Random sampling — simple, systematic and stratified

The three sampling methods that use chance: simple random sampling (every sample equally likely), systematic sampling (every kth unit after a random start), and stratified sampling (proportional allocation across groups), with how to carry each out and the advantages and drawbacks that examiners ask for.

Statistics · Statistical sampling

Random sampling — simple, systematic and stratified

Why it works

A sampling method is random when chance — not a person's choice — decides who's in the sample. That's what lets you argue a sample is unbiased. There are three on the AS course, and the exam wants you to carry each one out, *recognise it from a description, and state an advantage or disadvantage*.

Simple random sampling. Every possible sample of size

n

is equally likely, so every sampling unit has the same chance of being chosen. To do it you need a sampling frame: number every unit

1, 2, 3, \dots, N

, then pick

n

different numbers using a random source (a random number generator, random number tables, or drawing numbered tickets from a hat). Ignore repeats and any number outside

1

–

N

Advantages: free of bias; every unit equally likely; simple when $N$ is small.
Disadvantages: you must have a full sampling frame (a complete list), and it

becomes slow and expensive for a large population.

Systematic sampling. Choose units at a regular interval down an ordered list. The interval is

k = \frac{\text{population size}}{\text{sample size}} = \frac{N}{n},

and the start is a randomly chosen number between

1

and

k

. From there you take every

k

th unit. For example,

N = 800

n = 50

gives

k = 16

: pick a random start in

1

–

16

, say

7

, then take units

7, 23, 39, 55, \dots

Advantages: quick and simple, and works well for large populations.
Disadvantages: you still need an ordered list, and bias creeps in if the list

has a repeating pattern whose period lines up with

k

(e.g. every 16th house being a corner house).

The random start is what keeps systematic sampling random — without it, the sample is completely determined and it's no longer a random method.

Stratified sampling. When the population splits into distinct non-overlapping groups — strata — such as year groups, genders or shifts, you sample each stratum in proportion to its size. The number taken from a stratum is

\text{number from stratum} = \frac{\text{stratum size}}{\text{population size}} \times \text{sample size}.

Within each stratum you then use simple random sampling. Because the sample mirrors the make-up of the population, it represents those groups well.

Advantages: reflects the population structure, so it's representative of each

stratum; improves on simple random sampling when groups differ.

Disadvantages: you must know which stratum each unit is in and the size of each

stratum, and it only works when the population has clear strata.

A quick check: the numbers taken from all the strata should add back up to

n

(round sensibly so they do).