Sampling, bias & random numbers
You taste a spoonful to judge the whole pot — but only if you stirred first. A sample stands in for the population only if it's representative; a biased method is an unstirred pot, and no maths afterwards can fix it.
Random samples, bias, and random numbers
A population is every item of interest; a sample is a part of it. We sample because a full census is too expensive, slow, or destructive. A good sample is representative; a method that systematically over- or under-represents some members is biased, and the resulting sample is unrepresentative.
Bias is a fault in the method, not in the data — to spot it, ask “who gets left out or over-counted?” No later maths can undo a biased sample.
To diagnose bias, name the favoured group: “first 6 to arrive” over-counts the keen and those living close; “people who just bought the bar” already like it; “first 20 males” excludes females entirely. A leading question (“Do you agree everyone should vote?”) and self-selection (only those who care reply) are biases too.
A random sample of size is chosen so that every possible sample of size is equally likely — so every member has an equal chance. That needs a complete list, the sampling frame. If no list exists (every fish in the sea, every left-handed person), some members can never be chosen: that is selection bias.
Random sample = every size-n sample equally likely (needs a sampling frame). NOT “no pattern” or “picked carelessly” — that definition scores zero.
Using a table of random numbers
Number the population, then read digits in fixed-width blocks (the spacing in the table is only for legibility — treat the digits as one long line). Reject any value out of range and any repeat, and stop at your sample size.
Worked example
Start simple: reading a random-number table
Choose 6 from a population of 500, numbered –. Reading three digits down a column gives:
- are in range — keep.
- : out of range — reject (or, for 500, subtract 500: ).
- keep; the second is a repeat — reject.
- Six valid IDs reached, so stop.
Worked example
Calculator random numbers, both directions
150 students numbered 1–150. The calculator gives a number in ; multiply by 150 and round up to a student number.
- Number gives .
- Reverse: student needs , i.e. , so any works (e.g. ).
Unbiased estimates from a sample
A sample also estimates the population mean and variance. The unbiased estimates are and — note the , not .
Worked example
Unbiased estimates from totals
: ; . (9709/62/M/J/24 Q2)
A sample estimates μ by x̄ and σ² by s² with the n−1 divisor (the biased ÷n scores zero). A random sample makes every member equally likely.
Why divide by n−1, not n
The sample spreads around (which sits inside the data), not around the unknown true , so the squared deviations come out a touch too small. Dividing by instead of scales that bias away exactly, on average.
Examiner note
Common mistake