Standard Error of the Mean

What is standard error?

The standard error (SE) measures how much the sample mean varies from sample to sample. It's the standard deviation of the sampling distribution of the mean:

$SE = \frac{\sigma}{\sqrt{n}}$

Step-by-step

Given: $\sigma = 20$, $n = 64$.

$SE = \frac{20}{\sqrt{64}} = \frac{20}{8} = 2.5$

Interpretation

If you take many samples of size 64, the sample means will have a standard deviation of about 2.5. Most sample means will fall within $\pm 2 \times 2.5 = \pm 5$ of the true population mean.

How SE shrinks with larger $n$

| $n$ | $SE$ | |-----|------| | 16 | 5.0 | | 64 | 2.5 | | 256 | 1.25 | | 1000 | 0.63 |

To cut the SE in half, you need to quadruple the sample size (because $\sqrt{4n} = 2\sqrt{n}$).

SE vs SD

• SD ($\sigma$): describes the spread of individual data points.
• SE ($\sigma/\sqrt{n}$): describes the spread of sample means.
• SE is always smaller than SD (by a factor of $\sqrt{n}$).

Try it in the visualization

Adjust $\sigma$ and $n$. The bell curve for individual values (wide) and for sample means (narrow) are shown side by side. As $n$ increases, the sampling distribution shrinks dramatically.