From a Numbered List to a Random Sample
A sampling plan is not complete when it says “choose people at random.” Someone must carry out that instruction in a way that gives the intended members a fair chance of selection and can be checked afterward. Numbering the sampling frame and using a random-number generator makes the selection procedure explicit: the numbers identify frame members, while the generator determines which identifiers are selected.
The procedure matters as much as the final list. A list with the right number of distinct IDs is not necessarily a simple random sample if, for example, someone replaced an inconvenient selection or stopped too soon after a repeated draw. A careful record of the frame, random draws, and decisions about repeats makes these problems easier to detect.
The phrase “without replacement” means that after an ID is selected, it cannot be selected again as a second member of the sample. In a random-number-generator procedure, a repeated ID is therefore recorded as a repeat and ignored; the next draw is used instead. The generator must make independent draws that are uniform over the \(N\) frame numbers, or make each next draw conditionally uniform given the numbers drawn so far. Merely having each draw look uniform on its own is not enough.
A Practical Selection Procedure
Before drawing, fix the eligible frame and the requested sample size. If the members already have labels with gaps or irregular values, assign each frame member a unique index from 1 to \(N\), then use the index to look up the member’s original label. This avoids accidentally giving some members more chances because their labels are represented differently.
Check that it lists the intended eligible members, with no duplicate entries or missing index numbers. Assign one unique number to each member, from 1 through \(N\).
Set the generator to draw an integer uniformly from 1 through \(N\). State in advance that repeated IDs will be recorded but rejected, and that drawing continues until \(n\) distinct IDs have been accepted.
For each generated number, record it in order and mark whether it is accepted or rejected as a repeat. Stop when exactly \(n\) distinct IDs have been accepted.
Verify that each accepted ID is on the frame, maps to one member, and appears only once. Confirm that \(n\) distinct IDs were accepted and that no person was added, removed, or substituted outside the stated rule.
Use a generator option that produces integers uniformly over the required range. For example, taking the last digit of a random number is not automatically valid for a frame numbered 1 through 12: that digit cannot identify every member in the range. Follow the generator’s documented method for drawing from a specified integer range rather than improvising a conversion.
Worked Example: Selecting Four Members From Ten
Suppose a hypothetical program has a frame of ten members indexed 01 through 10, and the plan calls for \(n=4\). The random-number generator produces the sequence 07, 02, 07, 10, 01, 09. Apply the rule “accept the first four distinct IDs.”
The first draw, 07, is accepted. The second, 02, is also accepted. The third draw, 07, repeats an accepted ID, so it is recorded and rejected. The fourth draw, 10, is accepted, and the fifth draw, 01, gives the fourth distinct member.
The sample is therefore 07, 02, 10, and 01. The sixth draw, 09, is not needed because the required four distinct IDs have already been accepted. The audit finds four accepted IDs, all in the range 01–10, with no duplicates. A repeat in the draw log does not make the final sample a sample with replacement; it is a rejected draw under the stated without-replacement procedure.
Why Rejecting Repeats Produces a Simple Random Sample
The procedure has the desired probability property when its raw draws are independent and uniform over the frame numbers. The reason can be shown by considering the next accepted ID after some IDs have already been selected. Suppose \(k\) distinct IDs have been accepted. There are \(N-k\) eligible IDs not yet accepted. Repeated draws of the \(k\) already accepted IDs are rejected, so the next accepted ID must be one of the \(N-k\) remaining IDs.
For any particular remaining ID \(j\), the probability that it is the next accepted ID is the probability of zero or more rejected draws from the \(k\) previously accepted IDs, followed by a draw of \(j\). Independence and uniformity give:
Thus every remaining ID has the same chance of being the next accepted one. For any particular ordered sequence of \(n\) distinct IDs, the probability is consequently:
Each group of \(n\) members can appear in \(n!\) different orders. Since all those ordered sequences have the same probability, every group has probability \(\frac{n!}{N(N-1)\cdots(N-n+1)}=\frac{1}{\binom{N}{n}}\). That is the defining equal-probability property of a simple random sample of size \(n\). This argument depends on independence (or conditional uniformity at each draw), not merely on each draw being uniform when considered separately.
Worked Example: Tracking Repeats Until Three IDs Are Accepted
A hypothetical list has six indexed members, 1 through 6. A generator is set to draw independent uniform integers from 1 through 6. The requested sample size is three, and the recorded draw sequence begins 6, 4, 4, 2, 6, 1.
