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Mathematics · pattern F, Dataset

Variance, sample and population.

Paste your data for the variance with the sum of squared deviations shown, and switch between the sample and population divisors.

Inputs

Sample or population

Divide by n − 1 for a sample drawn from a wider population; by n only when the data is the entire population.

The formula used

sample s² = Σ(x − x̄)² ÷ (n − 1); population σ² = Σ(x − x̄)² ÷ n

Variance is in squared units, which is why standard deviation is usually reported instead. The algebra, however, runs on variance.

Variance

64.722

n = 10 · sample, ÷ (n − 1)

n
10
Sum
715.000
Mean
71.500
Median
71.000
Mode
no repeated value
Σ(x − x̄)²
582.500
Variance
64.722
Standard deviation
8.045
Standard error
2.5441
Minimum / maximum
59.00 / 85.00
Range
26.000
Q1 / Q3 / IQR
66.50 / 76.25 / 9.75

Your data — accent bars sit beyond one deviation

59.0085.00

Squared units are why variance is rarely quoted.

Variance of exam marks is in marks squared, which means nothing to a reader, so the square root is reported instead. Variance survives because it adds: the variance of a sum of independent variables is the sum of their variances, and standard deviations do not behave that way.

Questions about variance

Why divide by n − 1?
Because the sample mean sits closer to its own data than the population mean does, which biases the estimate low. Bessel's correction removes most of that bias.
Do variances add?
For independent variables, yes — which is why variance rather than standard deviation appears throughout statistical theory.
When is the population form right?
When your data is the entire population. In coursework it almost never is.

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