Key takeaway: A sampling distribution shows how much a sample statistic, for instance an average, would change from one random sample to the next when each sample has the same size.
Grade 12 – AP StatisticsAP Statistics Unit 2Statistics and Data
Definition of Sampling Distribution
A sampling distribution answers the question: if the same kind of sample were taken again and again, how would the statistic behave? Each sample yields one number, and the collection of all those numbers forms the distribution.
Its center shows whether the statistic is unbiased, and its spread shows the precision of the estimate. Three distributions must be kept apart: the population, one sample, and this distribution of the statistic. It is theoretical, since in practice we see only one sample.
Formula and notation
Take samples of size \(n\) from a population whose center is \(\mu\) and whose spread is \(\sigma\). The averages of those samples have
\[ \sigma_{\,\overline{x}}=\frac{\sigma}{\sqrt{n}},\qquad \mu_{\,\overline{x}}=\mu. \]
The second formula holds when the sample is under 10% of the population or when draws are independent.
Key facts
- The sampling distribution of a sample mean is centered at the population mean.
- Larger samples produce a narrower sampling distribution.
- Its shape approaches normal for large n, as the central limit theorem explains.
- It describes the statistic, never the individual values in the population.
Sampling Distribution example
A small population has the values 1, 5, 6 and 12. List every sample of size 2 taken without replacement, find each sample mean, and describe the resulting sampling distribution.
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The six samples are (1,5), (1,6), (1,12), (5,6), (5,12) and (6,12).
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Their means are 3, 3.5, 6.5, 5.5, 8.5 and 9.
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The six means add to 36, so their own average is \(\frac{36}{6}=6\). That equals the population mean, \(\frac{24}{4}=6\).
The six sample means 3, 3.5, 5.5, 6.5, 8.5 and 9 each have probability 1/6, and their distribution is centered at 6.
Common mistake
Watch out
A student says the spread of the six sample means, from 3 to 9, shows that the population values themselves range from 3 to 9.
Why it is wrong
The population values run from 1 to 12. The six numbers belong to a different distribution, that of the statistic, whose values are averages and are therefore less spread out. Keep the population, one sample and the sampling distribution separate.
Related terms and lessons
Browse the glossary: A–Z list · AP Statistics vocabulary · Statistics and Data