Standard Error Calculator
Paste your sample and get the standard error of the mean, with the standard deviation and a 95% confidence interval for the average.
- Sample std dev (n−1)
- 95% interval included
- Shows the working
- Free
Find the standard error
Separate with spaces, commas or new lines. At least two values.
This assumes your values are independent readings from the same population and that the averages of such samples are roughly normally distributed. With a small sample that is an assumption, not a fact.
Show the working
How to calculate the standard error
Standard deviation first
Work out the sample standard deviation, dividing by n − 1. For 2, 4, 4, 4, 5, 5, 7, 9 it is about 2.14.
Square root of the count
There are 8 values, and √8 is about 2.83.
Divide
2.14 ÷ 2.83 = 0.76. That is the standard error of the mean.
Standard deviation describes how spread out your readings are. Standard error describes how precisely you have pinned down the average, so it shrinks as you collect more data.
Questions people ask
What is the standard error of the mean?
A measure of how much the average of a sample is likely to wobble from the true average of the whole population. It is the sample standard deviation divided by the square root of the sample size.
What is the difference between standard deviation and standard error?
Standard deviation says how spread out the individual readings are; it does not shrink when you collect more data. Standard error says how precise your estimate of the average is; it gets smaller as the sample grows.
Why does the confidence interval use a t value instead of 1.96?
Because the spread is being estimated from the same small sample, which adds uncertainty of its own. The t multiplier accounts for that: it is well above 2 for a handful of values and drifts down towards 1.96 as the sample grows. This calculator looks it up by degrees of freedom (n − 1), stepping down to the next tabulated row so the interval errs on the wide side.
What does the 95% interval actually mean?
If you repeated the same sampling many times and built an interval each time, about 95% of those intervals would contain the real population mean. It is not a 95% chance that this particular interval contains it.
Does this use the sample or population standard deviation?
The sample form, dividing by n − 1. That is the right choice when your numbers are a sample drawn from a bigger group, which is almost always why someone wants a standard error.
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