Key takeaway: A simple random sample of size n is a sample chosen so that every possible group of n individuals from the population has the same chance of being the one selected.
Grade 12 – AP StatisticsAP Statistics Unit 1Statistics and Data
Definition of Simple Random Sample
A simple random sample (SRS) is the gold standard of fair selection. Two things hold at once: every individual has the same chance of being chosen, and every possible group of the chosen size has that same chance too. Picking names from a well-shaken hat comes close.
To draw one, list every individual, label them with numbers, and use a random number generator or table of random digits to choose labels without repeats. Because chance alone decides, the sample is not biased toward any type of person.
Formula and notation
With a population of \(N\) individuals and a sample of size \(n\), each possible sample has probability
\[ \frac{1}{\binom\,{N}{n}}. \]
Here \(\binom\,{N}{n}\) counts the groups of size \(n\). For \(N=30\) and \(n=4\) there are 27,405 equally likely samples.
Key facts
- Each individual has an equal chance, but equal individual chances alone do not guarantee an SRS.
- Repeated labels are skipped, because no individual can be chosen twice.
- Random selection protects against selection bias but does not guarantee a perfect match to the population.
- A sampling frame, the list of all individuals, is needed before drawing the sample.
Simple Random Sample example
A teacher numbers her 30 students 01 to 30 and wants a simple random sample of 4. The random digits read 17 52 08 17 41 29 03 66. Which students are chosen?
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Read two digits at a time and keep only labels from 01 to 30.
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17 is kept. 52 is skipped (too large). 08 is kept. The second 17 is a repeat, so it is skipped. 41 is too large.
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29 is kept and 03 is kept. Four students are now chosen, so stop.
The sample is students 17, 08, 29 and 03.
Common mistake
Watch out
A principal picks 30 students from every ninth name on an alphabetical list and states that this is a simple random sample.
Why it is wrong
Every-ninth-name selection is a systematic sample. Neighbors on the list such as two consecutive names can never end up together, so not every group of 30 has an equal chance. A simple random sample must give every group the same probability.
Related terms and lessons
Browse the glossary: A–Z list · AP Statistics vocabulary · Statistics and Data