Revise Sampling techniques and data collection for IB Mathematics: Applications and Interpretation (4.1)

In any statistical investigation the population is the entire group you want to know something about — 'all students at this school', 'every apple in the orchard', 'all the fish in the lake'. Measuring the whole population is a census: perfectly accurate in principle, but usually too slow, too expensive, or physically impossible. Instead you study a sample, a subset chosen to stand in for the population, and generalise your findings back to the whole. The quantity you calculate from the population (such as its true mean) is a parameter; the quantity you calculate from the sample and use to estimate it is a statistic.

We sample because it is faster, cheaper, and sometimes the only option: testing every match in a factory by striking it would leave nothing to sell. A good sample is representative — its structure mirrors the population's — so the trade for that saving is small. A bad sample distorts the population, and then even flawless arithmetic afterwards yields a wrong answer.

Before choosing how to collect data, classify what you are collecting, because it shapes every later choice of graph and statistic. Data is first split into qualitative and quantitative, and quantitative data is then split into discrete and continuous.

A quick test for discrete versus continuous: ask whether values between two neighbours can occur. Between 2 and 3 children there is nothing — discrete. Between 2 m and 3 m there are infinitely many heights — continuous. Money is a common grey area: strictly it comes in fixed steps (cents), so it is discrete, but with large amounts it is often treated as continuous.

The way you choose a sample decides whether it represents the population. You must be able to describe five methods, apply them, and give an advantage and a limitation of each. The first three are probability (random) methods, in which selection is governed by chance; the last two are non-probability methods, in which human choice enters — and with it, the risk of bias.

Stratified sampling is the one method that reliably produces calculation marks, so it deserves a clean routine. The idea: each stratum should contribute the same fraction of the sample as it makes up of the population. The number taken from a stratum is therefore its share of the population multiplied by the total sample size.

Number from a stratum $= \dfrac{\text{stratum size}}{\text{population size}} \times \text{total sample size}$. \n Round each result to the nearest whole number, then adjust the largest stratum if necessary so the parts sum to the required total.

Bias is a systematic tendency for a sampling method to over- or under-represent parts of the population, so estimates are consistently wrong in one direction. It is the central danger of data collection, and the reason the IB rewards criticism as much as calculation. Crucially, bias is not cured by a bigger sample — a larger convenience sample simply repeats the same distortion on a grander scale. It is cured by a better method.

Reliability and validity are the two lenses for judging a data-collection method. A method is reliable if repeating it under the same conditions gives consistent results, and valid if it genuinely measures what it claims to. They are independent qualities: a miscalibrated scale is reliable but not valid, giving the same wrong mass every time. Biased sampling threatens both — it makes results systematically inaccurate (a validity failure) and, if the source of bias shifts between studies, inconsistent too.

In Paper 2 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an accuracy mark (A) — and an accuracy mark depends on the method mark it follows. Follow-through (FT) means an earlier slip need not cost you the marks that depend on it, provided the later step is carried out correctly on your own figures. The engine also accepts any equivalent form and any correctly-rounded value. But that protection only exists if the method is on the page. Study how each mark below is earned by a specific line.

Data collection is the foundation the rest of the statistics course is built on. The sample you gather here becomes the raw material of descriptive statistics — the means, medians, quartiles and standard deviations of the next lesson — and the honesty of every one of those summaries is inherited from the honesty of your sampling. Later, the idea that a sample estimates a population parameter grows into confidence and hypothesis testing, where representative sampling is the assumption that makes the whole theory valid. Get the first step right — a representative sample, a clear method, and an open acknowledgement of bias — and everything that follows rests on solid ground.