What Is Standard Error?
The standard error measures how much a sample statistic, such as the mean, would vary from one sample to another. For a mean it equals the sample standard deviation divided by the square root of n.
Choose raw data, an SD and n, or a proportion. The calculator returns the standard error, shows the steps and plots how it shrinks as the sample grows.
Standard error is not standard deviation. The SD describes how spread out the individual values are. The standard error describes how precisely you have estimated the mean.
How to Use the Standard Error Calculator
- Choose raw data, SD and n, or a proportion.
- Enter the values.
- Read the standard error.
- Check the chart against n.
Standard Error Formula
Standard error of the mean
Standard error of a proportion
Standard Error Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| SE | Standard error | |
| s | Sample standard deviation | |
| n | Sample size | |
| p̂ | Sample proportion |
Standard Error Worked Example
With s = 8 and n = 50 the standard error is 8 ÷ √50 = 8 ÷ 7.071 = 1.131. For a proportion of 0.4 with n = 50 it is √(0.4 × 0.6 ÷ 50) = 0.0693.
With the example values loaded in the calculator above, the steps are:
- Mean = 23.75; sample SD s = 6.541079; n = 8.
- SE = 6.541079 ÷ √8 = 2.3126207.
- Quadrupling the sample size halves the standard error, because it depends on 1 ÷ √n.
Result: Standard error = 2.3126207.
How to Read a Standard Error
The standard error tells you how precisely the sample mean estimates the population mean. A smaller value means the sample means would cluster more tightly around the true mean if you repeated the study.
Because it depends on 1 ÷ √n, precision grows slowly. To halve the standard error you need four times as many observations. A rough 95% interval for the mean is the estimate plus or minus about two standard errors, so a standard error of 2.31 gives a margin near 4.6 for large samples. For small samples use a t value instead of 2.
Where Standard Error Is Used
Confidence intervals
The margin of error is a critical value times the standard error. Compute the standard error first, then build the interval around the mean.
Error bars on charts
Plot the mean with its standard error to show how precisely each group mean is known. Say in the caption which one the bars show.
Planning sample size
See how much a larger sample shrinks the standard error before you collect data.
Reporting results
Quote the mean with its standard error in a table, as many journals expect.
Common Standard Error Mistakes to Avoid
- Standard deviation describes the spread of the data. Standard error describes the precision of the mean. They answer different questions.
- Use the sample standard deviation with n − 1, unless the population SD is known.
- The formula assumes independent observations. Repeated measures on the same person are not independent.
- For a proportion use √(p̂(1 − p̂) ÷ n), not the formula for a mean.
Standard Error Terms and Definitions
- SEM
- The standard error of the mean.
Standard Error FAQs
How do you calculate the standard error of the mean?
Divide the sample standard deviation by the square root of the sample size: SE = s ÷ √n. For s = 6.54 and n = 8 the standard error is 6.54 ÷ 2.83 ≈ 2.31.
What is the difference between standard deviation and standard error?
Standard deviation measures how spread out the individual values are. Standard error measures how far the sample mean is likely to be from the population mean. The standard error is always smaller than the standard deviation when n is above 1.
How do I reduce the standard error?
Increase the sample size, or reduce the variability in your measurements. Quadrupling n halves the standard error, so gains get expensive.
What is the standard error of a proportion?
It is √(p̂(1 − p̂) ÷ n). For p̂ = 0.4 and n = 50 it is √(0.24 ÷ 50) ≈ 0.0693, or about 6.9 percentage points.