Statistics and Probability
The statistics half tests the three grouped-data formulae and their vocabulary; the probability half tests classical definition, complementary events, and simple counting. Both are scoring if the table is read correctly.
Core rules
- Mean of grouped data: direct Σfx/Σf; assumed-mean a + Σfd/Σf; step-deviation a + h(Σfu/Σf).
- Median = l + [(n/2 − cf)/f] × h, where l is the lower limit of the median class, cf the cumulative frequency below it, f its frequency, h the class size.
- Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h for the modal class.
- Empirical relation: 3 × median = mode + 2 × mean.
- Probability P(E) = number of favourable outcomes / total equally likely outcomes, with 0 ≤ P(E) ≤ 1 and P(E) + P(not E) = 1.
Worked example
Classes 0–10, 10–20, 20–30, 30–40 have frequencies 5, 8, 12, 5. Find the median. n = 30, so n/2 = 15. Cumulative frequencies are 5, 13, 25, 30, so the median class is 20–30 with cf = 13, f = 12, h = 10. Median = 20 + [(15 − 13)/12] × 10 = 20 + 1.67 = 21.67.
Traps
- Comparing n/2 with the cumulative frequency of the class itself instead of the class before it.
- Confusing f₀ (preceding class) with f₁ in the mode formula.
- Using n rather than n/2 when locating the median class.
- Reporting a probability above 1 or negative after an arithmetic slip.
- For “at least one” questions, forgetting that the complement is the faster route.