Systematic Sampling Calculator
Select a systematic sample: choose every k-th member of an ordered population after a random start. Enter the population size N and the sample size n to get the sampling interval, the start, and every selected position — or paste your list to pick the items directly.
Need a simple random sample instead? Use the random number generator with No repeats. Not sure how big n should be? Start with the sample size calculator.
When a list is pasted, N is the number of lines and the selected items are shown.
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How Systematic Sampling Works
k = ⌊N / n⌋ (sampling interval, rounded down)
r = random whole number from 1 to k (start)
Sample = r, r + k, r + 2k, …, r + (n − 1)k
Rounding k down guarantees the last position, r + (n − 1)k, never runs past N, so you always get exactly n members. Each member has the same chance of selection, 1/k — that is what makes the method a probability sample despite the fixed spacing.
Worked Example: Auditing 40 of 500 Invoices
An auditor needs a sample of n = 40 from N = 500 numbered invoices.
- N / n = 500 / 40 = 12.5, so k = 12.
- A random start between 1 and 12 comes up as r = 7.
- Selected invoices: 7, 19, 31, 43, 55, …, 7 + 39 × 12 = 475.
Invoices 476–500 can never be chosen with this start, and each invoice's selection chance is 1 in 12 rather than exactly 40/500 — the small cost of rounding k down. Enter N = 500, n = 40, start = 7 above to reproduce the list.
Advantages and the Periodicity Trap
- Simple to carry out: one random number, then count — ideal for lists, production lines, or foot traffic ("every 10th customer").
- Even coverage: the sample is spread across the whole list, which often makes it more precise than a simple random sample when the list is ordered by a relevant variable.
- Periodicity risk: if the list has a repeating pattern whose cycle matches k — for example every 7th day being a Sunday with k = 7 — every selected member shares the same position in the cycle and the sample is badly biased. Shuffle the list first, or use a simple random sample, when such a pattern is possible.
Frequently Asked Questions
How do you calculate the sampling interval k?
Divide the population size by the sample size and round down: k = ⌊N/n⌋. With N = 1,000 and n = 50, k = 20, so you take every 20th member after a random start between 1 and 20.
What if N/n is not a whole number?
Round k down. That keeps every selected position within the population and still gives exactly n members. The trade-off is that a few members at the end of the list cannot be selected for some starts; if that matters, use circular systematic sampling or a simple random sample.
Is systematic sampling random?
The start is random, so every member has a known, equal chance (1/k) of selection, which makes it a probability sampling method. But the members are not selected independently — once the start is fixed, the whole sample is determined.
What is the difference between systematic and stratified sampling?
Systematic sampling takes every k-th member of one ordered list. Stratified sampling first splits the population into groups (strata) and then samples randomly within each group. Systematic sampling of a list sorted by a variable behaves like an implicit stratification on that variable.
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