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Standard Deviation Calculator
Sample or population standard deviation, variance and mean of a data set, with every step shown.
Calculated in your browser – nothing you enter is sent or stored.
This standard deviation calculator finds the sample or population standard deviation, variance, mean, median and range of a data set, and shows every step of the work in a table. Paste numbers separated by commas, spaces or new lines – including a column copied from Excel or Google Sheets.
How to Use the Standard Deviation Calculator
- Type or paste your numbers.
- Choose sample if the data is a sample from a larger group (most homework and research), or population if it is every member of the group.
- Read the result and follow the steps below the calculator.
Standard Deviation Formulas
- Population: σ = √[ Σ(x − μ)² ÷ N ]
- Sample: s = √[ Σ(x − x̄)² ÷ (n − 1) ]
The sample formula divides by n − 1 instead of n (Bessel's correction) because a sample's own mean sits closer to its values than the true population mean does; dividing by a smaller number corrects for that.
Worked Example
Data: 10, 12, 23, 23, 16, 23, 21, 16 (n = 8).
- Mean: 144 ÷ 8 = 18.
- Deviations squared: (10 − 18)² = 64, (12 − 18)² = 36, (23 − 18)² = 25 three times, (16 − 18)² = 4 twice, (21 − 18)² = 9.
- Sum of squares: 64 + 36 + 25 + 25 + 4 + 25 + 9 + 4 = 192.
- Variance: sample 192 ÷ 7 = 27.43; population 192 ÷ 8 = 24.
- Standard deviation: sample √27.43 = 5.237; population √24 = 4.899.
What Standard Deviation Tells You
Standard deviation measures how spread out values are around the mean, in the same units as the data. A small SD means values cluster near the average; a large one means they vary widely. For data that follows a normal (bell-shaped) distribution, the empirical rule applies:
| Range | Share of values |
|---|---|
| Mean ± 1 SD | about 68% |
| Mean ± 2 SD | about 95% |
| Mean ± 3 SD | about 99.7% |
For example, SAT section scores are designed to have a standard deviation of about 100, so most students score within 100 points of the average.
Related Measures
- Variance is the standard deviation squared.
- Standard error of the mean = SD ÷ √n – how precisely the sample mean estimates the population mean.
- Coefficient of variation = SD ÷ mean, useful for comparing spread between data sets with different units.
In Excel and Google Sheets, STDEV.S gives the sample standard deviation and STDEV.P the population one.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods – Measures of scale
- Khan Academy – Variance and standard deviation of a sample
Last reviewed: October 2026 by the ShrinkBee team. Results are estimates for planning and education, not financial, tax, legal or medical advice.
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