Standard deviation of a sample mean
Earlier in this section, we presented a formula for the standard deviation of a sample mean,
![]() |
= | ![]() |
This formula, with σ replaced by s, can estimate the standard deviation of the mean from a single random sample.
Independent random samples
This formula is only accurate if the sample is collected in such a way that each successive value is unaffected by other values that have already been collected. For example, if values are sampled at random with replacement from a population, any population value has the same chance of being selected whatever values were previously selected. This is called an independent random sample.
When statisticians use the term random sample, independence is implied unless dependence is otherwise mentioned.
Dependent random samples
When the sample values depend on each other, they are said to be dependent.
The worst type of dependency arises through bad sample design. For convenience or to save money, values are often sampled from 'adjacent' individuals. Since these values tend to be similar, there is less variability in the sample than in the underlying population — the sample standard deviation, s, underestimates σ.
To make matters worse, the mean of such a dependent sample is more variable than the mean of an independent sample of the same size. For both reasons,
The formula
can badly underestimate the variability (and hence accuracy) of the sample mean of dependent random samples.
Always check that a random sample is independently selected from the whole population before using the formula for the standard deviation of the sample mean.
Independent random sample
Researchers are interested in controlling starling numbers in a city. They randomly locate ten traps in the city then capture and weigh one bird in each trap. The diagram below shows the weights of the trapped birds.
Click Take sample once to see the weights from an independent random sample of ten starlings. The cross on the right shows the sample mean weight. A normal distribution is also shown whose standard deviation was obtained from the formula above .
Click Take sample about 20 more times to repeat the sampling and observe that the sample means have a distribution whose spread conforms (roughly) to the standard deviations that are obtained from the individual samples.
Sampling from one area
Now click Reset, drag the slider to give a correlation of 0.9, and take a single sample. This shows what might be obtained if the ten birds were all sampled from a trap in one part of the city. (If a trap in a single location is used, data collection would be cheaper.) Note that the sampled weights are similar since the birds come from the same part of the city. The formula therefore predicts a lower standard deviation for the sample mean than before.
Take about 20 more samples and observe that the sample means are actually more variable than for an independent random sample. For both reasons, the standard deviation from the formula badly overestimates the accuracy of a sample mean.
With all 20 samples visible, drag the slider to observe both effects better.
The more dependent the sample, ...