Sample Variance Calculator
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Frequently Asked Questions — Sample Variance Calculator
Everything you need to know about variance.
What is sample variance?
Sample variance (s²) measures how spread out values are in a sample taken from a larger population. The formula is s² = Σ(xᵢ − x̄)² / (n − 1). It divides by n−1 (not n) to provide an unbiased estimate of the true population variance. This correction — called Bessel's correction — compensates for using the sample mean instead of the true population mean.
How to calculate sample variance step by step?
To calculate sample variance: 1. Find the sample mean: x̄ = Σxᵢ / n. 2. Subtract the mean from each value: (xᵢ − x̄). 3. Square each deviation: (xᵢ − x̄)². 4. Sum all squared deviations: Σ(xᵢ − x̄)². 5. Divide by n − 1 (not n): s² = Σ(xᵢ − x̄)² / (n − 1). The n−1 is crucial — it's what makes this sample variance rather than population variance.
Why does sample variance divide by n−1 instead of n?
Dividing by n−1 is called Bessel's correction. When you calculate variance from a sample, the sample mean tends to be closer to the sample data points than the true population mean would be. This systematically underestimates variance. Dividing by n−1 corrects this bias, producing an unbiased estimator of the population variance. The number n−1 represents the degrees of freedom — you lose one degree because the mean constrains the data.
When should I use sample variance vs population variance?
Use sample variance (s²) when your data is a subset drawn from a larger population — this includes surveys, experiments, clinical trials, and most real-world scenarios. Use population variance (σ²) only when you have every single data point from the entire population — census data, a complete class of students, or every product in a batch. When in doubt, use sample variance.
What is the sample variance of 3, 7, 7, 19?
For the sample 3, 7, 7, 19: Mean = (3+7+7+19)/4 = 9. Deviations: −6, −2, −2, 10. Squared deviations: 36, 4, 4, 100. Sum of squares = 144. Sample variance (s²) = 144 / (4−1) = 144 / 3 = 48. Sample standard deviation = √48 ≈ 6.928. The high variance reflects the large gap between 19 and the other values.
How many data points do I need for sample variance?
You need at least 2 data points for sample variance. With only 1 data point, n−1 = 0 and division by zero is undefined. In practice, more data points yield a more reliable variance estimate. For normally distributed data, 30+ observations is a common threshold for the sample variance to stabilize.