Systematic Sampling With a Random Start

Why Use a Random Start?

Suppose a clinic has an ordered list of 120 eligible patients and needs to contact 12 of them for a hypothetical survey. Selecting every tenth patient seems straightforward, but the choice of where to begin matters. If staff always start with the tenth name, some positions on the list can never be selected. A random start gives each eligible starting position a chance to determine the sample, while the fixed interval makes selection practical.

Systematic sampling is a probability sampling design: selection uses a specified random mechanism, rather than convenience or personal choice. To construct it correctly, first define the sampling frame and desired sample size, then calculate the sampling interval, randomly choose a start, and follow the interval through the ordered list.

Calculating the Interval and Constructing the Sample

Let \(N\) be the number of units in the frame and \(n\) the desired sample size. The basic procedure is easiest to define when \(N\) is an exact multiple of \(n\). In that case the interval is the number of frame positions per desired sample member:

Here \(k\) is a positive integer. The random start \(r\) is chosen uniformly from the integers \(1,2,\ldots,k\). “Uniformly” means that each of these \(k\) positions has the same probability, \(1/k\), of being selected as the start. Once \(r\) is chosen, the selected positions are \(r, r+k, r+2k,\ldots\), stopping at position \(N\). Since \(N=nk\), this gives exactly \(n\) units.

Why Each Unit Has an Equal Inclusion Probability

In the divisible case, the possible starts are \(1,\ldots,k\), so there are \(k\) equally likely samples. Each sample contains \(n\) positions separated by \(k\). Every position from 1 through \(N=nk\) belongs to exactly one of these samples. For example, position 1 belongs to the sample starting at 1, position 12 in a frame with \(k=10\) belongs to the sample starting at 2, and position 120 belongs to the sample starting at 10.

More generally, divide a unit’s position by \(k\) and consider its remainder. Its position can be written uniquely as \(r+jk\), where \(r\) is one of \(1,\ldots,k\) and \(j\) is a nonnegative integer. Thus that unit is included precisely when the random start is \(r\). Since each possible start has probability \(1/k\), the unit’s inclusion probability is \(1/k\). Because \(k=N/n\), this is also \(n/N\).

This equal inclusion probability is an important design property, but it does not make systematic sampling the same as simple random sampling without replacement. In a simple random sample, every possible set of \(n\) units has the same chance of selection. In this systematic design, only the \(k\) sets generated by the possible starts can be selected. The random start determines which one of those sets is chosen.

Conditions and a Practical Boundary

The calculation and inclusion-probability argument above rely on specific conditions. The frame must have a known, fixed number \(N\) of positions; the desired sample size \(n\) must divide \(N\) exactly for this version of the procedure; and the random start must be selected uniformly from 1 through \(k=N/n\). The ordering must be set before the random selection, and the selection must follow the stated interval.

In practice, \(N\) is not always divisible by the desired \(n\). Then \(N/n\) is not an integer, and the exact procedure described here cannot be applied as written while also guaranteeing exactly \(n\) selections. Simply rounding the interval and still claiming that this procedure gives a fixed sample size with inclusion probability \(n/N\) is not justified. A nondivisible case requires a separately specified systematic-sampling method and its own account of sample size and selection probabilities. The key is to state the method rather than silently treat a rounded interval as the exact design.

Common Mistakes and Why the Design Matters

Systematic sampling can be convenient when a frame is already in a usable order: after the random start, the rule is simple to apply and audit. It also makes the random mechanism visible. Someone checking the procedure can verify the frame count, interval, random start, and sequence of selected positions.

• Starting at position 1 automatically. This is not a random start. In the divisible procedure, the start must be selected randomly from all positions 1 through \(k\), not fixed by habit.
• Choosing a start outside the allowed range. If \(k=10\), the start must be an integer from 1 to 10. Choosing position 17 as the start and then stepping by 10 is a different procedure.
• Using a noninteger interval as if it were exact. The formula \(k=N/n\) gives an integer interval only when \(N\) is a multiple of \(n\). A fractional result requires a separately defined method.
• Replacing a selected unit who cannot be reached. Replacing that unit with the next convenient person changes the selection procedure. Record the issue and follow a prespecified protocol instead of making an ad hoc substitution.
• Calling the result a simple random sample. Although every unit has the same individual inclusion probability in the divisible design, not every group of \(n\) units can be selected. The design is systematic sampling, not simple random sampling.

A well-documented systematic sample records the frame and its ordering, \(N\), \(n\), \(k\), the method used to generate the start, the selected positions, and any departures from the protocol. These details make it possible to distinguish the intended random design from the way it was actually carried out.