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σ² Variance Calculator

Calculate population and sample variance for any data set. Shows mean, each deviation from the mean, squared deviations, and standard deviation. Enter numbers separated by commas or spaces.

Variance Formulas

Population Variance
σ² = Σ(xᵢ − μ)² / N
Sample Variance (Bessel's correction)
s² = Σ(xᵢ − x̄)² / (N − 1)
Standard Deviation
σ = √σ²

How to Calculate Variance

  1. 1
    Choose Population or Sample
    Use population variance (σ²) for complete data sets. Use sample variance (s²) when your data is a subset of a larger population.
  2. 2
    Calculate the Mean
    Add all values and divide by the count. This is the reference point for measuring how spread out the data is.
  3. 3
    Find Squared Deviations
    Subtract the mean from each value (deviation), then square each deviation. Squaring removes negatives and amplifies larger differences.
  4. 4
    Average the Squared Deviations
    Divide the sum of squared deviations by N (population) or N−1 (sample). This is your variance. Take the square root for standard deviation.

Worked Example

Data set: 2, 4, 4, 4, 5, 5, 7, 9

N = 8 | Sum = 40 | Mean = 40 ÷ 8 = 5
Σ(xᵢ − 5)² = 9+1+1+1+0+0+4+16 = 32
Population Variance = 32 ÷ 8 = 4
Population SD = √4 = 2
Sample Variance = 32 ÷ 7 ≈ 4.571

Frequently Asked Questions

Variance measures how far a set of numbers is spread out from their mean. It is the average of the squared differences from the mean. A variance of 0 means all values are identical; a high variance means the data is widely spread.

Population variance (σ²) divides by N and is used when you have data for every member of a group. Sample variance (s²) divides by N−1 (Bessel's correction) and is used when your data is a sample from a larger population. The correction prevents underestimation of the true variance.

Squaring serves two purposes: it eliminates negative values (so positive and negative deviations don't cancel each other out), and it gives extra weight to larger deviations, making variance more sensitive to outliers than simpler measures like mean absolute deviation.

Standard deviation is simply the square root of variance. Variance is in squared units of the original data, which can be hard to interpret. Standard deviation is in the same units as the data, making it easier to understand in context.

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