Standard Deviation Calculator
Paste your data and get the standard deviation, variance and mean instantly — for a sample or a whole population. Unlike most calculators, this one shows the complete worked solution: the mean, every deviation, every squared deviation and the final square root, so you can check the method as well as the answer.
- Sample and population
- Full step-by-step working
- Mean, median, range & more
- Your data stays in your browser
Calculate standard deviation from your data set
Sample standard deviation
—s
Paste or type your numbers on the left and every statistic updates as you go.
- —Variance
- —Mean
- —Count
- —Standard error
Descriptive statistics
- Sum—
- Median—
- Minimum / maximum—
- Range—
- Sum of squared deviations—
- Mean absolute deviation—
- Coefficient of variation—
- Typical range (mean ± 1σ)—
Step-by-step solution
Step 1 — the formula
s = √( Σ(x − x̄)² / (n − 1) )
Step 2 — find the mean
Enter data to see the mean calculated from your values.
Step 3 — deviations and squares
| Value (x) | x − mean | (x − mean)² |
|---|---|---|
| — | — | — |
Step 4 — divide, then square root
The sum of squared deviations is divided by the degrees of freedom to give the variance, and the square root of the variance is the standard deviation.
Why this standard deviation calculator is better
Most calculators hand you a number. If you are studying statistics, submitting coursework or checking someone else's analysis, the number alone is not the useful part.
Every step, free
The formula, the mean, a full deviation table and the final square root are all shown — the part competitors put behind a subscription.
Sample and population in one tap
Switch between dividing by n − 1 and by N and watch both the result and the working update, so the difference stops being abstract.
Twelve statistics, not one
Variance, mean, median, range, standard error, mean absolute deviation and the coefficient of variation come with every calculation.
Paste from anywhere
Commas, spaces, tabs or a column copied straight out of a spreadsheet all parse correctly, with up to 5,000 values per data set.
Precision you control
Choose 2, 3, 4 or 6 decimal places so your answer matches the rounding your textbook, lab report or marking scheme expects.
Your data never leaves
All parsing and arithmetic happen in your browser. Nothing is uploaded, which matters when the numbers are patient records, grades or trial results.
How to calculate standard deviation
The tool does this instantly, but here is the method it follows — the same one you would use by hand.
Find the mean
Add every value together and divide by how many values there are. This average is the point all the deviations are measured from.
Subtract and square
Take each value away from the mean and square the result. Squaring removes the minus signs, so values above and below the mean cannot cancel each other out.
Average the squares
Add the squared deviations, then divide by n − 1 for a sample or by N for a population. That figure is the variance.
Take the square root
The square root of the variance is the standard deviation, which returns the answer to the same units as your original data.
What standard deviation tells you
It measures spread. A small value means the data clusters tightly around the mean; a large one means it is scattered.
- Students and coursework. Check homework answers and see exactly where your hand calculation diverged from the correct working.
- Lab and research reports. Report a mean with its standard deviation and standard error, the convention in almost every scientific paper.
- Quality control. Track how consistent a process is over time; rising spread signals a problem long before averages move.
- Finance and investing. Standard deviation of returns is the standard measure of volatility, and the basis of risk-adjusted comparisons.
- Teaching. Project the deviation table on screen so a class can see why the squaring step exists rather than taking it on trust.
- Comparing data sets. The coefficient of variation lets you compare spread between sets measured in completely different units.
Standard deviation FAQs
The questions that decide whether your answer is right or a whole formula out.
Should I use the sample or population formula?
Use the population formula only when your numbers cover every member of the group you are describing — every student in one class, every item produced in a batch. If the numbers are a subset used to estimate something wider, use the sample formula. Coursework and research almost always want the sample version, which is why it is the default here.
Why divide by n − 1 instead of n?
Because the mean has been estimated from the same data, deviations measured from it are slightly too small, so dividing by n would understate the true spread. Dividing by n − 1, known as Bessel's correction, compensates for that bias. The effect is large for small samples and negligible once you have hundreds of values.
What is the difference between variance and standard deviation?
Variance is the average of the squared deviations, so its units are squared — square centimetres, square dollars — which is hard to interpret. Standard deviation is the square root of variance, which brings it back to the original units and makes it directly comparable with the mean.
What counts as a high standard deviation?
There is no universal threshold, because it depends on the scale of your data. Compare it with the mean instead: the coefficient of variation expresses the standard deviation as a percentage of the mean, so a value of 5 is large next to a mean of 20 and trivial next to a mean of 2,000.
Can the standard deviation be zero or negative?
It can be zero, which happens when every value in the set is identical and there is no spread at all. It can never be negative, because it is the square root of a sum of squares. If you have calculated a negative value, a sign has been lost somewhere in the working.
What does the 68-95-99.7 rule mean?
For data that follows a normal distribution, roughly 68 percent of values fall within one standard deviation of the mean, 95 percent within two and 99.7 percent within three. It is a quick sanity check on spread, but it only holds for reasonably bell-shaped data, not for skewed sets.
How is standard error different from standard deviation?
Standard deviation describes the spread of individual values in your data. Standard error describes the precision of the mean itself, and equals the standard deviation divided by the square root of the sample size. Collecting more data shrinks the standard error but leaves the standard deviation roughly unchanged.
Can I paste a column straight from a spreadsheet?
Yes. Values separated by new lines, tabs, commas, semicolons or spaces are all recognised, so a copied spreadsheet column works without cleaning. Anything that is not a number is ignored, and the count tells you how many valid values were read.
Get the answer and the working
Paste your numbers, choose sample or population, and read the standard deviation alongside the full solution you can copy into a report or check against your own. Free, private and unlimited.