Variance Calculator Formula

Understand the math behind the variance calculator. Each variable explained with a worked example.

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Formulas Used

Population Variance

pop_variance = ss / 5

Sample Variance

samp_variance = ss / 4

Mean

data_mean = mn

Variables

VariableDescriptionDefault
v1Value 14
v2Value 28
v3Value 36
v4Value 45
v5Value 57
mnDerived value= (v1 + v2 + v3 + v4 + v5) / 5calculated
ssDerived value= pow(v1 - mn, 2) + pow(v2 - mn, 2) + pow(v3 - mn, 2) + pow(v4 - mn, 2) + pow(v5 - mn, 2)calculated

How It Works

How to Calculate Variance

Population Variance

sigma^2 = Sum of (xi - mean)^2 / N

Sample Variance

s^2 = Sum of (xi - mean)^2 / (n - 1)

Variance measures the average squared distance of each data point from the mean. The sample variance divides by (n-1) instead of n to correct for the bias introduced by estimating the population mean from the sample (Bessel's correction).

Worked Example

Find the variance of 4, 8, 6, 5, 7.

v1 = 4v2 = 8v3 = 6v4 = 5v5 = 7
  1. 01Mean = (4 + 8 + 6 + 5 + 7) / 5 = 30 / 5 = 6
  2. 02Squared deviations: (4-6)^2=4, (8-6)^2=4, (6-6)^2=0, (5-6)^2=1, (7-6)^2=1
  3. 03Sum of squares = 4 + 4 + 0 + 1 + 1 = 10
  4. 04Population Variance = 10 / 5 = 2
  5. 05Sample Variance = 10 / 4 = 2.5

Frequently Asked Questions

When should I use sample vs. population variance?

Use population variance when you have data for the entire group of interest. Use sample variance when your data is a subset drawn from a larger population, which is the more common scenario.

Why divide by n-1 for the sample variance?

Dividing by n-1 (Bessel's correction) produces an unbiased estimator of the population variance. Dividing by n would systematically underestimate the true variance.

What are the units of variance?

Variance is measured in squared units of the original data. If values are in meters, variance is in meters squared. Take the square root to get the standard deviation in original units.

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