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.
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.
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 µ = 1000 and standard deviation σ = 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, σ = 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 σ = 200 but the standard error decreases.
Warning
Be particularly careful when reading journal articles. Some papers use standard deviations (SD) 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.