Statistics and Probability
Statistics and Probability
This domain tests how data are collected, displayed, summarised, and interpreted, and how probability models uncertainty. Expect to read dot plots, box plots, histograms, and two-way tables; compare distributions using centre, spread, and shape; calculate probabilities from tables, tree diagrams, and everyday contexts; and distinguish experimental from theoretical probability.
Core rules and concepts
- Measures of centre: mean, median, mode. Measures of spread: range, interquartile range (IQR), standard deviation. The IQR is (Q_3-Q_1) and is resistant to outliers.
- For a box plot, the box shows the middle 50%, the line is the median, and whiskers extend to the minimum and maximum (or to fences if outliers are shown).
- Probability: (0\le P(A)\le1), (P(\text{not }A)=1-P(A)). For mutually exclusive events, (P(A\text{ or }B)=P(A)+P(B)).
- For non-mutually exclusive events, subtract the overlap: (P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)).
- Conditional probability restricts the sample space: (P(A\mid B)=\frac{P(A\text{ and }B)}{P(B)}).
Worked example
In a group of 60 students, 24 take physics, 18 take chemistry, and 8 take both. Find (P(\text{chemistry}\mid\text{physics})).
The condition "given physics" reduces the sample space to the 24 physics students. Of those, 8 also take chemistry: (P(\text{chemistry}\mid\text{physics})=\frac{8}{24}=\frac13).
If asked for (P(\text{physics or chemistry})), use the addition rule: (\frac{24}{60}+\frac{18}{60}-\frac{8}{60}=\frac{34}{60}=\frac{17}{30}).
Traps that cost marks
- Using 60 as the denominator for a conditional probability instead of the conditioned group.
- Assuming events are independent without evidence.
- Reversing (P(A\mid B)) and (P(B\mid A)).
- Describing a distribution by its mean alone while ignoring spread or outliers.
- Reading a histogram as if bar height always equals frequency when the bars have unequal widths.