Difference between standard error and standard deviation

When dealing with numerical data sets, many people get confused between the standard deviation of the sample and the standard error of the sample mean. We want to stress the difference between these.

- Standard deviation (SD)
- This describes the spread of values in the sample. The sample standard
deviation,
*s*, is a random quantity -- it varies from sample to sample -- but it stays the same on average when the sample size increases. - Standard error of the mean (SE)
- This is the standard deviation of the sample mean, , and describes its accuracy as an estimate of the population mean, . When the sample size increases, the estimator is based on more information and becomes more accurate, so its standard error decreases.

Not only is this true for sample means, but more generally...

The standard error of all common estimators
decreases as the sample size, n, increases. |

Common mistakes in interpretation

Students often use the standard error when they should use the standard deviation, and vice versa.

- Standard error does not describe the variability of individual values
- A new value has about 95% probability of being within 2
**standard deviations**of sample mean. - Standard deviation does not describe the accuracy of the sample mean
- The sample mean has about 95% probability of being within 2
**standard errors**of the population mean.

Be careful that you do not confuse the two
terms (or misinterpret the values). |

Theory (again)

To illustrate the distinction between the standard deviation and standard error, the diagram below shows a normal population with mean = 1000 and standard deviation = 200.

Use the slider to adjust the sample size. Note that the standard error decreases when the sample size gets bigger even though the population standard deviation stays the same.

From data (simulation)

The next diagram takes random samples of values from the above population.

Click **Take Sample** a few times and observe that the sample
standard deviation varies from sample to sample but usually has a value
close to the population standard deviation, = 200.

Observe also that the standard error (estimated using the sample standard
deviation, *s*) is much lower than the standard deviation.

Use the pop-up menu to increase the sample size. Observe that the sample standard deviation remains around = 200 but the standard error decreases.

Warning

Be particularly careful when reading journal articles. Some papers use standard deviations (SD) are used to describe the distribution of variables, but others give the standard errors (SE) of the means of the variables.

When tables of variables are shown in journal
papers, check whether the tables show mean ± SD
or mean ± SE. |