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Standard deviation is a measure of how spread out numbers are around their mean. It is the square root of the variance, the average squared distance from the mean. A population standard deviation divides by N, a sample standard deviation divides by n − 1, and both are in the same units as the data.
This standard deviation calculator works out both versions as you type, together with the variance, mean, count, sum, standard error and coefficient of variation. It also lists every deviation and shows each step, so you can check homework or a report line by line. Pasting a column from a spreadsheet works as it is; if your numbers are stuck in a paragraph of text, the paragraph to single line converter can tidy them first.
| Data | Mean | Population σ | Sample s |
|---|---|---|---|
| 10, 12, 23, 23, 16, 23, 21, 16 | 18 | 4.899 | 5.2372 |
| 2, 4, 4, 4, 5, 5, 7, 9 | 5 | 2 | 2.1381 |
| 1, 2, 3, 4, 5 | 3 | 1.4142 | 1.5811 |
| 4, 7, 13, 16 | 10 | 4.7434 | 5.4772 |
| 5, 5, 5, 5 | 5 | 0 | 0 |
| 1, 100 | 50.5 | 49.5 | 70.0036 |
The calculator follows these four steps, and the working panel under the tool fills them in with your own numbers.
Add all the values and divide by how many there are. For 10, 12, 23, 23, 16, 23, 21, 16 the sum is 144 and the mean is 144 / 8 = 18.
Subtract the mean from every value. Here the deviations are −8, −6, 5, 5, −2, 5, 3 and −2, and they always add up to 0.
Square each deviation so negatives stop cancelling positives, then add them. The sum of squares for this data is 192.
For a population divide by N: 192 / 8 = 24, and √24 ≈ 4.899. For a sample divide by n − 1: 192 / 7 ≈ 27.43, and the square root is about 5.237.
Population formulas use Greek letters (σ, μ, N); sample formulas use Latin letters (s, x, n).
| Statistic | Population | Sample |
|---|---|---|
| Mean | μ = Σx / N | x = Σx / n |
| Variance | σ² = Σ(x − μ)² / N | s² = Σ(x − x)² / (n − 1) |
| Standard deviation | σ = √σ² | s = √s² |
| Standard error of the mean | σ / √N | s / √n |
| Coefficient of variation | σ / |μ| × 100% | s / |x| × 100% |
| z-score of a value | (x − μ) / σ | (x − x) / s |
| Value x | x − 18 | (x − 18)² |
|---|---|---|
| 10 | −8 | 64 |
| 12 | −6 | 36 |
| 23 | 5 | 25 |
| 23 | 5 | 25 |
| 16 | −2 | 4 |
| 23 | 5 | 25 |
| 21 | 3 | 9 |
| 16 | −2 | 4 |
| Sum | 0 | 192 |
The only difference in the arithmetic is the divisor, but it changes the answer most for small data sets.
| Population | Sample | |
|---|---|---|
| Use it when | Your data is every member of the group (all 30 students in one class) | Your data is a subset that stands in for a bigger group (30 students from a whole school) |
| Divide the sum of squares by | N | n − 1 (Bessel correction) |
| Symbols | σ, σ², μ, N | s, s², x, n |
| Fewest values needed | 1 | 2 |
| Spreadsheet functions | STDEV.P, VAR.P | STDEV.S, VAR.S |
| Python statistics module | pstdev, pvariance | stdev, variance |
| Result for the worked example | 4.8989794855664 | 5.2372293656638 |
| Feature | Typical online calculator | This calculator |
|---|---|---|
| Results | After pressing Calculate | Live as you type |
| Sample and population | One at a time | Both, with the other type shown under the result |
| Working | Formula only, or a fixed example | Every step with your numbers, plus a full deviations table with z-scores |
| Extra statistics | Mean and variance | Standard error, coefficient of variation, median, minimum, maximum and range |
| Check against the normal curve | Not offered | Share of your data within 1, 2 and 3 standard deviations |
| Large or offset data | Often a naive sum-of-squares formula | Welford algorithm, accurate even for values like 1,000,000,004 |
| Keep or share | Retype the data | Copy results, download the table as CSV, or share a link that reopens your data |
Check a hand calculation step by step. The working panel shows the mean, the sum of squares and the division, so you can find exactly where an answer went wrong.
