Statistics Central Limit Theorem

Larger sample sizes provide means that are better estimates of the population mean.

In the gif below, you can see one population (shown twice in the first row).  The population mean is indicated by the vertical red line.  When samples have only five subjects (shown in the left column 2 down) they have means that pile up around the true population mean ( shown in the left column, 3rd graph down).  When sample size increases, as shown in the right column for samples with 25 subjects, the means lie closer to the true population mean.

(gif developed by William Arloff and Brenda Anderson)

The variability in sample means is reported as the standard error of the mean, which is calculated as the sample standard deviation divided by the square root of the sample size.  We can see the relationship between the sample size and the standard error of the mean in the diagram below.  The two points that are represented in the graph above are the two large green dots below.

In the graph above, you can also see the relationship between population standard deviation and standard error of the mean.  The blue line represents standard error of the means for varying sample sizes when the population standard deviation is 20, while the green line represents standard errors of the mean for varying samples sizes when the population standard deviation is 80.