What Makes a Sample “Simple Random”?
In Why Random Selection Matters, we saw that chance-based selection can reduce the opportunity for selection bias. A simple random sample, or SRS, is a particular kind of random sample with a precise defining property: every possible group of the chosen size has the same chance of being selected.
The word group matters. It is not enough that each individual has an equal chance of being selected. We must ask how the selection method treats all the possible samples as groups. Two methods could give every individual the same inclusion chance but allow some groups more often than others—or make some groups impossible.
For this definition, suppose the population contains \(N\) distinct individuals and the sample contains \(n\) distinct individuals. The number of possible groups of size \(n\) is \(\binom{N}{n}\), read “\(N\) choose \(n\).” If the sample is an SRS, each possible group has probability \(1/\binom{N}{n}\) of being selected. The calculation is useful for small populations; the definition itself applies whether or not we list all the groups.
An SRS is ordinarily a selection of \(n\) different individuals from the population. A selected individual is not chosen again to fill another place in that sample. Most importantly for the definition, we judge the final group—not the order of selection. If a method happens to select A then B, or B then A, those are two orders for the same sample group \(\{A,B\}\).
Equal Chances for Groups and Individuals
If every group of size \(n\) has an equal chance, every individual in the population also has the same chance of appearing in the sample. For example, if there are four individuals and the sample size is two, each individual appears in three of the six possible groups. Under an SRS, each of those groups has probability \(1/6\), so each person’s chance of inclusion is \(3/6=1/2\).
The reverse implication does not hold. Knowing that each individual has the same chance of being selected does not tell us that every group has the same chance. To identify an SRS, check the probabilities of the whole possible samples, not only the inclusion chances of individuals.
This definition is about the selection method, not whether a particular sample looks representative. An SRS might, by chance, include several individuals with a shared characteristic and none with another. That outcome does not automatically show the method was not an SRS. As in Sources of Variability in Collected Data, different samples can result from the same chance process.
Worked Example: Checking All Groups in a Small Population
Worked Example: Selecting Two Students
A fictional tutoring program has four students on its roster: Ana, Bo, Cam, and Dee. It selects two students using a procedure that gives each possible pair the same chance. Is the resulting sample an SRS, and what is each student’s chance of inclusion?
Identify the population and sample size: The population consists of the four students, so \(N=4\). The sample size is \(n=2\).
List the possible sample groups: The pairs are Ana–Bo, Ana–Cam, Ana–Dee, Bo–Cam, Bo–Dee, and Cam–Dee. There are six possible groups:
Check the defining property: The procedure assigns probability \(1/6\) to each of the six pairs. Since all possible groups of size two have the same chance, the selected pair is an SRS.
Find an individual’s inclusion chance: Ana appears in Ana–Bo, Ana–Cam, and Ana–Dee. Each of those three groups has probability \(1/6\), so:
The same count applies to Bo, Cam, and Dee: each student appears in three of the six equally likely pairs and therefore has probability \(1/2\) of being included. The group-level check establishes that this is an SRS; the individual calculation is a consequence of that check.
Why Equal Individual Chances Are Not Enough
Imagine a different selection rule for the same four students. The program randomly chooses one student from the pair Ana–Bo and one from the pair Cam–Dee. Each of the four students has a \(1/2\) chance of being chosen. But the only possible samples are Ana–Cam, Ana–Dee, Bo–Cam, and Bo–Dee. The pairs Ana–Bo and Cam–Dee can never be selected.
The rule gives equal chances to individuals, but it does not give every possible pair an equal chance: two pairs have probability zero, while four pairs are possible. Therefore, it is not an SRS of two students from the four-person population. The sample still comes from a chance-based method; it simply does not meet the more specific SRS definition.
This distinction is useful when evaluating descriptions of how data were collected. Hearing “everyone had the same chance” may tell us something about individual inclusion chances, but we should still ask whether every group of the specified size could be selected and whether the groups had equal probabilities.
