Sampling, bias and random numbers
We usually want information about a whole population but can measure only a sample. The selection method decides whether the sample can represent the population.
A sample is evidence about a population
Definition
2 marksWhat is meant by population and sample?
Model answer: A population is the complete set of items or people of interest. A sample is the part of that population that is observed.
Definition
1 markWhat is meant by random sample of size n?
Model answer: A sample selected so that every possible sample of size n has an equal chance of being chosen.
Population → frame → sample
Population
all 804 students
↓ list every member
Sampling frame
001 Ana · 002 Ben · 003 Chen · … · 804 Zara
↓ random valid IDs
Sample
567 · 109 · 665 · 021
Reject out-of-range values and repeated IDs.
- A sampling frame is the list from which members are selected.
- A representative sample reflects the population well enough for the intended comparison.
- A method is biased when it systematically favours some members or excludes others.
Find who has a different chance of selection
To explain why a method is unsatisfactory, identify the affected group and state how its chance of selection differs.
- Surveying the first people to arrive excludes later arrivals.
- An online voluntary poll favours people who notice the poll and choose to reply.
- A frame that omits some members gives those members zero chance.
Key idea
Worked example
Write a complete one-mark explanation
Method: interview the first 30 students entering the library.
Model answer: Students who do not enter the library at that time have no chance of selection, so the sample may not represent all students.
Apply a random-number rule exactly
If 804 students are numbered 001 to 804, read random digits in groups of three. Reject 000, values above 804 and any repeated ID.
Worked example
Follow the current-paper digit stream
After 567 and 109, the next groups are 843, 109, 665 and 021.
- 843 is outside the frame, so reject it.
- 109 has already been selected, so reject it.
- 665 and 021 are the next two valid IDs.
(9709/62/M/J/24 Q2(a))
A population is numbered 001 to 240. Starting from the first group, use 198, 246, 031, 198, 207, 000 to select the first three different valid IDs.
Show worked answer
Accept 198; reject 246; accept 031; reject repeated 198; accept 207.
Common mistake