4.1 Populations, samples and data collectionIB Maths: Analysis and Approaches SL: Revision notes

The population is the whole group you want to know about; a sample is the part you actually collect data from. A random sample is one in which every member of the population has an equal chance of being chosen, and choices are independent.

Data are discrete if they are counted and can only take particular values (number of people in a household, shoe sizes), and continuous if they are measured and can take any value in a range (time, mass, electricity use). Continuous data are always recorded to some accuracy, but that does not make them discrete.

• Simple random: number every member of the population and use a random number generator to pick distinct numbers. Needs a complete list (a sampling frame).
• Systematic: from an ordered list, pick every $k$th member after a random start, where $k = \frac{\text{population size}}{\text{sample size}}$. Can be biased if the list has a repeating pattern.
• Stratified: split the population into groups (strata) and randomly sample from each in proportion to its size: number from a stratum $= \frac{\text{stratum size}}{\text{population size}} \times \text{sample size}$.
• Quota: fill a set number from each group, but choose people non-randomly (e.g. whoever passes). Quick, but open to bias.
• Convenience: use whoever is easiest to reach. Cheap and quick, but usually biased.

Example: 330 DP1 and 270 DP2 students, sample of 40 stratified by year: $\frac{330}{600} \times 40 = 22$ from DP1 and 18 from DP2.

A sampling method is biased if it tends to over- or under-represent some part of the population, so the results are systematically wrong. When explaining bias, always name who is left out or over-represented and why that matters for the variable being measured; 'it is not random' alone earns little credit.

The reliability of a data source depends on who collected it, how, and why: a sample from a railway station at 8 a.m. says little about people who drive or work nights. Stratified sampling improves the estimate when the variable differs between groups (e.g. commuting time by district), because each group is represented in the right proportion.

An outlier is a value more than $1.5 \times \text{IQR}$ below the lower quartile or above the upper quartile: $x < Q_1 - 1.5 \times \text{IQR} \quad\text{or}\quad x > Q_3 + 1.5 \times \text{IQR}.$ Being an outlier does not automatically mean a value is wrong. Decide in context:

• A genuine extreme (a very tall player in a basketball squad) should be kept.
• An error (a negative electricity reading, a distracted participant, a typing mistake) should be corrected if possible or removed.

Missing data should be left out of calculations, not replaced by 0, which would distort the mean. Report how many values were excluded.