Standard deviation calculator for samples and populations
This standard deviation calculator measures how spread out a set of numbers is. Paste or type your data and it instantly returns the sample and population standard deviation, both variances, the mean, sum, count, standard error of the mean and coefficient of variation — with every formula shown using your own numbers.
It accepts a plain list of values or a frequency table (each value with the number of times it occurs), so it works for raw survey responses, exam marks, lab measurements and grouped textbook exercises alike. A step table lists x, x − x̄ and (x − x̄)² for every value, which is exactly the working most teachers and exam boards ask students to show.
How to use the Standard Deviation Calculator
- Choose “Data list” to enter raw values, or “Frequency table” to enter value, frequency pairs.
- Paste or type your numbers. Commas, spaces, tabs and new lines all work as separators.
- Pick “Sample (n − 1)” if your data is a sample from a larger group, or “Population (N)” if it includes every member.
- Read the standard deviation in the result panel, with variance, standard error and other statistics underneath.
- Scroll to the formulas and step table to see the working, then use “Copy results” or “Copy table” to paste them elsewhere.
Standard Deviation Calculator features
Sample and population side by side
Both versions of standard deviation and variance are always shown, so you can see the effect of dividing by n − 1 instead of N.
Frequency table input
Enter grouped data as value, frequency pairs instead of typing each repeated value out.
Full step table
Deviation and squared deviation for every value, with totals — ready to copy into Excel or Google Sheets.
Standard error and CV
The standard error of the mean (s / √n) and coefficient of variation are calculated automatically.
Accurate two-pass method
The mean is found first and the squared deviations second, avoiding the rounding errors of the “shortcut” formula on large values.
Private and instant
Calculations run in your browser as you type, and your data never leaves your device.
Sample vs population standard deviation
The population standard deviation σ is used when your data contains every member of the group you care about — for example, the marks of all 30 students in one class. It divides the sum of squared deviations by N: σ = √( Σ(x − μ)² / N ).
The sample standard deviation s is used when your data is a sample drawn from a bigger population — such as 200 people surveyed out of a whole city. It divides by n − 1 instead: s = √( Σ(x − x̄)² / (n − 1) ). This is called Bessel’s correction. A sample tends to sit closer to its own mean than to the true population mean, so dividing by n − 1 corrects the variance estimate so it is not too small on average.
If you are unsure, use the sample version — it is the default in most statistics courses, in Excel’s STDEV.S and in Google Sheets’ STDEV. Excel’s STDEV.P gives the population value.
Worked example: 2, 4, 4, 4, 5, 5, 7, 9
There are n = 8 values with a sum of 40, so the mean is x̄ = 40 / 8 = 5. Subtract the mean from each value to get the deviations −3, −1, −1, −1, 0, 0, 2, 4, then square them: 9, 1, 1, 1, 0, 0, 4, 16. Their total, the sum of squares, is SS = 32.
Population variance is 32 / 8 = 4, so σ = √4 = 2. Sample variance is 32 / 7 ≈ 4.5714, so s ≈ 2.1381. The standard error of the mean is s / √8 ≈ 0.7559, and the coefficient of variation is s / x̄ ≈ 42.76%. These are exactly the numbers the calculator shows when it first loads.
What the results mean
To find just the mean, median and mode, use the average calculator. To round your final answer to a set number of decimal places or significant figures, use the rounding calculator, and for converting test scores the marks percentage calculator is handy.
- Variance is the average squared distance from the mean. Standard deviation is its square root, which puts the spread back into the same units as your data.
- The 68–95–99.7 rule: for roughly bell-shaped (normal) data, about 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three.
- Standard error (SE) describes how precisely the sample mean estimates the population mean. It shrinks as the sample grows, because it divides by √n.
- Coefficient of variation (CV) is the standard deviation as a percentage of the mean. It lets you compare spread between data sets with different units or scales, but it is only meaningful for data measured on a ratio scale with a true zero.
- Frequency tables use the same formulas with each term weighted by its frequency f: x̄ = Σfx / Σf and SS = Σf(x − x̄)².
Standard Deviation Calculator FAQ
How do I calculate standard deviation by hand?
Find the mean, subtract it from each value, square each result, add the squares together, divide by n − 1 for a sample (or N for a population) and take the square root. The standard deviation calculator shows each of these steps with your numbers.
Should I use sample or population standard deviation?
Use population standard deviation only when your data includes every member of the group you are describing. If the data is a sample used to learn about a larger group — which is the usual case in surveys, experiments and coursework — use the sample standard deviation.
Why is the sample standard deviation bigger than the population one?
It divides by n − 1 instead of n, which makes the result slightly larger. The difference is noticeable for small samples and becomes negligible as the sample grows.
How do I find the standard deviation of a frequency table?
Switch the input to “Frequency table” and enter one value and its frequency per line, such as “4, 3”. The calculator weights each squared deviation by its frequency and uses n = Σf.
Can standard deviation be negative?
No. It is the square root of an average of squared numbers, so it is always zero or positive. It is zero only when every value is identical.
Does this match Excel’s STDEV.S and STDEV.P?
Yes. The sample result matches STDEV.S (and the older STDEV), and the population result matches STDEV.P. Variances match VAR.S and VAR.P.