Standard Error Calculator
Find the standard error of the mean and a confidence interval from your data, or from the mean, standard deviation and sample size.
Standard error of the mean
0.011136
- Mean
- 9.262
- Lower limit of the interval
- 9.23681
- Upper limit of the interval
- 9.28719
- Standard deviation
- 0.035214
- Sample size
- 10
With 10 values, the standard error of the mean is 0.0111. The 95% confidence interval for the mean is 9.2368 to 9.2872.
Calculation breakdown
| Step | Formula | Result |
|---|---|---|
| Standard error | 0.0352 ÷ √10 | 0.011136 |
| t value | t for 95% confidence with 9 degrees of freedom | 2.2622 |
| Margin of error | 2.2622 × 0.0111 | 0.02519 |
| Lower limit | 9.262 − 0.0252 | 9.23681 |
How it works
Paste your data, or switch to the mean, standard deviation and sample size. Choose a confidence level.
Worked example
The NIST example: N = 195, mean 9.261460, standard deviation 0.022789, 95% confidence:
- Standard error: 0.022789 ÷ √195 = 0.001632
- t value with 194 degrees of freedom: 1.9723
- Margin of error: 1.9723 × 0.001632 = 0.003218
- Interval: 9.258242 to 9.264679
Rounding
Results are shown to six decimal places. Calculations use the unrounded values.
Sources
Frequently asked questions
What is the standard error of the mean?
It is the standard deviation of the sample divided by the square root of the sample size, s ÷ √N. It estimates how much a sample mean would vary from sample to sample. A larger sample gives a smaller standard error.
What is the difference between standard error and standard deviation?
Standard deviation describes the spread of the individual values in your data. Standard error describes the uncertainty in the mean of those values. The standard error is the standard deviation divided by the square root of the sample size, so it is always smaller.
How do you calculate a confidence interval for the mean?
NIST gives the interval as the mean plus or minus a t value times the standard error, where the t value depends on the confidence level and N − 1 degrees of freedom. For N = 195 and 95% confidence the t value is 1.9723.
Why does the calculator use a t value, not 1.96?
The population standard deviation is not known, so the sample standard deviation is used and the t distribution applies. With a large sample the t value approaches 1.96 for 95% confidence, and with a small sample it is larger, giving a wider interval.
Does the calculator match published examples?
Yes. It reproduces the NIST e-Handbook example with mean 9.261460, standard deviation 0.022789 and N = 195, giving the 95% interval 9.258242 to 9.264679.
Results are for information only. The interval assumes a random sample from a roughly normal population.