SigmaCalc

Variance Calculator

The squared spread behind standard deviation.

Dataset (comma / space / new-line separated)

Sample variance (n−1): 19.789
Population variance (n): 17.81

What you entered

Dataset: 10 values

1
Mean
Σx ÷ 10= 12.7
2
Sum of squared deviations
Σ(x − mean)²= 178.1
3
Sample variance (÷ n−1)
Σ(x−mean)² ÷ 9= 19.7889
4
Population variance (÷ n)
Σ(x−mean)² ÷ 10= 17.81

Result

Sample variance: 19.789

This dataset has a sample variance of 19.789 — the square root of variance is the standard deviation, the more commonly-quoted measure of spread.

Sample and population variance for worked datasets

Variance is the average squared distance from the mean. Sample variance divides by n minus 1; population variance divides by n.

Dataset	Mean	Sample variance	Population variance	Sample SD
2, 4, 4, 4, 5, 5, 7, 9	5.00	4.571	4.000	2.138
10, 12, 23, 23, 16, 23, 21, 16	18.00	27.429	24.000	5.237
1, 2, 3, 4, 5	3.00	2.500	2.000	1.581
5, 10, 15, 20, 25, 30	17.50	87.500	72.917	9.354
70, 75, 80, 85, 90, 95, 100	85.00	116.667	100.000	10.801
3, 7, 7, 19, 24, 30, 32, 45	20.88	215.268	188.359	14.672

Variance is in squared units, which is what makes it awkward to interpret directly: a variance of 116.667 on exam marks is 116.667 squared marks, a quantity with no natural meaning. That is why standard deviation, its square root, is usually reported instead. Variance earns its place in the maths rather than the reporting - variances of independent variables add, standard deviations do not, which is why it underpins ANOVA, regression and portfolio theory. Squaring also means a value twice as far from the mean contributes four times as much, so variance is highly sensitive to outliers.

Variance is standard deviation, squared

Variance is the average of the squared deviations from the mean — taking its square root gives standard deviation, the more intuitive 'typical distance from the mean' measure in the same units as your data.

Why square the deviations at all

Squaring makes every deviation positive (so they don't cancel out when summed) and disproportionately penalizes large deviations, which is exactly the behavior most statistical models — from finance to machine learning — are built around.

Frequently asked questions

A portfolio returned 8%, 12%, −3%, 15%, 6% over 5 years. How risky was it?

Mean = 7.6%. Squared deviations from mean: 0.16, 19.36, 112.36, 54.76, 2.56. Sum = 189.2. Sample variance (÷4) = 47.3 %-squared. Sample SD = √47.3 = 6.9 percentage points. Higher variance = more unpredictable returns.

How is variance different from standard deviation?

Variance is in squared units (points², dollars², etc.) — hard to interpret directly. Standard deviation is its square root, back in original units. SD is more intuitive for communication; variance is used internally in formulas like ANOVA, regression, and portfolio theory.

Why not just use the range or average absolute deviation?

Range (max − min) uses only 2 values and is extremely sensitive to outliers. Average absolute deviation is more robust but harder to work with mathematically. Variance has nice algebraic properties (additivity for independent variables) that make it the foundation of most statistical theory.