Standard deviation tells you how spread out a set of numbers is: a small value means the data clusters tightly around the average, and a large value means it is scattered far and wide. This free standard deviation calculator does the whole job in your browser — paste in any list of numbers and it works out the population standard deviation (σ), the sample standard deviation (s), both variances, the mean, median, sum, minimum, maximum and range, updating live as you type. Nothing is uploaded, there is no sign-up, and there is no limit on how many numbers you can process: a homework set of eight values and a spreadsheet export of eight thousand are both fine.
How to use this standard deviation calculator
Type or paste your numbers into the box. Commas, spaces, tabs, semicolons and line breaks all work as separators, so you can copy a column straight out of Excel or Google Sheets and drop it in. Results appear immediately and refresh with every keystroke. The two headline numbers are the standard deviations: use the population result when your numbers are the complete set you care about (every exam score in the class, every day of last month), and the sample result when your numbers are a sample drawn from a bigger group you want to make conclusions about (100 surveyed customers representing all your customers).
The formula, in plain language
For a population, σ = √( Σ(x − μ)² / N ): subtract the mean from every value, square each difference, average the squares, then take the square root. For a sample, s uses the same steps but divides by n − 1 instead of n. That small change, called Bessel’s correction, compensates for the fact that a sample’s own mean is usually closer to its values than the true population mean is, which would otherwise make the spread look smaller than it really is. Squaring the standard deviation gives the variance, which this calculator also reports for both cases.
A worked example
Try the classic textbook data set 2, 4, 4, 4, 5, 5, 7, 9 (the “Try an example” button loads it). The mean is 5 and the median is 4.5. The squared differences from the mean add up to 32, so the population variance is 32 ÷ 8 = 4 and the population standard deviation is exactly 2. Dividing by n − 1 instead gives a sample variance of about 4.571 and a sample standard deviation of about 2.138. If your statistics course or textbook gives both answers for this data set, this is why — and now you can check which one a question is asking for before you answer it.
What the other results mean
The mean is the ordinary average. The median is the middle value when the numbers are sorted, which resists distortion from one extreme outlier. The range is simply maximum minus minimum — a crude spread measure that, unlike standard deviation, is completely controlled by the two most extreme values. Comparing the mean and median is a quick sanity check: when they sit close together the data is fairly symmetric, and when the mean is dragged far from the median, a few very large or very small values are pulling it.
Frequently asked questions
What is the difference between population and sample standard deviation?
The population version (σ) divides by N and describes a complete data set. The sample version (s) divides by n − 1 and estimates the spread of a larger population from a smaller sample; it is the one used in most research, surveys and lab work.
Why is the sample standard deviation always a bit larger?
Because it divides the same total of squared differences by a smaller number (n − 1 instead of n). The correction deliberately inflates the estimate slightly to make up for the information lost by not measuring the whole population.
Can standard deviation be zero or negative?
Never negative — it is a square root of an average of squares. It is zero only when every value in the set is identical, meaning there is no spread at all.
How many numbers can I paste in?
As many as you like. The calculation runs entirely on your device, so long lists from spreadsheets work fine and your data never leaves the page.
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Versión en español: Calculadora de Desviación Estándar — Desviación Típica Poblacional y Muestral. You may also like: Average Calculator — Mean, Median, Mode & Range of Any Number Set · Percentage Increase Calculator — Percentage Change Between Two Numbers.
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