Worked Example: Equal Inclusion Chances, Unequal Pair Chances
Worked Example: Sampling Four Garden Plots
A community garden has four plots labeled A, B, C, and D. A selection procedure chooses two plots. The probabilities for the possible pairs are listed below. Determine whether the sample is an SRS, and check whether the four plots have equal chances of inclusion.
| Pair selected | Probability |
|---|---|
| A–B | \(1/4\) |
| C–D | \(1/4\) |
| A–C | \(1/8\) |
| A–D | \(1/8\) |
| B–C | \(1/8\) |
| B–D | \(1/8\) |
Check that the probabilities describe a complete selection rule: The probabilities add to \(1/4+1/4+1/8+1/8+1/8+1/8=1\). There are six possible pairs because \(\binom{4}{2}=6\).
Check whether it is an SRS: In an SRS, each pair would have probability \(1/6\). Here, A–B and C–D each have probability \(1/4\), while the other pairs each have probability \(1/8\). The probabilities are unequal, so this is not an SRS.
Check individual inclusion chances: Plot A is included in A–B, A–C, and A–D. Its probability of inclusion is \(1/4+1/8+1/8=1/2\). Plot B is included in A–B, B–C, and B–D, also for a total of \(1/4+1/8+1/8=1/2\). Plot C is included in C–D, A–C, and B–C, for \(1/4+1/8+1/8=1/2\). Plot D is included in C–D, A–D, and B–D, also for \(1/2\).
Conclude: All four plots have the same chance of inclusion, but the possible pairs do not have equal probabilities. This example confirms that equal chances for individuals are not enough to establish an SRS.
Worked Example: Auditing a Selection Description
Worked Example: Choosing Two Clinic Times
A fictional clinic has five appointment periods to review: Monday morning, Monday afternoon, Tuesday morning, Tuesday afternoon, and Wednesday morning. A staff member says, “We selected two periods at random.” The written selection rule says that the pair Monday morning and Monday afternoon is chosen half the time, while each of the other nine possible pairs is chosen with probability \(1/18\). Does this rule produce an SRS of size two?
Count the possible groups: There are five periods and the sample size is two, so:
Check the group probabilities: For an SRS, each of the ten pairs would have probability \(1/10\). The stated rule gives one pair probability \(1/2\), and gives each other pair probability \(1/18\). Those values are not equal. The probabilities still total \(1/2+9(1/18)=1\), but the selection rule is not an SRS.
Explain why “at random” is insufficient: The staff member’s phrase does not establish that all pairs are equally likely. In fact, the written rule gives one pair a much greater chance than each other pair. A careful description should state the rule and compare the chances assigned to all possible groups.
Conclude: The clinic’s selection is described as involving chance, but it is not a simple random sample of two appointment periods because the ten possible pairs do not have equal chances of selection.
Common Mistakes and Full-Credit Communication
- Checking people but not groups. Equal chances for individuals do not establish an SRS. Full-credit wording says that every possible group of the stated size must have the same chance.
- Assuming every random sample is an SRS. An SRS is a specific kind of random sample. Explain whether the defining equal-group-chance condition is satisfied.
- Confusing possible groups with selection order. The pair Ana–Bo is one group, not two different samples because Ana might be chosen first or second. Evaluate the final groups without regard to order.
- Assuming an SRS must look representative. A particular sample can differ from the population by chance. The definition describes the method’s probabilities, not the appearance of one result.
- Using “random” without explaining the method. Saying “the sample was random” is not enough to verify an SRS. State what the possible groups are and how their chances compare, when that information is available.
- Forgetting the sample size. The definition applies to all possible groups of the chosen size. A claim about an SRS of two does not answer whether a method would give all groups of three equal chances.
A concise, complete explanation might say: “There are six possible pairs of students. Each pair has probability \(1/6\), so every group of size two has the same chance of being selected. Therefore, the method produces an SRS.” If probabilities differ, identify that difference and conclude that the method does not meet the SRS definition.
Check Your Understanding
Use the defining property of an SRS to explain each answer.
- A population has six individuals, and a sample of size two is selected. How many possible groups are there? If the method is an SRS, what is the probability of each group?
- A method gives each individual the same chance of inclusion, but some groups of the required size can never be selected. Is it an SRS? Explain.
- For four individuals and a sample size of two, each of the six pairs is assigned probability \(1/6\). What is the inclusion probability for one individual, and why?
- A procedure chooses one person from each of two fixed pairs. Explain why equal chances for all individuals do not necessarily make the resulting sample an SRS.
- Why does observing that one SRS does not closely resemble the population fail, by itself, to show that the selection method was not an SRS?