4.1 Sampling, data quality and outliersIB Maths: Applications and Interpretation SL: Revision notes

A population is the whole group being studied; a sample is a smaller part of it used to draw conclusions about the whole. A random sample gives every member of the population an equal chance of selection. At SL, a data set given in a question is treated as the population unless stated otherwise. Discrete data can take only separate values, usually counted (number of siblings, goals scored). Continuous data can take any value in a range, usually measured (height, time, mass). Example: the number of cars passing a school in an hour is discrete; the time a car takes to pass a point is continuous.

A data source is reliable if it is trustworthy: it comes from a reputable organisation, uses a clear method and a suitably large sample, and gives similar results when repeated. Bias is a systematic tendency for a sample to differ from the population, for example when some members have no chance of being chosen, when a survey question is leading, or when only some people reply (non-response bias). A larger sample does not remove bias: a large sample from the wrong group is still biased.

Simple random sampling: number every member, then use random numbers (GDC) to pick distinct members. Fair, but needs a full list and may miss small groups. Systematic sampling: choose every $k$th member from an ordered list, starting at a random place, where $k=\frac{\text{population}}{\text{sample size}}$. Quick, but biased if the list has a repeating pattern that matches $k$. Stratified sampling: split the population into groups (strata) and sample each in proportion to its size: $\text{sample from a group}=\frac{\text{group size}}{\text{population}}\times\text{sample size}.$ Example: 600 members (240 juniors, 210 adults, 150 seniors), sample of 80. Juniors $=240\times\frac{80}{600}=32$, adults $=28$, seniors $=20$. Check $32+28+20=80$.

Convenience sampling: choose whoever is easiest to reach (e.g. the first people to walk past). Cheap and fast, but often biased and not representative. Quota sampling: the interviewer must find a set number of people in each group (e.g. 20 men and 20 women) but chooses them non-randomly. It reflects the group structure like stratified sampling, but selection within groups is not random, so bias is still possible. No sampling frame is needed. Effectiveness: random and stratified samples are the most representative; convenience samples are the least.

Data may contain recording errors (typing mistakes, impossible values such as 400 hours in a week, a stopwatch left running) or missing data. Sensible responses: check the source and correct the value; remove an item that cannot be corrected; follow up non-respondents or replace them; or report the issue and treat conclusions with caution. Never silently keep an impossible value. Missing data matter most when the missing items are different from the rest, because then the remaining data are biased.

An outlier is a data item more than $1.5\times\text{IQR}$ from the nearest quartile: a value is an outlier if it is below $Q_1-1.5\times\text{IQR}$ or above $Q_3+1.5\times\text{IQR}$, where $\text{IQR}=Q_3-Q_1$. Example: $4,5,5,6,7,8,9,25$. Then $Q_1=5$, $Q_3=8.5$, $\text{IQR}=3.5$, upper boundary $8.5+1.5(3.5)=13.75$. Since $25>13.75$, $25$ is an outlier. Lower boundary $5-5.25<0$, so there are no low outliers. In context, an outlier may be a valid part of the sample (an exceptionally fast runner) or an error. Check before removing. Outliers are shown as crosses on box and whisker diagrams (4.2) and they inflate the standard deviation and range (4.3).