Report repeated measurements as mean ± standard deviation, or use the standard error for error bars. Round the final figures with the significant figures calculator.
See how tightly a class scored around the average, and use the z-score column to find which scores sit unusually high or low.
Track how much a measured size or weight varies between items. A rising standard deviation is often the first sign that a process is drifting.
The coefficient of variation compares variability between data sets with different units or sizes, such as house prices in two cities.
Check how steady your sleep, spending or running times are from week to week. A small standard deviation means a steady routine.
A few details explain most surprises. To keep a data set for later, paste it into the online notepad; data copied from a document with styling can be cleaned with the remove text formatting tool.
Standard deviation is in the units of your values, such as centimetres or dollars. Variance is in squared units, which is why it is harder to read.
Dividing by n − 1 instead of N gives a slightly bigger answer. The gap shrinks as the data set grows.
Each deviation is squared, so one extreme value can raise the result a lot. Compare the range and the median, or check the z-score column for values beyond ±3.
Bessel correction makes the sample variance unbiased, but its square root still slightly underestimates the population standard deviation.
It only makes sense for data measured from a true zero, such as weight or price, and it is undefined when the mean is 0.
The calculator uses Welford running-mean method instead of the shortcut formula with Σx², which can lose most of its digits when the values are large and close together.
It describes the numbers you enter. It does not test hypotheses or give confidence intervals that depend on assumptions about your data.
Enter each value separately. Grouped data, such as a frequency table or class intervals, needs to be expanded into a list first.
Write 1000 rather than 1,000, and use a full stop for decimals. A decimal comma, as in 2,5, is read as two numbers.
Larger lists are cut at 100,000 values with a notice. The deviations table shows the first 200 rows, the CSV download has every row, and the share link holds up to 3,000 values.
The definitions and methods on this page come from these references, last checked on November 9, 2026. For more tools of this kind, see the free online calculators page.
Short answers to what people ask most.
Standard deviation measures how far the values in a data set typically sit from their mean. A small value means the numbers are bunched together, and a large value means they are spread out.
Find the mean, subtract it from each value, square the results and add them up. Divide that sum by N for a population or by n minus 1 for a sample, then take the square root.
A population standard deviation divides by N and is used when your data covers every member of the group. A sample standard deviation divides by n minus 1 and is used when your data is a sample taken from a larger group.
Deviations from the sample mean are a little smaller on average than deviations from the true mean, so dividing by n would underestimate the spread. Dividing by n minus 1, known as Bessel correction, makes the sample variance an unbiased estimate of the population variance.
Use population when the numbers are the whole group you care about, such as every test score in one class. Use sample when the numbers stand in for a bigger group, such as 50 customers surveyed out of thousands.
Variance is the average squared distance from the mean, and standard deviation is its square root. Standard deviation is in the same units as your data, while variance is in squared units.
The standard error is the standard deviation divided by the square root of the number of values. It estimates how much the mean itself would vary from one sample to the next.
The coefficient of variation is the standard deviation divided by the mean, shown as a percentage. It compares spread between data sets with different units or scales, and it is not meaningful when the mean is zero or close to zero.
No. It is the square root of an average of squared numbers, so the smallest possible value is 0, which happens only when every value is the same.
For data that is roughly normally distributed, about 68 percent of values fall within 1 standard deviation of the mean, about 95 percent within 2 and about 99.7 percent within 3. The calculator shows how close your own data comes to this pattern.
Paste or type numbers separated by commas, spaces, tabs, semicolons or new lines, so a column copied from a spreadsheet works as it is. Do not use thousands separators, because a comma always starts a new value.
No. The calculation runs in your browser and nothing is sent to a server. Your data appears only in the page address when you use the share link.