Standard Deviation Calculator
Paste a list of numbers to get the sample or population standard deviation, variance, mean, sum of squares, coefficient of variation and a 95% confidence interval for the mean — with every step of the working shown.
Written by Suhaib Hassan. The formulas follow the NIST/SEMATECH Engineering Statistics Handbook. Our methodology.
It uses the two-pass method — find the mean, then sum the squared deviations from it — which avoids the rounding errors of the shortcut formula.
Most calculators return a bare number. This one shows the deviation table, says which divisor it used, and plots your data against the matching bell curve.
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Values used
Total of all values
Σ(x − mean)²
SD ÷ |mean| × 100
SD ÷ √n
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Your data against a normal curve
The curve is a normal distribution with your mean and standard deviation; dots are your values. The shaded band is one standard deviation either side of the mean.
Step-by-step working
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| # | Value (x) | Deviation (x − mean) | Squared deviation |
|---|
Population vs sample standard deviation
The two formulas differ only in the divisor. The sample version divides by n − 1 so that the variance estimate is not biased low when the mean is itself estimated from the same data.
Population SD (σ) divisor N
Use when your list is every member of the group you care about — every employee, every unit in a batch.
Sample SD (s) Bessel-corrected, divisor n − 1
Use when your list is a sample drawn from a larger population — the usual case in surveys and experiments.
Why n − 1? Deviations are measured from the sample mean, which sits closer to the data than the true population mean does, so Σ(x − x̄)² tends to be smaller than Σ(x − μ)². Dividing by n − 1 instead of n corrects that on average: the sample variance s² is then an unbiased estimator of σ². (The square root s is still very slightly biased for small n.)
Symbols glossary
What each symbol on this page means.
| Symbol | Name | Meaning |
|---|---|---|
| s | Sample standard deviation | Square root of the sample variance (divisor n − 1). |
| σ | Population standard deviation | Square root of the population variance (divisor N). |
| x̄ , μ | Sample mean, population mean | The arithmetic average of the values. |
| n , N | Sample size, population size | How many values there are. |
| SS | Sum of squares | Σ(x − mean)², the total squared distance from the mean. |
| CV | Coefficient of variation | Standard deviation as a percentage of the mean; compares spread across different scales. |
| SE | Standard error of the mean | s ÷ √n — how much a sample mean is expected to vary. |
Worked example: checking a batch of ball bearings
Eight bearings are measured (mm): 25.002, 25.008, 24.996, 25.004, 25.000, 24.998, 25.006, 25.002. How consistent is the process? (Load this data from the examples menu to see it above.)
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Mean
Add the eight values and divide by 8.
x̄ = 200.016 ÷ 8 = 25.002 mm - 2
Squared deviations
Deviations are 0, +0.006, −0.006, +0.002, −0.002, −0.004, +0.004, 0.
Σ(x − x̄)² = 0.000112 - 3
Sample variance and SD
The bearings are a sample from a production run, so divide by n − 1 = 7.
s² = 0.000112 ÷ 7 = 0.000016 → s = 0.004 mm
Where standard deviation is used
Any time you need to describe how spread out measurements are, not just where their centre is.
Quality control
Process capability and control charts compare the standard deviation of a measurement with the tolerance the part must meet.
Consistency of a processFinance
The standard deviation of an asset's returns is a common volatility measure and the risk axis in mean–variance analysis.
Volatility & riskResearch & medicine
Studies report results as mean ± SD to show how much individuals differ, and use the SD to plan sample sizes.
Describing variabilityWhat is standard deviation?
Standard deviation measures how far values typically sit from their mean. A small standard deviation means the values cluster tightly around the average; a large one means they are spread out. It is expressed in the same units as the data, which is why it is used more often than variance.
How to calculate standard deviation
- Find the mean of the values.
- Subtract the mean from each value and square the result.
- Add the squares to get the sum of squares (SS).
- Divide SS by n − 1 (sample) or N (population) to get the variance.
- Take the square root of the variance.
By Suhaib Hassan · Reviewed 28 September 2026 · How CalculatePilot verifies formulas →
The 95% confidence interval for the mean
For a sample, the interval is x̄ ± t × s ÷ √n, where t is the Student t critical value with n − 1 degrees of freedom (2.262 for n = 10). It is shown only for the sample setting, because a population mean is known exactly and needs no interval.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods — 1.3.5.6 Measures of scale
- NIST/SEMATECH e-Handbook of Statistical Methods — 1.3.5.2 Confidence limits for the mean
- ISO 3534-1:2006, Statistics — Vocabulary and symbols — Part 1: General statistical terms and terms used in probability.
Frequently asked questions
Why do we divide by n − 1 for a sample?
Because the deviations are measured from the sample mean rather than the unknown true mean, the raw sum of squares runs low. Dividing by n − 1 (Bessel's correction) makes the sample variance an unbiased estimate of the population variance.
What is the difference between standard deviation and standard error?
Standard deviation describes the spread of individual values. Standard error of the mean, s ÷ √n, describes how much the sample mean itself would vary from sample to sample, so it shrinks as the sample grows while the standard deviation does not.
How much do outliers affect standard deviation?
A great deal, because deviations are squared. One extreme value can inflate the standard deviation substantially. If outliers are a concern, compare it with the interquartile range, which is far less sensitive.
Can standard deviation be negative?
No. It is the square root of a sum of squares, so it is zero or positive. It is zero only when every value is identical.
Should I choose sample or population?
Choose population only if your list contains every member of the group you are describing. If it is a sample used to learn about a larger group, choose sample. When in doubt, sample is the standard choice.
What does the coefficient of variation tell me?
It expresses the standard deviation as a percentage of the mean, so you can compare the relative spread of data measured on different scales. It is only meaningful when the mean is not close to zero and the data is on a ratio scale.
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