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What does this calculator estimate?
The standard deviation calculator computes the sample or population standard deviation, variance, mean, and sum for a comma-separated set of numbers, plus the standard error and a 95% margin of error.
Sample vs population standard deviation
Use the population standard deviation (σ) when your data is the entire group you care about. Use the sample standard deviation (s) when your data is just a sample of a larger population — it divides by n − 1 to correct for the fact that a sample tends to underestimate true spread. The two values are close for large datasets.
Reading the deviation table
The step-by-step table shows each value, its deviation from the mean (x − mean), and that deviation squared. Squaring the deviations removes negative signs so they can be averaged, which is the core of how variance and standard deviation are built up.
Standard error and margin of error
The standard error of the mean (s/√n) measures how precisely the sample mean estimates the population mean — it shrinks as the sample grows. Multiplying by 1.96 gives a 95% margin of error, the ± interval around the sample mean that would contain the population mean in about 95% of samples.
How this calculator works
Formula
Standard deviation measures the spread of values around the mean. For a sample: s = √(Σ(xᵢ − x̄)² ÷ (n − 1)). For a population: σ = √(Σ(xᵢ − μ)² ÷ n). The standard error of the mean is s/√n, and the 95% margin of error is 1.96 × that.
Worked example
For the values 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5. The sample standard deviation is about 2.138, the population standard deviation is 2.000, and the standard error of the mean is about 0.756.
Assumptions to verify
- Values are entered as a comma- or space-separated numeric list.
- Choose sample when the data is a subset of a larger population; choose population when it represents the whole population.
- At least two values are required.
Frequently asked questions
What is a standard deviation?
It measures how spread out a set of numbers is around their average. A small standard deviation means the values cluster tightly; a large one means they vary widely. It is the square root of the variance.
What is the difference between sample and population standard deviation?
Population standard deviation (σ) uses every member of the group and divides by n. Sample standard deviation (s) uses a subset of a larger population and divides by n − 1 to correct a small-sample bias. Choose the one that matches your data.
How do I calculate standard deviation by hand?
Find the mean, subtract it from each value and square the result, sum those squares, divide by n (population) or n − 1 (sample), and take the square root. The deviation table here shows each of those steps.
What is the standard error of the mean?
It is the sample standard deviation divided by the square root of the sample size (s/√n). It measures how precisely the sample mean estimates the population mean and gets smaller as the sample gets larger.
What does a 95% margin of error mean?
It is 1.96 times the standard error. The interval ± margin of error around your sample mean would contain the true population mean in about 95% of repeated samples of the same size.
Cite this tool
BoringToolsKit. “Standard Deviation Calculator.” boringtoolskit.com/standard-deviation-calculator/ (reviewed August 2026). Free to reference in articles, syllabi, and answer posts with a link.
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