From the SRS Definition to a Selection Method
In Simple Random Sample Defined, we learned that an SRS gives every possible group of the chosen size an equal chance of selection. A random number table gives us a practical way to choose that group from a numbered list. The table supplies digits; a clear, consistent rule tells us how to use them.
The main idea is to label every individual, read the table in a set direction using groups of digits of the right length, and accept labels that belong to the population and have not already been selected. We skip numbers that cannot identify a new individual. We do not change the rule because we dislike a result or because a selected label seems unusual.
The method assumes that the digits are random and that we follow the rule consistently. When each eligible label is equally likely at each draw, and selected individuals are not selected again, the procedure gives each possible group of the required size the same chance. That is the group-level property required for an SRS.
Set Up the Table Procedure
First, check the sampling frame: the list of individuals from which the sample will be selected. As discussed in Population, Sampling Frame, and Sample, the frame should correspond adequately to the population of interest. Give each individual exactly one label, with no missing or duplicated labels.
Choose labels with the same number of digits. For a roster of 18 people, label them 01 through 18. Two digits are used even for labels below 10, so person 4 is 04. This keeps the groups of digits aligned. For a roster of 250, use three-digit labels, 001 through 250.
Before reading, decide where to start in the table and which direction to move. If the instructions specify a row, column, or starting position, follow them. Otherwise, use a chance method to choose a starting point. Then read in the stated direction, continuing to the next line in the same direction when necessary. Do not look at values first and then pick a starting point that produces a preferred sample.
Next, group the digits into blocks matching the label length. With two-digit labels, read two digits at a time; with three-digit labels, read three at a time. Keep the groups in order. A group is accepted only if it matches a label in the roster that has not yet been selected.
- Accept a label in the roster if it has not appeared in the sample yet.
- Skip a value outside the roster’s label range, such as 73 when labels run from 01 to 18.
- Skip a label already selected. This prevents selecting the same individual twice.
- Stop as soon as the requested sample size has been reached.
For example, with labels 01 through 18, 04 identifies an individual, but 00, 19, and 73 do not. Even though 00 is made of two digits, it is not a roster label. The number of digits in a group does not make every possible value valid.
A Reliable Step-by-Step Routine
Give every individual one unique label using the same number of digits.
Choose a starting point and direction before examining the values you will use.
Use as many digits per group as there are digits in each label.
Accept a valid, not-yet-selected label; skip out-of-range values and repeats.
Stop when you have the requested number of different labels, then use the roster to identify those individuals.
Worked Example: Selecting Four People from a Roster
Worked Example: A Community Garden Survey
A fictional community garden has 18 registered volunteers. The coordinator wants an SRS of four volunteers to answer a survey about tool storage. Each volunteer is labeled 01 through 18. Starting at a previously chosen location in a random number table, the coordinator reads two digits at a time from left to right. The digit groups encountered are:
Apply the rule in order: The first group, 73, is outside the range 01 through 18, so skip it. The next group, 04, is a valid label and is selected. The group 18 is valid and not yet selected, so select it too.
The next group is another 18. Since that volunteer is already selected, skip the repeat. The group 92 is out of range, so skip it. The next group, 06, is valid and new; select it. The group 00 is not a roster label, so skip it. The group 11 is valid and new; select it. Four different volunteers have now been selected, so stop. There is no need to use the remaining groups 17 and 03.
Report the sample: The selected labels are 04, 18, 06, and 11. Using the roster, the coordinator can identify the four volunteers. The order in which the labels were selected does not change the sample group.
The procedure did not treat an invalid value or a repeat as a reason to restart. It continued through the table using the same rule. This is important: restarting after seeing an inconvenient result could alter the selection chances.
Worked Example: Using Three-Digit Labels
Worked Example: Sampling Equipment Records
A fictional repair center has 250 equipment records and wants a sample of four records for a routine audit. The records are numbered 001 through 250. A random number table is read in three-digit groups from a predetermined starting point. The groups are:
Read each group in sequence: The first group, 251, is too large because the largest roster label is 250, so skip it. Select 007; it identifies a record in the roster. Select 099 as well. The next group, 007, is a repeat, so skip it rather than counting the same record twice.
