Standard Deviation Calculator
Find the standard deviation for population or sample data.
How standard deviation works
The standard deviation measures how spread out the values are around the mean. A small standard deviation means the values are close to the mean; a large one means they are widely scattered.
Population and sample use different formulas. The population standard deviation divides by N; the sample standard deviation divides by n−1 (Bessel's correction) because it estimates the population value from a sample.
The variance is the standard deviation squared. Both are reported so you can see the relationship between the two measures of spread.
The formulas
Population: σ = √(Σ(x−μ)² ÷ N) · Sample: s = √(Σ(x−x̄)² ÷ (n−1))
Subtract the mean from each value, square the differences, sum them, divide by N (population) or n−1 (sample), and take the square root. Use the population formula when your data covers the entire group, and the sample formula when it is a sample.
Worked example
Values 2, 4, 4, 4, 5, 5, 7, 9
The mean is 5. The sum of squared deviations is 32. As a population: √(32 ÷ 8) = 2. As a sample: √(32 ÷ 7) ≈ 2.14.
Frequently asked questions
What is the difference between population and sample standard deviation?
Population divides by N and describes the entire group. Sample divides by n−1 (Bessel's correction) and estimates the population value from a sample, which avoids underestimating spread.
Why do I need at least two values for a sample?
With n−1 as the denominator, a single value would divide by zero. You need at least two observations to estimate a sample standard deviation.
What does a standard deviation of zero mean?
Every value is identical to the mean — there is no spread at all.
Related research methods
- Descriptive Statistics
When to use this tool
Use this to summarise the spread of a set of continuous values and to report variability alongside a mean.
Assumptions
- For the sample standard deviation, values are treated as a sample from a larger population.
Limitations
- The standard deviation is sensitive to outliers.
- It describes spread around the arithmetic mean, which may not be the best summary for skewed data.
How to interpret the result
The standard deviation is the typical distance of values from the mean, in the original units.
How to report the result
Report the mean and standard deviation together — for example, 'M = 5.0, SD = 1.2'.
Next steps in your analysis
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