Standard Deviation Calculator
Standard deviation answers one question: typically, how far from the average does a value sit? This reports both the sample and the population figure, because picking the wrong one is the single most common mistake in introductory statistics.
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Both conventions, side by side
n or n − 1: the only decision that matters here
If your numbers are every member of the group you care about — all twelve machines on the line, all thirty students in the class — divide by n. That is the population standard deviation, and it is exact.
If your numbers are a sample standing in for a larger group you cannot measure, divide by n − 1. This is Bessel's correction. It exists because the sample mean is itself estimated from the same data, which makes the deviations systematically too small. Dividing by the smaller number nudges the estimate back up to where it belongs.
The gap shrinks fast as the sample grows: at n = 5 the sample SD is about 12% larger, at n = 50 about 1%, at n = 500 about 0.1%. On small samples the choice genuinely changes your answer, which is why both are reported above.
Standard deviation versus standard error
These get swapped constantly, and they measure different things. Standard deviation describes how spread out the individual values are. Standard error describes how much the sample mean itself would bounce around if you repeated the whole study. Standard error is SD divided by the square root of n, so it shrinks as you collect more data — while the standard deviation does not, because the underlying spread of the population is whatever it is.
Error bars on a chart should say which one they are. Standard error bars look far tighter and make results look far more certain.
Coefficient of variation: comparing spread across units
An SD of 4 means nothing until you know the scale. CV expresses the standard deviation as a percentage of the mean, so a blood-pressure series and a share-price series become comparable. It is only meaningful when the values are all positive and the mean is well away from zero — near zero, CV explodes.
Frequently Asked Questions
Should I use sample or population standard deviation?
Why divide by n − 1 instead of n?
What does a standard deviation of zero mean?
Is standard deviation affected by outliers?
Sources
Official publications only. Links open the original document in a new tab.
- National Institute of Standards and Technology Special Publication 811 — Guide for the use of the SI Significant figures and rounding conventions
- National Institute of Standards and Technology CODATA fundamental physical constants Values of the fundamental physical constants