Statistics and Probability
Statistics and probability tests the collection, presentation and interpretation of data and the calculation of chance. Typical items involve frequency tables, grouped data, cumulative frequency curves, histograms, scatter diagrams, tree diagrams and probability without replacement.
Core rules
- Mean of a frequency distribution = Σfx ÷ Σf; for grouped data, x is the class midpoint.
- Median position = (n + 1)/2 for raw data; on a cumulative frequency curve, read across at n/2. Quartiles are read at n/4 and 3n/4, and IQR = upper − lower quartile.
- Histogram bars: frequency density = frequency ÷ class width, plotted on the vertical axis.
- Probability = favourable outcomes ÷ total outcomes, and 0 ≤ P ≤ 1.
- Mutually exclusive events: P(A or B) = P(A) + P(B). Independent events: P(A and B) = P(A) × P(B).
- Tree diagrams: multiply along a branch, add between branches. For 'without replacement', both the number of favourable items and the total decrease.
Worked example
A bag contains 5 red and 3 blue counters. Two are drawn without replacement. Find the probability that both are red. Reasoning: P(first red) = 5/8. After removing one red, 4 red remain among 7 counters, so P(second red given first red) = 4/7. Multiply: 5/8 × 4/7 = 20/56 = 5/14 ≈ 0.357. The draws would be independent only if the counter were replaced, in which case the answer would be (5/8)² = 25/64.
Traps that cost marks
- Using class frequencies rather than midpoints to estimate a grouped mean.
- Reading a histogram as if bar height were the frequency.
- Forgetting to reduce the denominator in a without-replacement problem.
- Adding probabilities when the events happen in succession.
- Reading the median at n/2 on raw data instead of on the cumulative frequency curve.