Variance Calculator

When should I use population vs. sample variance?

Use population variance (σ²) when your data includes every member of the group you care about — e.g., the heights of all employees in a 50-person company. Use sample variance (s²) when your data is a subset drawn from a larger population — e.g., 50 people surveyed from a city of 1 million. Sample variance uses N−1 to correct for the underestimation bias that occurs when estimating spread from a sample.

What is Bessel's correction (the N−1 denominator)?

A sample's values tend to cluster closer to the sample mean than to the true population mean (because the sample mean is chosen to minimize the sum of squared deviations within the sample). This causes the simple N denominator to underestimate the true variance. Dividing by N−1 instead of N corrects for this bias, producing an unbiased estimator of the population variance.

What does standard deviation tell me that variance does not?

Standard deviation is the square root of variance, so it is in the same units as the original data (e.g., meters, dollars). Variance is in squared units, which is harder to interpret. A standard deviation of 5 cm means the typical data point is about 5 cm from the mean — immediately intuitive. Variance of 25 cm² is mathematically convenient but less directly interpretable.

How is variance related to the normal distribution?

For a normal (bell-curve) distribution, variance σ² completely specifies the spread. About 68% of data falls within ±1σ, 95% within ±2σ, and 99.7% within ±3σ (the empirical rule). These percentages are fixed regardless of the mean — only σ determines how wide or narrow the bell curve is.