Standard Deviation Calculator
Calculate the sample and population standard deviation of up to 10,000 numbers, together with the mean, both variances and the standard error of the mean. Paste values separated by commas, spaces, semicolons or new lines and choose how many decimal places to show. The calculation runs on mathjs and clearly labels which formula divides by n and which by n - 1. Made for lab reports, statistics classes and quality checks.
Frequently asked questions
Should I use the sample or the population standard deviation?
Use the population standard deviation when your list contains every member of the group you care about, such as all scores of one class. Use the sample standard deviation when the list is a sample drawn from a larger population, which is the usual case in research. The tool shows both so you can pick the right one.
What formulas are used?
The population standard deviation is the square root of the sum of squared deviations from the mean divided by n. The sample standard deviation divides by n - 1 instead, which is Bessel's correction. Both are computed with the mathjs std function.
Why does the sample result need two values?
With a single value n - 1 is zero, so the sample variance and standard deviation are undefined. The tool still shows the population values, which are 0 for a single number. Add at least one more value to get the sample statistics and the standard error.
What is the standard error of the mean?
It is the sample standard deviation divided by the square root of the number of values. It estimates how much the sample mean would vary between repeated samples of the same size. It shrinks as you collect more data.
How do I enter the data?
Type or paste numbers separated by commas, spaces, semicolons or line breaks. Use a period for decimals, since commas separate values. Scientific notation such as 4.2e-5 is accepted, and anything that is not a number is reported.
How many numbers can I use?
Up to 10,000 values per calculation. Longer lists are rejected with a message rather than truncated. This covers a typical spreadsheet column or a long series of measurements.
What is the relationship between variance and standard deviation?
The standard deviation is the square root of the variance. Variance is in squared units, for example square meters, while the standard deviation is in the same unit as the data. The tool shows both for the sample and the population.
How precise are the results?
The calculation uses double precision floating point numbers, about 15 to 16 significant digits. Results are rounded to the decimal places you select, 6 by default and up to 15. Trailing zeros are dropped from the display.
What does a standard deviation of zero mean?
It means every value in the list is identical, so there is no spread around the mean. Any difference between values gives a positive standard deviation. A larger value means the data is more spread out.
Does the order of the numbers matter?
No, the standard deviation depends only on the values, not on their order. Duplicates are counted every time they appear. Negative numbers and decimals are handled like any other value.