Standard deviation calculator
MathNo. 67 / 84
Get the mean, population and sample standard deviation, both variances and the arithmetic steps from one pasted list of numbers.
spread is just distance that went to statistics school.
work surface
Math, not advice — results are indicative.Read the disclaimer.
most-used utilities
Exam results
See how tightly a class clustered around the average, and whether one score did all the damage.
Machine tolerance
Measure the spread of a batch of parts and decide whether the process is drifting or just noisy.
Monthly returns
Compare a fund's average month with its swings, because the average alone never mentions the bad months.
pro tips
Use the sample figure when the numbers are a subset of something bigger. Most real measurements are, unless you genuinely measured every unit.
Standard deviation wears the same unit as the data. A spread of 4 points on a test is not the same thing as 4 dollars on an invoice.
A result of zero means every value is identical. Check the list before calling it perfect consistency.
what it does
Paste a list of numbers and the widget returns the mean, the population and sample standard deviations, both variances and the count. The sample figure gets the big type because most lists are a sample of something larger.
Population means you have every value that exists: the whole class, every machine on the line, all twelve months of the year. Sample means you have a subset and want to estimate the spread of the rest, so the divisor becomes n minus one to correct for the guesswork.
The steps box walks through the arithmetic: add the values, divide by the count for the mean, square each distance from that mean, add those squares, divide by n or n minus one, then take the square root. With the default list the squared distances add up to 200.8333, which splits into 33.4722 for the population variance and 40.1667 for the sample version.
One value has no spread, so the sample standard deviation is left blank and the population figure is zero. Figures are rounded to four decimals for reading, text is skipped, and the count stays visible so the sample size never gets lost.
quick answers
Should I use the population or sample figure?
Use population only when the list contains every value you care about. For a subset used to estimate a larger group, the sample figure is the honest choice.
Why divide by n minus one for a sample?
Because the sample mean sits closer to the sampled values than the true mean would, which makes the spread look smaller than it is. Dividing by n minus one corrects that bias.
Can standard deviation be negative?
No. It is a square root of a variance, and variance is built from squared distances, so the result is zero or positive.