What the Standard Deviation Calculator does
This calculator finds the standard deviation and variance of a set of numbers - both the sample version, which divides by n - 1, and the population version, which divides by n - together with the mean, the coefficient of variation and the standard error, and it lays out every step and the table of squared deviations so you can check homework or a report line by line.
Paste a list of numbers in any common format, or switch to a frequency table when your data comes as "value and how many times". The arithmetic uses Welford's method, which stays accurate even for large numbers that sit close together, and it all runs in your browser: nothing you paste is sent anywhere.
How to use it
- Choose how your data is written: a plain list of numbers, or value-and-frequency pairs (one pair per line, such as
4, 3for "the value 4 occurs three times"). - Paste or type the numbers. Commas, spaces, tabs, semicolons and new lines all separate values, so a column copied from a spreadsheet works directly. Anything that is not a number is listed in red and left out, never guessed.
- Say whether the numbers are a sample or the whole population. The big answer switches between s and σ; the other one is still shown in the tiles.
- Read the step-by-step working and the deviations table. Choose the number of decimal places to match your course or report, then copy the results, download the table as CSV or print the page.
Reading the results
The standard deviation is in the same units as your data. A mean of 74 marks with a standard deviation of 9.9 marks means a typical score sits roughly ten marks from the average; for data shaped like a bell curve, about two thirds of values fall within one standard deviation of the mean.
The coefficient of variation (CV) divides the standard deviation by the mean, so it compares spread between data sets measured on different scales. It is meaningless when the mean is zero or when the data can be negative, and the calculator shows a dash then.
The standard error of the mean, s / √n, is not the spread of the data: it is how much the mean itself would vary between repeated samples of the same size. It shrinks as n grows even when the data are just as spread out.
Worked example: ten test marks
Marks: 62, 75, 81, 58, 90, 73, 68, 77, 85, 71. They add up to 740, so the mean is 740 / 10 = 74.
The deviations from 74 are -12, 1, 7, -16, 16, -1, -6, 3, 11 and -3. Squared, they are 144, 1, 49, 256, 256, 1, 36, 9, 121 and 9, which add up to 882.
As a sample of the class's ability, divide by n - 1 = 9: the variance is 882 / 9 = 98 and the standard deviation is √98 = 9.8995. If these ten are the entire group you care about, divide by 10 instead: the variance is 88.2 and σ = 9.3915.
The sample coefficient of variation is 9.8995 / 74 = 13.38%, and the standard error of the mean is 9.8995 / √10 = 3.1305 marks.
Formulas and scoring rules
- Mean
x̄ = Σ f·x / n, n = Σ fFor a plain list every frequency f is 1.- Sum of squared deviations
SS = Σ f·(x − x̄)²Accumulated with the weighted Welford update rather than Σx² − n·x̄², which loses accuracy when values are large and close together.- Sample variance and standard deviation
s² = SS / (n − 1); s = √s²Undefined for a single value.- Population variance and standard deviation
σ² = SS / n; σ = √σ²- Coefficient of variation
CV = SD / |x̄| × 100%- Standard error of the mean
SE = SD / √n- Median
middle value of the sorted data; the average of the two middle values when n is even
Sample or population: which standard deviation do I need?
Use the population formula only when your numbers are every member of the group you are describing - all 30 pupils in this class, every invoice issued in March. Use the sample formula whenever the numbers stand in for a larger group: 30 pupils chosen to represent a year group, a batch of measurements of a quantity you could keep measuring.
Dividing by n - 1 (Bessel's correction) makes the sample variance an unbiased estimate of the population variance, because the deviations are measured from the sample's own mean, which sits closer to the data than the unknown true mean. The difference matters for small n and fades as n grows: for 10 values the two SDs differ by about 5%, for 1,000 values by 0.05%. Spreadsheets follow the same split: STDEV.S is the sample version and STDEV.P the population one.
Limitations: what the result does not prove
- The standard deviation describes spread but not shape. Two data sets can share a mean and SD while one is symmetric and the other heavily skewed or bimodal; look at the values, or a chart, before relying on the "two thirds within one SD" rule.
- It is sensitive to outliers: a single typo such as 900 instead of 90 can double it. Check the deviations table for a row that dwarfs the others.
- The standard error assumes independent observations drawn from the same population. It does not account for clustered, paired or time-series data.
- Numbers are parsed with a full stop as the decimal mark. A comma is always treated as a separator, so write 1234.5, not 1.234,5 or 1,234.5.
Privacy: where your data goes
Everything you paste, type or drop is processed in this browser tab. It is not uploaded, logged, stored or sent to analytics. Session recording and tag-manager scripts are switched off on this page.
Standards and sources
- NIST/SEMATECH e-Handbook of Statistical Methods - Measures of scale
- NIST Statistical Reference Datasets - Univariate summary statistics
- Welford, B. P. (1962) - Note on a method for calculating corrected sums of squares and products, Technometrics
Frequently asked questions
What is the difference between sample and population standard deviation?
The population version divides the sum of squared deviations by n and describes a complete group. The sample version divides by n - 1 and estimates the spread of the larger population your numbers were drawn from, which is why it comes out slightly larger.
How do you calculate standard deviation step by step?
Find the mean, subtract it from every value, square each difference, add the squares, divide by n - 1 for a sample (or n for a population) to get the variance, then take the square root. The calculator shows each of these steps with your own numbers.
How do I find standard deviation from a frequency table?
Switch the input to value and frequency and enter one pair per line. Each squared deviation is multiplied by its frequency before adding, and n is the total of the frequencies, not the number of rows.
Is variance the same as standard deviation squared?
Yes. The variance is the average squared deviation and the standard deviation is its square root, which brings the figure back to the original units: a variance of 98 square marks is a standard deviation of 9.9 marks.
What does the coefficient of variation tell me?
It is the standard deviation as a percentage of the mean, so you can compare the variability of things measured on different scales, such as the weights of mice and of elephants. It only makes sense for data on a ratio scale with a positive mean.
Can the standard deviation be negative or zero?
It can never be negative, because it is the square root of an average of squares. It is zero only when every value is identical. A single value has a population SD of zero and an undefined sample SD.
Why does my calculator give a different standard deviation from Excel?
Usually because one used the sample formula and the other the population one: Excel's STDEV and STDEV.S divide by n - 1, while STDEV.P divides by n. Check which button or function you used and set the matching option here.
Last reviewed by the A2Z.Tools team against the sources listed above.