The group 250 is valid and new, so select it. The group 000 does not identify a record in the range 001 through 250, so skip it. Select 143, which is the fourth distinct valid label. Stop there.
Report the sample: The sample consists of records 007, 099, 250, and 143. Three-digit labels make the grouping unambiguous; for instance, 007 is one label, not separate labels 00 and 7.
Notice that 250 is included because it is the largest valid label, while 251 is not. The rule is based on the roster’s actual label range, not on whether a value looks close to the range.
Worked Example: Continuing Across a Table Row
Worked Example: Choosing Students for a Library Review
A fictional school has 20 students in a library advisory group. The students are labeled 01 through 20, and a sample of five is needed. The sampling instructions specify a starting point in a random number table and say to read rightward, continuing at the beginning of the next row if needed. The two-digit groups encountered across the row boundary are:
Follow the specified path: Select 06. Skip 21 because it is outside the range 01 through 20. Select 14. The next 06 is a repeat, so skip it. Select 03 and then 19. At this point, four distinct students have been selected.
The instructions say to continue into the next row in the same direction. The next group in that row is 12, so select it. The five selected labels are 06, 14, 03, 19, and 12; now stop. A table’s row boundary does not end the selection process unless the instructions say it does. Continue by the stated route until the sample size is reached.
Explain the design: Each group of digits was considered in its original order, and the coordinator neither rearranged the values nor restarted at the top of the table. Each selected label points to a different student, so the result is a sample of five distinct individuals.
Why Skipping Repeats Matters
A random number table can produce the same label more than once. In an ordinary SRS of distinct individuals, the same individual cannot fill two places in the sample. Therefore, a repeated label does not increase the sample size; it is skipped and the reading continues.
For example, if labels 04 and 18 have already been selected, encountering 18 again does not select a fifth person or replace 04. The sample remains a group of distinct individuals. Continue through the table until the requested number of different valid labels has been reached.
Skipping repeats is different from choosing a sample “without randomness.” The random table determines the order in which candidate labels appear. Repeats simply cannot add a new individual to the sample. Similarly, out-of-range values are not mapped to some valid label; they are skipped. Mapping or changing values without a specified method could give some individuals different chances.
After selecting the labels, use the sampling frame to connect each label to the correct person or object. Keep the roster and the selected labels together so that the process can be checked and repeated if needed.
Common Mistakes and AP Exam Tips
- Using inconsistent digit groups. If labels use two digits, read two digits at a time throughout. Changing group size partway through can change which labels are read.
- Forgetting leading zeros. A label such as 04 must stay two digits when the table is read. Writing labels with inconsistent lengths can make the groups ambiguous.
- Accepting numbers that are not on the roster. With labels 01 through 18, a group such as 73 is not a person. State the valid range, then skip values outside it.
- Counting a repeated label twice. A sample of five people must contain five different people. Skip an already-selected label and keep reading.
- Changing direction or restarting selectively. Decide how to read before examining the digits, and follow that rule even if early values are invalid or repeated.
- Stopping too early. Keep reading until the requested number of valid, distinct labels has been selected—not until the table contains that many digit groups.
- Claiming that a sample “looks representative” proves it is an SRS. As in Simple Random Sample Defined, the definition concerns the chances assigned to possible sample groups, not the appearance of one selected group.
For a full-credit explanation, identify the label range and the number of digits read at a time; state how the starting point and direction are determined; explain that out-of-range labels and repeats are skipped; and report when the requested sample size is reached. If asked why the procedure is an SRS, connect the chance-based selection of distinct labels to the equal-chance property for possible groups.
Check Your Understanding
Use the table procedure and the stated roster ranges to answer each question.
- A roster has 27 individuals labeled 01 through 27. How many digits should each table group contain? Which of these values should be skipped: 09, 27, 28, and 00?
- A sample needs three individuals from labels 01 through 15. The groups read are 04, 16, 04, 12, 03. Which labels make up the sample, and why is 04 counted only once?
- A list has 125 objects labeled 001 through 125. A three-digit group is 125. Should it be accepted or skipped? Explain using the label range.
- Why should a sampler decide the starting point and reading direction before examining the table values?
- If the requested sample size has already been reached, should the sampler continue selecting labels from later groups? Explain.