Paste your data and get the standard deviation both ways — sample and population — plus variance, standard error and the 1σ range, with the working shown.
A sample — part of a larger group. This is the usual choice.
Commas, spaces or new lines. Paste a spreadsheet column straight in.
Sample standard deviation
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Variance—
Mean—
Count (n)—
Sum of squares—
Standard error—
Coefficient of variation—
Range within 1σ—
Range within 2σ—
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What standard deviation measures
Standard deviation answers one question: how far, typically, does a value sit from the mean? A small SD means the numbers cluster tightly; a large one means they are scattered. Two data sets can share an identical mean and behave completely differently — 49, 50, 51 and 10, 50, 90 both average 50, but the second has thirty times the spread.
The calculation, step by step
Find the mean of all the values.
Subtract the mean from each value to get its deviation.
Square every deviation, so negatives do not cancel positives.
Add the squares together — that is the sum of squares.
Divide by n − 1 for a sample, or n for a population. That is the variance.
Take the square root. That is the standard deviation.
Sample or population — why n − 1?
If your numbers are the entire group you care about — every employee in the company, every part in the batch — divide by n. If they are a sample drawn from something bigger and you want to estimate the wider group, divide by n − 1. A sample's own mean sits slightly closer to its own points than the true mean does, which biases the estimate downward; dividing by the smaller number corrects it. This is Bessel's correction, and for small data sets the difference is substantial.
For data that follows a normal bell curve, roughly 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. Those bands are shown above so you can sanity-check the number against the data you actually pasted in. If a lot more or a lot fewer values fall inside the 1σ band than about two-thirds, your data probably is not normally distributed and the standard deviation alone will not describe it well.
Standard deviation vs standard error
They are easy to confuse. Standard deviation describes the spread of the data. Standard error describes the precision of the mean you calculated from it, and equals SD ÷ √n — so it shrinks as you collect more data, while the standard deviation does not. Quote the SD when describing variability; quote the standard error when putting error bars on an average.
The coefficient of variation (SD ÷ mean) makes spread comparable across different scales and units, which is useful when comparing the consistency of two things measured in different quantities. For the centre of your data rather than its spread, use the average calculator.
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Frequently asked questions
Should I use sample or population standard deviation?
Use population (÷ n) only when your numbers are the complete group you care about. Use sample (÷ n − 1) when they are a subset and you want to estimate the wider group — which is the case in most real analysis, and is the default here.
What is the standard deviation formula?
SD = √( Σ(x − mean)² ÷ (n − 1) ) for a sample, or √( Σ(x − mean)² ÷ n ) for a population. Square each value's distance from the mean, average those squares, then take the square root to get back to the original units.
What is a good standard deviation?
There is no universal answer — it depends entirely on the scale and the context. A useful comparison is the coefficient of variation (SD ÷ mean), shown above: below about 10% is generally tight, above 30% is quite scattered, but the right threshold varies by field.
Why square the deviations instead of just taking absolute values?
Squaring keeps negatives from cancelling positives, penalises large deviations more heavily, and produces a quantity with clean mathematical properties that underpins most of statistics. Taking the square root at the end returns the answer to the original units.
What is variance, and how does it relate?
Variance is the standard deviation squared — it is the average squared deviation from the mean. It is what the maths works with, but because it is in squared units it is harder to interpret directly, which is why the standard deviation is what usually gets reported.