Standard Error Calculator
Calculate the standard error of the mean and its confidence interval.
What the Standard Error Calculator does
The standard error measures how much a sample mean would vary if you repeated the study. It is the standard deviation divided by the square root of the sample size, and it drives every confidence interval you will ever build.
Formula
SE = s ÷ √nConfidence interval = Mean ± t × SEDegrees of freedom = n − 1
Inputs explained
| Input | Unit | Required | Notes |
|---|---|---|---|
| Input method | one of 2 options | Yes | — |
| Data set | text | In some modes | Separate values with commas, spaces or new lines. Shown Input method is Raw data. |
| Standard deviation | number | In some modes | Accepts more than 0. Shown Input method is Summary statistics. |
| Sample size (n) | number | In some modes | Accepts 2 or more. Shown Input method is Summary statistics. |
| Sample mean | number | Optional | Shown Input method is Summary statistics. |
| Confidence level | one of 6 options | Yes | — |
How to use it
- Choose Input method and Confidence level.
- Fill in the remaining inputs the form shows for your choice.
- Select Calculate.
Worked example
A sample of 36 with a standard deviation of 12 and a mean of 85.
- SD
- 12
- Sample size
- 36
- Mean
- 85
- Confidence
- 95%
SE = 12 ÷ 6 = 2, giving a 95% interval of about 80.94 to 89.06.
Reading the result
- "Relative standard error" expresses SE as a percentage of the mean, letting precision be compared across measurements on very different scales — an SE of 2 means different things for a mean of 85 versus a mean of 2.
- The sensitivity table shows the square-root relationship in the direction useful for planning: it lets you read off roughly what sample size would be needed to hit a target SE, rather than only observing that a bigger sample helps.
Assumptions and limitations
- The t-based confidence interval assumes the underlying population is approximately normal, or that the sample is large enough for the central limit theorem to apply — for a small, heavily skewed sample this interval can be less reliable than the formula alone suggests.
- In "summary" mode, the entered standard deviation and mean are trusted exactly as given — the calculator cannot verify they came from the same dataset.
Common mistakes
- Assuming the confidence interval shown describes where 95% of individual data points fall — it describes uncertainty in the estimate of the mean, a narrower and different claim than a prediction interval for individual observations.
Frequently asked questions
What is the difference between standard deviation and standard error?
Standard deviation describes spread within your data. Standard error describes uncertainty in your estimate of the mean, and it falls as the sample grows.