Paste or type your numbers below. You get the standard deviation, the variance, the mean and the rest of the summary immediately — along with the full working, so you can check the arithmetic or reproduce it by hand.
Separate values with commas, spaces, tabs or new lines — paste a column straight from a spreadsheet and it will parse. Decimals and negatives are fine.
The shaded bands are one, two and three standard deviations either side of the mean. 5 of 8 values — 63% — fall inside the innermost band.
Show the working, step by step
What the standard deviation actually tells you
The mean tells you where a set of numbers sits. The standard deviation tells you how tightly
they cluster around that point. Two datasets can share an identical mean and describe
completely different situations: 50, 50, 50 and 0, 50, 100 both
average 50, but the first has a standard deviation of 0 and the second of about 41.
Because it is calculated by squaring the distance of each value from the mean and then taking a square root at the end, the standard deviation comes back in the same units as your original data. If you measured heights in centimetres, the standard deviation is in centimetres. That is its practical advantage over the variance, which is in centimetres squared and so is harder to reason about directly.
As a rough guide, for data that follows a normal distribution roughly 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. The dot plot above shades the one-standard-deviation band on your own numbers and reports what fraction actually landed inside it — which is often a useful reality check on whether your data is anywhere near normal.
The formula
The sample and population standard deviations differ in one place only: the denominator.
| Quantity | Sample | Population |
|---|---|---|
| Symbol | s | σ |
| Mean | x̄ = Σx / n | μ = Σx / N |
| Variance | s² = Σ(x − x̄)² / (n − 1) | σ² = Σ(x − μ)² / N |
| Standard deviation | s = √s² | σ = √σ² |
Everything else — squaring the deviations so that values above and below the mean cannot cancel, summing them, then taking the square root to get back to the original units — is identical. The formula page works through why each step is there, and the notation page covers what σ, s, μ and x̄ each refer to.
Working one through by hand
Take the eight values 2, 4, 4, 4, 5, 5, 7, 9, treated as a complete population.
The mean is 40 ÷ 8 = 5. Subtracting 5 from each value gives the deviations
−3, −1, −1, −1, 0, 0, 2, 4. Squaring those gives 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32.
Dividing by N = 8 gives a variance of 4, and the square root of 4 is a standard deviation
of exactly 2.
Treat the same eight numbers as a sample and only the last two steps change: 32 ÷ 7 = 4.571, and √4.571 = 2.138. The sample figure is larger, as Bessel's correction always makes it. Paste those numbers into the calculator above and switch between the two modes to watch the denominator change.
Where people go wrong
- Averaging standard deviations. To combine the spread of several groups you must average the variances, weighted by degrees of freedom, and take the square root at the end. Averaging the standard deviations themselves gives a different and wrong answer. Use the pooled SD calculator.
- Reaching for population because the word "population" appears in the problem. What matters is whether your numbers are the whole group, not what the group is called. A survey of 500 residents of a city is a sample, not a population.
- Comparing standard deviations across different units or wildly different means. An SD of 5 means something quite different against a mean of 10 than against a mean of 10,000. Use the relative standard deviation for that comparison.
- Assuming the 68–95–99.7 rule always applies. Those percentages hold for a normal distribution. On skewed data — incomes, waiting times, most count data — they can be badly off.
Related calculators
-
Variance calculator
The same computation stopped one step earlier, before the square root.
-
Relative SD (%RSD)
Spread as a percentage of the mean, for comparing across different units.
-
Z-score calculator
How many standard deviations a single value sits from the mean.
-
Standard error
How much a sample mean would wobble if you sampled again.
-
Pooled SD
Combine the spread of two or more groups into one figure.
-
Grouped data SD
For data that arrives as class intervals and frequencies.
Common questions
Should I choose sample or population?
Choose population only when your numbers are the entire group you care about — every student in one class, all twelve months of last year's sales. Choose sample when the numbers are a subset you are using to say something about a larger group.
Sample is the right answer far more often than people expect, and it is the default here. If you are unsure, use sample: it gives the slightly larger, more cautious estimate. The full comparison is here.
Why does the sample formula divide by n − 1 instead of n?
Because a sample's own mean sits closer to its own data than the true population mean does. Measuring spread around that too-convenient centre makes the data look less varied than the population really is. Dividing by n − 1 rather than n inflates the result just enough to cancel that bias out.
This is Bessel's correction. The effect is large for small samples — with n = 5 it raises the variance by 25% — and negligible once n is in the hundreds.
What counts as a high standard deviation?
There is no universal threshold, because standard deviation carries the units of your data. An SD of 3 is enormous for adult body temperature in °C and trivial for house prices in dollars.
To compare spread across different units or very different means, use the relative standard deviation, which expresses the SD as a percentage of the mean and so has no units.
Can I paste a column from Excel or Google Sheets?
Yes. Copy the cells and paste them straight into the box — the line breaks and tabs are handled. Anything that is not a number, such as a header row or a currency symbol, is skipped and reported above the results so you can see exactly what was ignored.
Is my data sent anywhere?
No. The whole calculation runs in your browser in JavaScript. Nothing you type is uploaded, logged or stored, and the page works offline once it has loaded.
How precise is the result?
Calculations run at full IEEE-754 double precision and use Welford's algorithm, which stays accurate on data with a large mean and a small spread — the case where the common textbook shortcut loses most of its significant digits. Displayed values are rounded to six significant figures. The method is documented in full.