Formula & Calculator
Sample Variance
Estimates population variance from a sample, using Bessel's correction (n-1) to produce an unbiased estimate.
Interpretation
s² = Σ(xᵢ−x̄)²/(n−1). Unbiased estimator of population variance from a sample. Denominator n−1 (Bessel's correction). Used in inferential statistics.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| s² | Sample variance | |
| xᵢ | Each data value | |
| x̄ | Sample mean | |
| n | Sample size |
What it means
Sample variance (s²) is the sum of squared deviations from the sample mean, divided by n−1 instead of n. This correction (Bessel’s correction) makes the sample variance an unbiased estimator of the population variance, because using n would underestimate the variance, especially for small samples. The sample variance is the basis for the t‑statistic, confidence intervals, and ANOVA. It is the standard measure of dispersion in sample data. Understanding the distinction between population and sample variance is crucial for correct statistical inference and for avoiding bias in estimates.
Worked example
Sample Variance – Two Examples
Real‑World| Parameter | Value |
|---|---|
| Data | 2,4,6,8,10 |
| x̄ | 6 |
| n | 5 |
| Parameter | Value |
|---|---|
| Data | 5,5,5,5 |
| x̄ | 5 |
| n | 4 |
Common mistakes
- Sample variance: Uses n−1 (degrees of freedom) in the denominator – do not use n.
- Mean x̄: The sample mean – not the population mean.
- Units: Variance is in squared units of the data.
- Unbiased estimator: s² is an unbiased estimator of σ².
- N‑1 vs. N: Using N would bias the estimate downwards.
Applications
Sample variance (s²) estimates the population variance from a sample, using (n−1) in the denominator to provide an unbiased estimate. This correction is crucial in inferential statistics. Engineers use sample variance to estimate process variability from limited data, to construct confidence intervals, and to perform hypothesis tests. In research, it measures how well the sample represents the population. In quality control, it helps set control limits. By using sample variance, professionals can make reliable inferences about the population without measuring every unit. This formula is a cornerstone of statistical analysis, enabling decision‑making under uncertainty and ensuring that conclusions are based on sound principles.
- Estimation of population variability from sample data
- Confidence interval construction for means and variances
- Hypothesis testing (t‑tests, F‑tests)
- Quality control – sample‑based process monitoring
- Research and experimental data analysis
Frequently Asked Questions
s² = Σ(xᵢ−x̄)² / (n−1). It estimates the population variance from a sample using Bessel's correction (n−1) to account for the fact that the sample mean is used instead of the true mean.
Because dividing by n−1 makes the sample variance an unbiased estimator of the population variance. Using n would give a biased (too low) estimate.
For larger samples, the difference between dividing by n and n−1 becomes negligible. But for small samples, the correction is important to avoid underestimating variability.
The sample standard deviation s is the square root of the sample variance s². It brings the measure back to the original units.
It is used to compute the t‑statistic (for one‑sample t‑tests) and in F‑tests for comparing variances. It is also used in confidence interval calculations.
For normally distributed data, the standard error of s² is approximately σ² √(2/(n−1)). It measures the sampling variability of the variance estimate.
A sample variance of zero means all values are identical (no variability). This is rare in real data.
Sample variance is highly sensitive to outliers because the deviations are squared. A single extreme value can inflate s² dramatically.
In quality control, the sample variance (or range) is used to monitor process variability. Control charts for variance (e.g., S‑charts) detect changes in spread.
No, variance is always non‑negative because it is a sum of squared terms divided by a positive number.