facts that explain how variance behaves — and the mistakes that trip people up.

σ² ≥ 0

Never negative

Variance averages squared numbers, so it can never be below zero. A value of exactly zero means every data point is identical.

σ = √σ²

Standard deviation is its root

Take the square root of the variance to get the standard deviation, which is expressed in the original units rather than squared ones.

Var(x + c) = Var(x)

Shifting does nothing

Adding the same constant to every value moves the mean but leaves the spread — and so the variance — completely unchanged.

Var(c·x) = c²·Var(x)

Scaling squares the factor

Multiplying every value by a constant multiplies the variance by the square of that constant, because deviations get squared.

units²

Watch the units

Variance lives in squared units. Reporting a height variance as “cm” instead of “cm²” is a common slip.

n vs n − 1

Pick the right divisor

The most frequent mistake is using n on sample data. When in doubt, treat your data as a sample and divide by n − 1.

Related measures

Variance and its neighbours

Variance sits at the centre of a small family of spread statistics — each is one short step away.

s = √s²

Standard deviation

The square root of variance, back in the original units. The most quoted measure of spread in everyday reporting.

Σ(x − x̄)²

Sum of squares

Variance before the division — the total squared deviation. It appears throughout regression and ANOVA.

Cov(x, y)

Covariance

Generalises variance to two variables, measuring how they move together. Variance is just a variable's covariance with itself.

FAQ

Common questions

Variance measures how spread out a set of numbers is around their mean. It is the average of the squared deviations from the mean — a small variance means the data clusters tightly around the average, while a large variance means it is widely dispersed.

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