Enter your data set and get the standard deviation instantly, with a full step-by-step solution showing exactly how the answer was calculated.
Enter numbers only, separated by commas, spaces, or line breaks.
Sample Standard Deviation
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Count (n)
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Mean
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Variance
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Sum
Step-by-Step Solution
Standard deviation tells you how spread out a set of numbers is from its average (mean). A low standard deviation means the values are clustered close to the mean; a high standard deviation means they're spread over a wider range.
How to use the standard deviation calculator
Enter your data values, separated by commas, spaces, or line breaks.
Choose whether your data is a sample (a subset of a larger population) or the entire population.
Click Calculate to see your result and a full step-by-step breakdown.
Not sure whether to use sample or population? If your data represents everyone or everything you care about, use population. If it's a smaller group meant to represent a larger one, use sample — this is the far more common case in real research and coursework.
What is standard deviation?
Standard deviation (usually written as σ for a population or s for a sample) measures the typical distance between each data point and the mean of the data set. It's the square root of variance, which makes it easier to interpret because it's expressed in the same units as your original data.
Population standard deviation
σ = √(Σ(x−μ)² ÷ N)
Where μ is the population mean, N is the population size, and Σ means "sum of."
How to find population standard deviation by hand
Find the mean of all values in the population.
Subtract the mean from each value to get each deviation.
Square each deviation so negative and positive differences don't cancel out.
Average the squared deviations — this is the population variance.
Take the square root of the variance to get the population standard deviation.
Sample standard deviation
s = √(Σ(x−x̄)² ÷ (n−1))
Where x̄ is the sample mean, n is the sample size, and the denominator uses n−1 (Bessel's correction) instead of n.
How to find the sample standard deviation by hand
Find the mean of your sample.
Subtract the mean from each value.
Square each deviation.
Sum the squared deviations, then divide by n−1 (not n) — this is the sample variance.
Take the square root of the sample variance to get the sample standard deviation.
Why n−1 instead of n? Using n−1 corrects for the fact that a sample tends to underestimate the true population variance. This adjustment, called Bessel's correction, makes the sample standard deviation a better estimate of the population value.
Standard deviation vs. variance
Variance and standard deviation measure the same underlying idea — spread — but variance is expressed in squared units (which makes it harder to interpret directly), while standard deviation converts that back into the original units by taking the square root. If you also need to work with variance directly, see the TheCalcBright calculator index for related tools.
For a deeper walkthrough of the underlying math, Khan Academy's standard deviation lesson is a solid free reference.
TB
About This Calculator
Written and maintained by the TheCalcBright Team. Every formula on this site is checked against standard statistics references, and calculators are tested against known worked examples before publishing. Have feedback? Contact us.
Cite This Calculator
TheCalcBright Team. (2026). Standard Deviation Calculator. TheCalcBright. https://thecalcbright.com/standard-deviation-calculator/
Frequently Asked Questions
What's the difference between sample and population standard deviation?
Population standard deviation uses every member of the group you're studying and divides by N. Sample standard deviation uses a subset meant to represent a larger population and divides by n−1 instead, which corrects for the tendency of samples to underestimate true variability.
Can standard deviation be negative?
No. Standard deviation is always zero or positive, because it's calculated from squared deviations and a square root. A standard deviation of zero means all values in the data set are identical.
What does a high or low standard deviation mean?
A low standard deviation means your data points are clustered close to the mean. A high standard deviation means they're spread out over a wider range. There's no universal "good" or "bad" value — it depends entirely on the context of your data.
Do I need to know the mean before calculating standard deviation?
Yes, the mean is the first step in every standard deviation formula. This calculator computes the mean automatically as part of the step-by-step solution, so you don't need to calculate it separately first.
Can I use this calculator for grouped or frequency data?
This calculator is designed for raw, ungrouped data values. For grouped data with frequencies, the calculation method differs slightly since each value is weighted by how often it occurs.