Statistics and Probability
Statistics and Probability tests how data are collected, tabulated and summarised, and how uncertainty is quantified. Expect questions on mean, median, mode and quartiles for grouped data, cumulative frequency curves, range, variance and standard deviation, plus probability of single and combined events, including counting with permutations and combinations.
Summarising grouped data
- Use class midpoints x with mean = Σfx/Σf.
- The median class is where cumulative frequency reaches N/2; Q₁ sits at N/4 and Q₃ at 3N/4; interquartile range = Q₃ − Q₁.
- The modal class is the class with the highest frequency, not its midpoint.
Dispersion
Variance = Σf(x − x̄)²/Σf and the standard deviation is its square root. The range is the difference between the largest and smallest values.
Probability
- P(A) = favourable outcomes ÷ total equally likely outcomes; 0 ≤ P(A) ≤ 1; P(A′) = 1 − P(A).
- Mutually exclusive: P(A or B) = P(A) + P(B).
- Independent: P(A and B) = P(A) × P(B).
- Without replacement, both the total and the favourable count fall after each draw.
Counting
nPr = n!/(n − r)! for ordered arrangements; nCr = n!/[r!(n − r)!] for selections where order does not matter.
Worked example
A bag holds 5 red and 3 blue balls. Two are drawn without replacement. Find the probability that both are the same colour. P(RR) = 5/8 × 4/7 = 20/56 and P(BB) = 3/8 × 2/7 = 6/56. Since RR and BB are mutually exclusive, add: 26/56 = 13/28.
Traps
- Using 5/8 twice instead of 4/7 for the second red ball (replacement error).
- Adding probabilities for events joined by "and" when they are independent.
- Quoting the midpoint of the modal class as the mode of grouped data.
- For "at least one", forgetting the complement method: 1 − P(none).