A standard deviation calculator measures how spread out a list of numbers is around its mean. Formula: Mean = sum/n. SS = sum((x – mean)^2). Sample SD = sqrt(SS/(n-1)). Population SD = sqrt(SS/n). The values 10, 12, 15, 18 and 20 have a mean of 15 and a sample standard deviation of about 4.1231.
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How to use the standard deviation calculator
- Enter at least two numbers.
- Choose sample or population.
- Read the mean and standard deviation.
How the standard deviation calculator works
The standard deviation calculator measures how spread out a list of numbers is around its mean. Enter at least two numbers separated by commas or spaces and choose sample or population. It finds the mean, adds up the squared distance of each value from the mean, divides and takes the square root:
- Sample: s = √( Σ(x − x̄)² ÷ (n − 1) )
- Population: σ = √( Σ(x − μ)² ÷ n )
The result shows the mean and the standard deviation to 6 decimal places. Sample is the default.
Standard deviation calculator worked example
Take the values 10, 12, 15, 18 and 20. The mean is 75 ÷ 5 = 15. The deviations are −5, −3, 0, 3 and 5, and their squares add to 25 + 9 + 0 + 9 + 25 = 68. For a sample, 68 ÷ 4 = 17, and √17 ≈ 4.123106. For a population, 68 ÷ 5 = 13.6, and √13.6 ≈ 3.687818. The standard deviation calculator shows the one you choose.
| Values | Mean | Sample SD | Population SD |
|---|---|---|---|
| 10, 12, 15, 18, 20 | 15 | 4.123106 | 3.687818 |
| 1 to 10 | 5.5 | 3.02765 | 2.872281 |
| 2, 4, 4, 4, 5, 5, 7, 9 | 5 | 2.13809 | 2 |
| 5, 5, 5, 5 | 5 | 0 | 0 |
| 1, 3 | 2 | 1.414214 | 1 |
Sample or population?
Use population when your list contains every member of the group you care about, such as the scores of all 25 students in one class. Use sample when the numbers are a subset used to estimate a larger group, such as 50 customers surveyed out of thousands. Dividing by n − 1, called Bessel’s correction, makes the sample variance an unbiased estimate of the population variance. The two results get closer as n grows.
What the number means
Standard deviation is in the same units as the data. A small value means the numbers cluster near the mean; a large value means they spread out. For data that is roughly bell-shaped, about 68% of values fall within one standard deviation of the mean and about 95% within two. A list of identical values has a standard deviation of 0.
Variance and related measures
Variance is the square of the standard deviation; for the example sample it is 17. The coefficient of variation divides the standard deviation by the mean, giving a unitless comparison: 4.123 ÷ 15 ≈ 27%. The range, largest minus smallest, is a quicker but rougher measure of spread, and it is very sensitive to a single extreme value.
Outliers
Because deviations are squared, one extreme value can raise the standard deviation a lot. Adding 100 to the example list raises the sample result from about 4.1 to about 34.9. If your data has outliers or is strongly skewed, also look at the interquartile range, which ignores the extremes.
Entering data
Separate values with commas, spaces or line breaks. Semicolons also work. Anything that is not a number, such as a typo like 1O instead of 10, is left out and listed under “Ignored” with the count, so you can fix it. Thousands separators such as 1,200 are understood when you separate values with spaces, semicolons or new lines; in comma-only input, each comma separates values.
Where it is used
A standard deviation calculator is used in many fields. Teachers use spread to see whether test scores are bunched or scattered. Manufacturers track it to keep parts within tolerance. Investors use the standard deviation of returns as a measure of volatility, and lab staff use it to judge how repeatable a measurement is.
Tips
- Pick sample unless you have the whole population.
- Compare the result with the mean to judge spread.
- Look for outliers before drawing conclusions.
- Use the standard deviation calculator result to find z-scores.
Related converters and calculators
Related calculators:
- Z-score calculator: standard scores
- Confidence interval calculator: estimates
- Sample size calculator: surveys
- Statistics calculator: summary
Sources
NIST/SEMATECH e-Handbook: Measures of scale; OpenStax Introductory Statistics 2e, 2.7: Measures of the spread of the data.
Formula and method
Mean = sum/n. SS = sum((x - mean)^2). Sample SD = sqrt(SS/(n-1)). Population SD = sqrt(SS/n).
Choose sample (n-1) or population (n) mode. Requires at least two numbers.
Questions about this calculator
What is the formula?
The square root of the average squared deviation, using n − 1 for a sample.
Sample or population?
Sample unless you have every member of the group.
What does 0 mean?
All values are identical.
How many values do I need?
At least two.
What is variance?
The standard deviation squared.
Check the assumptions and units before relying on a result. Financial, tax and health estimates do not replace advice for your individual circumstances.
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