Systematic Random Sampling

A Regular Way to Select Individuals

In Simple Random Sample Defined, every possible sample of a given size has an equal chance of selection. A simple random sample can be selected using a random number table or randInt, as in the earlier tutorials on those methods. There is another practical way to select individuals from an ordered list: choose a random starting position, then take individuals at a fixed interval.

This method is called systematic random sampling. It can spread a sample across a list without requiring a separate random number for every selected person. The method depends on the order of the sampling frame, so that order must be checked for patterns that might affect the results.

For the basic procedure in this tutorial, suppose the sampling frame contains \(N\) individuals and the desired sample size is \(n\), with \(N\) divisible by \(n\). The interval is \(k=N/n\). Choose a random starting position from 1 through \(k\), and then count forward by \(k\) positions until the sample is complete.

The random start is chosen from the first \(k\) positions, not from the entire list. If the start is \(r\), the selected positions are \(r, r+k, r+2k,\ldots\). Because \(N=nk\), this selects exactly \(n\) individuals.

How to Select a Systematic Random Sample

The procedure is simple, but each part matters. Use the frame positions consistently, and do not change the random start or skip a selected position just because another person seems easier to contact. If a selected individual does not respond, record the nonresponse rather than quietly substituting someone else.

If \(N\) is not divisible by \(n\), \(N/n\) is not a whole number. There are ways to adapt systematic selection to such a situation, but the simple counting rule above does not directly apply. For a clear plan using this basic procedure, choose a sample size and frame size that give a whole-number interval, or specify and explain an appropriate adjustment rather than rounding the interval without checking the resulting sample.

Worked Example: Selecting Participants From a Roster

What the Random Start Does—and Does Not Do

When \(N=nk\) and the start is chosen uniformly from positions 1 through \(k\), every individual position has the same chance, \(1/k\), of being selected. For instance, each position belongs to exactly one of the \(k\) possible starting sequences. Since \(k=N/n\), this individual inclusion chance is \(n/N\).

However, equal chances for individuals do not mean systematic sampling is the same as an SRS. With \(N=12\) and \(n=3\), the interval is 4. The only possible systematic samples are positions 1, 5, and 9; positions 2, 6, and 10; positions 3, 7, and 11; or positions 4, 8, and 12. There are just four possible samples from the random start. An SRS of size 3 could produce many other groups of three individuals.

So, as in Why Random Selection Matters, the random start limits the opportunity for deliberate selection. It does not guarantee that a particular sample will mirror the population perfectly. Nor does it make every possible group of \(n\) individuals equally likely. That distinction matters when describing the design: systematic sampling uses chance, but it is not generally an SRS.

Why the Order of the Frame Matters

An ordered frame can be helpful when it spreads selected individuals across the list. But a repeating pattern in the frame can line up with the sampling interval. If that happens, a systematic sample may repeatedly select individuals in the same part of the pattern.

A periodic pattern is a feature that repeats at regular intervals. For example, a production record might list items in the order they came off a line, with a particular type of item recurring every fifth position. If the sampling interval is also 5, every selected item will occupy the same place in that cycle. A random start chooses which place in the cycle is sampled, but it does not mix the places together.

Check what the list order means, not just whether the list is numbered. Ask whether rows repeat by shift, location, machine, day, or another feature related to the measured variable. A pattern in the frame is a concern when its regularity can line up with the interval and the repeating feature is related to the question being studied.

Worked Example: A Repeating Production Pattern

When Systematic Sampling Can Be Useful

Systematic sampling can be convenient when there is a complete, ordered list and selecting individuals at regular intervals is easier than generating and tracking many separate labels. A roster, a sequence of records, or a list of items may be workable frames. The design also makes it straightforward to verify that the intended number of individuals has been selected.

Convenience is not enough by itself. As in Population, Sampling Frame, and Sample, the frame must cover the population of interest adequately. A list that leaves out a group cannot give that group a chance to be selected. Also check for duplicates, outdated entries, or an order that is connected to the variable. A random start does not correct undercoverage or make an unsuitable frame suitable.

If the frame has no concerning periodic pattern, systematic sampling can be a practical chance-based design. If there is a pattern that aligns with the interval, an SRS or another design may be a better choice. In either case, describe the actual method used so readers can tell what supports generalizing to the population represented by the frame.

Common Mistakes and AP Exam Tips

• Choosing the start from the whole list. For the basic procedure, the random start must be one of positions 1 through \(k\), not any position from 1 through \(N\).
• Forgetting to calculate the interval. State \(k=N/n\) and show the calculation when \(N\) is divisible by \(n\). Then select every \(k\)th position after the random start.
• Starting at a random position and then choosing convenient individuals. The rest of the sample must follow the fixed interval. Substituting people because they are easier to reach changes the selection method.
• Calling every systematic sample an SRS. A random start does not make every possible sample of size \(n\) equally likely. Explain that systematic sampling selects at regular intervals from a random start.
• Ignoring the frame’s order. Mention what the list is ordered by and check whether a repeating pattern related to the variable could align with \(k\).
• Claiming random selection removes every source of bias. A random start does not repair an incomplete frame, nonresponse, poor measurement, or a problematic periodic pattern.

For a full-credit description, identify the population and frame, calculate the interval, state that the start is chosen at random from the first interval, and explain that every \(k\)th individual is selected. If the list has a possible cycle, describe how its length relates to the interval and why that could affect the sample.