Statistics and Probability
The exam tests mean, median and mode for grouped data, ogive reading, and classical probability of simple events.
Core rules
- Mean by direct method: Σfᵢxᵢ/Σfᵢ; step deviation uses assumed mean a and class width h: x̄ = a + h(Σfᵢuᵢ/Σfᵢ).
- Median: l + [(n/2 − cf)/f] × h, where l is the lower limit of the median class, cf the cumulative frequency before it, f its frequency and h the class width.
- Mode: l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h.
- Empirical relation: 3 × median ≈ mode + 2 × mean.
- Probability P(E) = favourable outcomes/total equally likely outcomes, with 0 ≤ P(E) ≤ 1 and P(not E) = 1 − P(E).
Worked example
Classes 0–10, 10–20, 20–30, 30–40 have frequencies 5, 8, 12, 5, so n = 30 and n/2 = 15. Cumulative frequencies are 5, 13, 25, 30, so the median class is 20–30 with l = 20, cf = 13, f = 12, h = 10. Median = 20 + [(15 − 13)/12] × 10 = 20 + 1.67 = 21.67.
Traps
Using n instead of n/2 in the median formula; taking the cumulative frequency of the median class rather than the class before it; picking the modal class by frequency rather than by the largest frequency; reporting probability above 1; forgetting that outcomes must be equally likely.