Populations, censuses and samplesEdexcel International A Level Further Maths: Revision notes

A population is the whole set of items or people of interest, such as all $4200$ households in a council area. A census observes or measures every member of the population. A sample is a selection of members of the population, used to find out about the whole. The population does not have to be people: it could be all the fuses made in a day or all the bags of flour in a warehouse.

A sampling unit is a single member of the population that can be selected for the sample, for example one household or one fuse. A sampling frame is a list of all the sampling units in the population, for example an electoral register or the serial numbers $1$ to $5000$. A good sampling frame is complete and has no duplicates. A random sample is chosen by numbering the units in the frame and selecting numbers with a random number generator.

A census gives the exact population values and has no sampling variability. It is also unbiased if everyone responds. But it is expensive and slow, it needs a lot of data processing, and it is impossible if the testing destroys the item or the population is infinite. A sample is cheaper and quicker, needs less data processing and may be the only choice when testing is destructive. But the results are subject to sampling variability, may be biased if the sample is not representative, and may be less accurate. A small sample may miss rare groups. In an exam, always refer to the context. For example, testing fuses until they blow destroys them, so a census would leave none to sell.

A parameter is a fixed number describing a population, such as $\mu$, $\sigma$ or the proportion $p$. Parameters are usually unknown. A statistic is any quantity calculated only from the values in a random sample, with no unknown parameters. The sample mean $\bar X$, sample maximum and sample proportion $\hat p$ are statistics. The expression $\bar X-\mu$ is not a statistic because $\mu$ is unknown.

Different random samples give different values of a statistic, so a statistic is a random variable. Its probability distribution over all possible samples of a given size is its sampling distribution. Example: three discs numbered $1$, $2$, $3$ are drawn with replacement, twice. There are $9$ equally likely pairs. The larger value $M$ has $\mathrm{P}(M=1)=\frac19$, $\mathrm{P}(M=2)=\frac39$, $\mathrm{P}(M=3)=\frac59$. The sample mean has values $1,1.5,2,2.5,3$ with probabilities $\frac19,\frac29,\frac39,\frac29,\frac19$. To build a sampling distribution, list every possible sample, find the statistic for each, then total the probabilities of the equal values.