What is the difference between population and sample standard deviation?

Population standard deviation divides by N (total count), while sample standard deviation divides by N-1. Use sample when your data is a subset of a larger population.

What does standard deviation tell you?

Standard deviation measures how spread out the values are from the mean. A low value means data points are close to the mean; a high value means they are spread over a wider range.

Standard Deviation Calculator

Enter up to 10 values to calculate both the population standard deviation (sigma) and sample standard deviation (s). Also shows variance, mean, and data count.

Standard deviation measures how spread out a set of numbers is from their mean. A low standard deviation says the values cluster tightly around the average; a high one says they're scattered. It's one of the most-used descriptive statistics in science, engineering, finance, and quality control because so many real-world distributions are approximately normal, and "how unusual is this value?" is naturally answered in standard-deviation units.

This calculator computes both population standard deviation (σ, divides by N) and sample standard deviation (s, divides by N−1). Use sample when your data is a subset of a larger population — which is the case in almost every real-world dataset. Use population only when you have measured every member of the group you care about.

The math is straightforward but the interpretation is where most people slip. A standard deviation has the same units as the underlying data: if you're measuring heights in inches, the SD is in inches. Saying "the SD is 3 inches" is meaningful; saying "the SD is 3" without units isn't.

Inputs

Value 1

Value 2

Value 3

Value 4

Value 5

Value 6 (0 to skip)

Value 7 (0 to skip)

Value 8 (0 to skip)

Value 9 (0 to skip)

Value 10 (0 to skip)

Results

Population Std Dev (σ)

4.898979

Sample Std Dev (s)

5.237229

Population Variance

Sample Variance

27.428571

Mean

Count

Last updated: April 16, 2026

Formula

Mean: x̄ = (Σ xᵢ) / N Population variance: σ² = Σ(xᵢ − x̄)² / N Population standard deviation: σ = √σ² Sample variance: s² = Σ(xᵢ − x̄)² / (N − 1) Sample standard deviation: s = √s² The N−1 divisor (Bessel's correction) compensates for using the sample mean instead of the true population mean, which slightly underestimates true variance. Example: data 10, 12, 23, 23, 16, 23, 21, 16 Mean: 144 / 8 = 18 Deviations: −8, −6, +5, +5, −2, +5, +3, −2 Squared: 64, 36, 25, 25, 4, 25, 9, 4 = 192 Population σ² = 192 / 8 = 24, σ ≈ 4.90 Sample s² = 192 / 7 ≈ 27.43, s ≈ 5.24

How to use this calculator

Enter your data values in the input fields. Leave any unused fields at 0 — the calculator will detect and skip them.
Read both the population and sample standard deviations. For most real datasets, use the sample value.
Compare to the mean: SD as a fraction of the mean (the "coefficient of variation") helps when comparing variability across different-scale measurements.
For larger datasets than 10 points, use a spreadsheet (=STDEV.S for sample, =STDEV.P for population in Excel and Google Sheets).

Worked examples

Test scores

Five test scores: 78, 82, 85, 90, 95 Mean: 86 Sample SD: ≈ 6.6 Most scores fall within 1 SD of the mean (79–93), which is 4 of 5 scores. This matches the normal-distribution rule of thumb: ~68% of data falls within ±1 SD.

Stock returns

Annual returns for a stock over 10 years: +12%, −8%, +25%, +6%, +18%, −15%, +9%, +22%, +3%, −5% Mean: 6.7% Sample SD: ≈ 12.6% A 12.6% SD is high — the stock is volatile. Compare to the S&P 500's long-run SD of roughly 15% annual returns. The same average return with a much lower SD would be a much "smoother" investment.

When to use this calculator

Use standard deviation when: - Describing the spread of any quantitative dataset - Comparing variability across groups - Setting quality-control tolerances ("within ±3 SD") - Calculating z-scores (how unusual a value is) - Building confidence intervals or running t-tests

For skewed data (income, lifetimes, time-to-failure), standard deviation can be misleading because most observations might be on one side of the mean. In those cases, prefer median + interquartile range (IQR) for description.

Standard deviation assumes equally-weighted observations. For weighted samples (e.g., survey data with weights), use the weighted standard deviation formula.

Common mistakes to avoid

Confusing variance with standard deviation. SD is the square root of variance. They're proportional but not equal.
Using population SD when sample SD is appropriate. Almost all real-world data is a sample of a larger population.
Treating SD as a fraction of the mean automatically. SD has units; coefficient of variation (SD / mean) is dimensionless.
Applying the "68/95/99.7" empirical rule to non-normal data. The rule comes from the normal distribution; for skewed or heavy-tailed data it can be very wrong.
Comparing SDs across different units of measurement. SD in dollars and SD in percent aren't comparable.
Reporting SD without sample size. A sample of n=4 can have a misleadingly precise-looking SD.

Frequently Asked Questions

Sources & further reading

NIST/SEMATECH e-Handbook of Statistical Methods — U.S. National Institute of Standards and Technology

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