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Simple Random Sample — Definition, Formula & Examples

A simple random sample (SRS) is a sample chosen so that every possible group of nn individuals from the population has the same chance of being selected. It is the gold standard sampling method because it eliminates selection bias by design.

A simple random sample of size nn from a population of size NN is a subset chosen such that each of the (Nn)\binom{N}{n} possible samples is equally likely to be the one obtained.

Key Formula

P(any specific sample)=1(Nn)P(\text{any specific sample}) = \frac{1}{\binom{N}{n}}
Where:
  • NN = Total number of individuals in the population
  • nn = Number of individuals selected for the sample
  • (Nn)\binom{N}{n} = The number of ways to choose n individuals from N

How It Works

To draw an SRS, assign every member of the population a unique number. Then use a random number generator (or a table of random digits) to select nn distinct numbers; the corresponding individuals form your sample. Because the selection is entirely random, every individual has the same probability nN\frac{n}{N} of being chosen, and — crucially — every combination of nn individuals is equally likely. This two-part condition is what separates an SRS from other random sampling methods.

Worked Example

Problem: A teacher wants to choose an SRS of 3 students from a class of 20 to participate in a survey. How many equally likely samples exist, and what is the probability that any one specific group of 3 is chosen?
Count possible samples: Use the combination formula to find how many groups of 3 can be formed from 20 students.
(203)=20!3! 17!=20×19×183×2×1=1140\binom{20}{3} = \frac{20!}{3!\,17!} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = 1140
Find the probability: In an SRS each of those 1140 possible samples is equally likely.
P(specific sample)=11140≈0.000877P(\text{specific sample}) = \frac{1}{1140} \approx 0.000877
Answer: There are 1,140 equally likely samples. The probability that any particular trio is chosen is 11140\frac{1}{1140}, or about 0.088%.

Why It Matters

AP Statistics free-response questions frequently ask you to describe an appropriate sampling method and explain why it reduces bias. Recognizing the precise definition of an SRS — every sample of size nn equally likely, not just every individual equally likely — is essential for earning full credit and for understanding why formulas for sampling distributions and margin of error work.

Common Mistakes

Mistake: Saying "every individual has an equal chance of being chosen" and treating that alone as the definition of an SRS.
Correction: That condition is necessary but not sufficient. In an SRS, every possible *group* of size nn must be equally likely. Systematic sampling can give each individual an equal chance without being an SRS.

Related Terms