# Sample Variance Calculator

Free sample variance calculator. Enter your data set to calculate sample variance (s²), population variance (σ²), standard deviation, and mean. Uses Bessel's correction.

## What this calculates

Calculate sample variance and population variance directly from your data values. Enter up to 8 values and get both variance types, standard deviations, and the mean.

## Inputs

- **Data Set (comma-separated)** — This field is used as a fallback. Enter your data below.
- **Value 1** — First data value.
- **Value 2** — Second data value.
- **Value 3** — Third data value (optional).
- **Value 4** — Fourth data value (optional).
- **Value 5** — Fifth data value (optional).
- **Value 6** — Sixth data value (optional).
- **Value 7** — Seventh data value (optional, enter 0 to skip).
- **Value 8** — Eighth data value (optional, enter 0 to skip).
- **Include zeros in calculations?** — Toggle on if zeros are actual data values, not empty placeholders.

## Outputs

- **Sample Variance (s²)** — Variance using n-1 (Bessel's correction).
- **Population Variance (σ²)** — Variance using n.
- **Sample Std Dev (s)** — Square root of sample variance.
- **Population Std Dev (σ)** — Square root of population variance.
- **Mean (x̄)** — Arithmetic mean of the data set.
- **Count (n)** — Number of data values used.
- **Sum of Squared Deviations** — Σ(xᵢ - x̄)² used in the variance calculation.

## Details

Variance measures how far data values spread from their average. This calculator works directly from data values rather than summary statistics.

**Sample Variance (s²):**
s² = Σ(xᵢ - x̄)² / (n - 1)

**Population Variance (σ²):**
σ² = Σ(xᵢ - μ)² / N

**Step-by-step process:**

1. Calculate the mean (x̄) of all values
2. Subtract the mean from each value to get deviations
3. Square each deviation
4. Sum all squared deviations
5. Divide by (n - 1) for sample variance or by n for population variance

**Why the Difference?**
Sample variance uses (n - 1) because a sample tends to underestimate the true population variance. This correction (called Bessel's correction) makes the sample variance an unbiased estimator of the population variance.

## Frequently Asked Questions

**Q: When do I use sample variance vs. population variance?**

A: Use sample variance (dividing by n-1) when your data is a subset of a larger population. Use population variance (dividing by n) only when your data includes every member of the group you are studying. Most real-world data is a sample, so sample variance is the safer default.

**Q: Why does sample variance divide by n-1 instead of n?**

A: This is Bessel's correction. When you estimate the mean from the same data, you lose one degree of freedom. Dividing by (n-1) compensates for this, making the sample variance an unbiased estimator. If you divided by n, the sample variance would systematically underestimate the true population variance.

**Q: What does the include zeros toggle do?**

A: By default, zero values are treated as empty placeholders and excluded from the calculation. If zero is a real data value in your set (like a test score of 0 or a temperature reading of 0), toggle this on so zeros are included in the mean and variance calculations.

**Q: What if I have more than 8 data values?**

A: For larger data sets, you can use the related Variance Calculator which accepts summary statistics (Σx, Σx², and n). Compute those sums in a spreadsheet and enter them directly.

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Source: https://vastcalc.com/calculators/statistics/sample-variance
Category: Statistics
Last updated: 2026-04-08