Variance Calculator

Calculate sample or population variance, standard deviation, mean, and deviation breakdowns step by step with Bessel's correction support.

Dataset Parameters

Live calculation

Enter 2 or more numbers separated by commas, spaces, or newlines.

Sample Datasets:

Bessel's correction (dividing by n − 1) produces an unbiased estimate of population variance from a sample.

Sample Variance (s²)
8.2857(n = 8)
Standard Deviation (s)2.8785
Mean (x̄): 14.5Sum of Squares (SS): 58
Sum (∑x)116
Range (Max − Min)9
Minimum10
Maximum19

Squared Deviation Breakdown

Value (xᵢ)Deviation (xᵢ − x̄)Squared ((xᵢ − x̄)²)
12-2.56.25
150.50.25
14-0.50.25
10-4.520.25
194.520.25
172.56.25
13-1.52.25
161.52.25
Sum of Squared Deviations (SS):58

What Is Statistical Variance?

Variance measures the dispersion and spread of data points around their arithmetic mean. A low variance indicates that values are clustered tightly near the mean, while a high variance indicates extensive scatter.

1. Sample Variance (Unbiased Bessel's Correction)
s²=
∑ (xᵢ − x̄)²n − 1
2. Population Variance (Complete Universe)
σ²=
∑ (xᵢ − μ)²N
Step-by-Step Calculation Breakdown (Example: [4, 6, 8])
Step 1: Compute Arithmetic Mean (x̄)
• Sum = 4 + 6 + 8 = 18
• Mean x̄ = 18 ÷ 3 = 6.000
Step 2: Calculate Deviations and Sum of Squared Deviations (SS)
• (4 − 6)² = (−2)² = 4
• (6 − 6)² = (0)² = 0
• (8 − 6)² = (+2)² = 4
• Sum of Squared Deviations (SS) = 4 + 0 + 4 = 8.000
Step 3: Divide by Degrees of Freedom
• Sample Variance s² = 8 ÷ (3 − 1) = 8 ÷ 2 = 4.000
• Population Variance σ² = 8 ÷ 3 = 2.667
• Sample Standard Deviation s = √4 = 2.000

Population vs. Sample Variance

When calculating variance, selecting the proper denominator is essential:

  • Sample Variance (n − 1): Used when analyzing a representative sample from a larger universe. Dividing by n − 1 compensates for underestimating variability.
  • Population Variance (N): Used only when every single entity in the targeted universe has been recorded without sampling.

Frequently Asked Questions

What is variance in simple terms?
Variance measures how spread out values are around the mean. If values cluster tightly, variance is low. If values are widely scattered, variance is high. It is calculated from squared deviations from the arithmetic average, so larger gaps contribute disproportionately. Variance is foundational for standard deviation, confidence intervals, ANOVA, and regression analysis.
What is the difference between sample and population variance?
Population variance divides the sum of squared deviations by N because you possess data for the complete universe. Sample variance divides by n − 1 (Bessel's correction) to eliminate downward bias when estimating population variance from an incomplete sample.
Why are deviations squared in variance?
Squaring ensures that positive and negative deviations do not cancel out to zero, while simultaneously penalizing large outliers quadratically. Because squaring alters the units of measurement (e.g. dollars²), standard deviation (the square root of variance) is used for direct interpretation in original units.
Can variance be negative?
No. Real variance is always greater than or equal to zero because it is computed by summing non-negative squared deviations. A variance of zero indicates that every single value in the dataset is identical.
How does variance relate to standard deviation?
Standard deviation is simply the square root of variance: s = √(s²). While variance is mathematically convenient for proofs and decomposition of variation, standard deviation is intuitive because it shares the same unit of measurement as the raw data.
When can variance be misleading?
Variance is sensitive to extreme outliers and asymmetric skewness. Two datasets may have identical variance but vastly different shapes (e.g. bimodal vs bell curve). Always pair variance with the median, interquartile range (IQR), or visual plots.

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