Standard Error Calculator
Not the standard deviation. The other one.
Work out Standard Error. Not the standard deviation. The other one. States the assumption instead of hiding it.
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Standard error of the mean
0.0866025
Mean 12.05, 95% interval 11.8452 to 12.2548
The standard deviation describes how much individual measurements scatter. The standard error describes how much the mean of a sample this size would scatter if you repeated the whole exercise — a different and usually much smaller quantity, and the one that belongs on an error bar. It is the standard deviation divided by the square root of n, and that square root is the most important fact about sample size: quadrupling the sample only halves the error. Planning a study around a linear improvement is how budgets get set wrong. The interval uses the t distribution rather than the normal, which is the correct choice when the population standard deviation is estimated from the same data. For small samples the difference is substantial, and it converges on 1.96 as the sample grows.
How the Standard Error Calculator works
Standard error of the mean from raw measurements, with the 95% confidence interval it implies. The standard deviation describes how measurements scatter; the standard error describes how the mean would scatter — and that is what belongs on an error bar.
Also known as: standard error from data · sem vs standard deviation · what goes on an error bar · confidence interval from measurements
Frequently asked questions
What is the difference between standard deviation and standard error?
Standard deviation describes the spread of individual measurements. Standard error describes how much the sample mean would vary if you repeated the whole exercise — a different and usually much smaller quantity.
Which goes on an error bar?
Usually the standard error, if the bar is meant to show uncertainty in the mean. Papers vary and often fail to say which, which makes their figures uninterpretable — always label it.
How much does a bigger sample help?
Only as the square root. Quadrupling the sample halves the standard error, not quarters it. Planning a study around a linear improvement is how research budgets get set wrong.
Why does this use the t distribution?
Because the population standard deviation is being estimated from the same data, which adds uncertainty. The t interval is wider than the normal one, substantially so for small samples, and converges on it as n grows.
What does a 95% confidence interval actually mean?
That the procedure captures the true mean 95% of the time across repeated sampling. It is not a 95% probability that the true mean lies in this particular interval — a subtle distinction that trips up almost everyone.
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