The first 6 is accepted, and 4 is accepted. The next 4 repeats an accepted ID and is rejected. The next 2 is accepted, so the sample is now complete: it contains 6, 4, and 2. The later 6 and 1 are not used.
There are three distinct accepted IDs, each within the frame, so the sample size and range checks pass. Under the independent uniform draw rule, each of the six IDs had probability \(1/6\) of being the first accepted ID. After 6 was accepted, each of the five remaining IDs had probability \(1/5\) of being next; after 4 was accepted, each of the four remaining IDs had probability \(1/4\) of being next. The repeats affect how many raw draws are needed, not the equal chance of each possible three-member group.
Auditing the Result
An audit checks both the final sample and the steps used to produce it. The final list alone cannot show whether the correct generator range was used or whether a selected member was quietly substituted. Keep enough information to compare the procedure with its stated plan: the frame version, \(N\), \(n\), generator settings, ordered draw log, repeat decisions, and final accepted IDs.
- Check the frame: Does every index from 1 through \(N\) correspond to exactly one eligible member?
- Check the count: Are exactly \(n\) distinct IDs accepted? A repeated raw draw does not count as another selected member.
- Check the range and mapping: Is every accepted ID valid, and does it map to the intended frame member?
- Check duplicates and omissions: Is any accepted ID listed twice? Are any selected members missing from the final record? The sample need not contain every frame member; “omission” here means a missing required sample entry or a gap in the frame’s numbering or documentation, not an absent member who was simply not selected.
- Check departures: Were repeats handled as planned, and were any numbers skipped, rerun, or replaced because of who they identified?
A random-number generator may be software-based and use a reproducible seed. Recording the seed and software settings can help reproduce the same output, but reproducibility alone does not establish that the frame was complete or that the specified selection rule was followed.
Worked Example: Finding an Incomplete Draw
A hypothetical roster is indexed 01 through 12. A plan requests five members and says to accept the first five distinct IDs from the generator stream. The draw log is 03, 11, 03, 07, 12, 02, but the saved sample contains only 03, 11, 07, and 12. The person recording the sample says, “I stopped after the first five draws.”
The audit shows why that is a departure from the plan. Among the first five raw draws, 03 appears twice, so only four distinct IDs have been accepted. The required stopping condition is five distinct accepted IDs, not five raw draws. The next draw, 02, is new and valid; following the stated rule gives the sample 03, 11, 07, 12, 02.
The saved list is short by one member, even though all four listed IDs are valid and distinct. The correction is to continue the existing draw stream and include 02, not to choose a convenient fifth member or silently start a new draw sequence. If the original log were unavailable, the selection would need to be documented as incomplete rather than represented as a completed sample under the stated procedure.
Common Mistakes and Why the Checks Matter
A common mistake is to treat every generated number as a selected member. If a number repeats, counting it again creates fewer than \(n\) distinct members. The correction is to distinguish raw draws from accepted IDs and stop only when the accepted count reaches \(n\).
Another mistake is to assume that any set of \(n\) different IDs is automatically a simple random sample. Distinctness is necessary, but not sufficient: the selection must follow a probability procedure that gives every size-\(n\) group the same chance. For example, a generator stream with uniform marginal draws but a dependence pattern that forces certain pairs would not justify the equal-probability conclusion. Use independent uniform draws, conditional uniform draws, or a valid random-permutation method.
Finally, a random sample is not required to look balanced or typical. It may happen to contain several members with a feature that seems unusual, or none with another feature. That appearance alone does not show that the drawing procedure failed. The audit evaluates the frame and the stated random procedure; it does not judge randomness by whether the final list looks like someone’s expectations.
Check Your Understanding
Use the stated sampling rule in each question, and distinguish raw draws from accepted sample members.
- A frame has 15 indexed members and the requested sample size is 4. The generator gives 09, 12, 09, 03, 15. Which IDs are accepted, and when should the procedure stop?
- Why is it not enough to know that every raw draw is individually uniform over the frame numbers to conclude that all ordered sequences of distinct accepted IDs are equally likely?
- A frame has eight members and the final saved list contains five entries, one of which is repeated. What checks are needed before calling it a sample of size five without replacement?
- For a frame indexed 1 through 10, explain why taking the last digit of each generator output is not automatically a valid way to select an ID uniformly.
- A completed random sample does not include one particular member of the frame. Does that alone indicate an omission error? Explain the difference between a member not selected and a missing required sample entry.