Standard Error Calculator
Work out the standard error of the mean from a raw list of numbers, or from a sample standard deviation and sample size.
How to use this calculator
Choose Raw data if you have the individual sample values, or Summary data if you already know the sample standard deviation and sample size.
For raw data, type your values separated by commas and press Calculate. The tool works out the mean, sum of squares, sample standard deviation, and standard error for you. For summary data, enter the standard deviation and sample size directly. Use Reset to clear the fields and start again.
What is standard error?
The standard error (SE) measures how much a sample mean is likely to differ from the true mean of the population it was drawn from. A small SE means the sample mean is a close, reliable estimate of the population mean; a large SE means there's more variability, and the sample mean could be further off.
Standard error is closely related to standard deviation, but they answer different questions. Standard deviation describes the spread within a single sample. Standard error describes how much the sample mean itself would vary if you repeated the sampling process. As sample size grows, standard error shrinks, because larger samples give a more stable estimate of the population mean.
Standard error formula
where s is the sample standard deviation and n is the sample size.
SE = √(Σ(x − x̄)² / (n − 1)) / √n
Worked examples
Mean = 72 / 6 = 12. Sum of squared differences = 314. Sample standard deviation = √(314 / 5) ≈ 7.925. Standard error = 7.925 / √6 ≈ 3.235.
Standard error = 36.78 / √49 = 36.78 / 7 ≈ 5.254.
Frequently asked questions
What's the difference between raw and summary data?
Raw data is the actual list of sample values. Summary data skips the individual values and gives you the sample standard deviation and sample size directly.
Why does a bigger sample reduce the standard error?
Standard error scales inversely with the square root of sample size. A larger sample averages out random variation more effectively, so the sample mean sits closer to the true population mean.
Is standard error the same as standard deviation?
No. Standard deviation measures spread within one sample. Standard error measures how much the sample mean would vary across repeated samples. Standard error is always smaller than the standard deviation, except when the sample size is one.
What counts as a "good" standard error?
There's no universal cutoff — it depends on the scale of your data and how precise your analysis needs to be. In general, a smaller SE relative to the mean indicates a more reliable estimate.