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Variance

General

Statistical Variance

The average of squared differences from the mean, measuring how spread out a dataset is. Standard deviation is simply the square root of variance.

Definition

Variance measures how spread out a set of numbers is around its average, calculated as the mean of the squared differences between each value and the dataset's mean. A small variance means values cluster tightly around the average; a large variance means they're spread widely.

It's the mathematical foundation underneath standard deviation, which is just the square root of variance brought back into the original unit scale. The Standard Deviation Calculator computes variance as an intermediate step before taking that square root.

Formula

Variance (ĻƒĀ²) = Ī£(xįµ¢ āˆ’ Ī¼)Ā² / n

Where xįµ¢ is each value, Ī¼ (mu) is the mean, and n is the number of values (or n āˆ’ 1 for sample variance).

Worked Example

A dataset of test scores: 70, 75, 80, 85, 90, with a mean of 80.

  • Squared deviations: (70āˆ’80)Ā² = 100, (75āˆ’80)Ā² = 25, (80āˆ’80)Ā² = 0, (85āˆ’80)Ā² = 25, (90āˆ’80)Ā² = 100
  • Sum: 100 + 25 + 0 + 25 + 100 = 250
  • Variance: 250 / 5 = 50
  • Standard deviation: āˆš50 ā‰ˆ 7.07

That 7.07 figure is in the same units as the original scores, unlike the variance of 50, which is in squared points.

Key Things to Know

  • Variance is always non-negative. Since deviations are squared, the result can never be negative, zero only happens when every value is identical.
  • Larger deviations are penalized disproportionately. A value twice as far from the mean contributes four times as much to variance, not twice, because of the squaring.
  • Sample variance divides by nāˆ’1, not n. This adjustment, called Bessel's correction, corrects for bias when estimating a population's variance from a limited sample.
  • Variance and standard deviation carry the same information, different units. Choose standard deviation when you want interpretability, variance when you're doing further statistical calculations that require it directly.
  • Outliers inflate variance heavily. A single extreme value can dominate the variance calculation because of the squaring, even if the rest of the dataset is tightly clustered.

Related Calculators

Standard Deviation Calculator ā†’Calculate mean, variance, and standard deviation for any dataset instantly. Enter comma-separated numbers to get population and sample statistics.

Frequently Asked Questions

Why use variance instead of just looking at the range?
Range only looks at the two extreme values and ignores everything else, while variance accounts for every data point's distance from the mean. Two datasets can share the same range but have very different variances depending on how the middle values are distributed.
Why are the differences squared instead of just averaged?
Squaring prevents positive and negative deviations from canceling each other out, and it penalizes larger deviations more heavily than smaller ones. Without squaring, the differences would always sum to zero around the mean.
What's the difference between population variance and sample variance?
Population variance divides by the total count (n), while sample variance divides by n minus 1, a correction that accounts for the extra uncertainty in estimating a population from a sample. Most real-world statistical work uses sample variance unless you truly have the entire population.
Why is standard deviation more commonly reported than variance?
Variance is in squared units, which makes it hard to interpret intuitively, a variance of 64 doesn't map cleanly onto the original scale. Standard deviation, the square root of variance, brings the number back into the same units as the original data.
Can variance be zero?
Yes, if every value in the dataset is identical, there's no spread at all, and variance comes out to exactly zero.