Statistics and Probability: Mean, Median, Mode and Classical Probability
The final domain covers the three measures of central tendency for grouped data - mean by direct, assumed-mean and step-deviation methods, median and mode from frequency tables - plus classical probability on coins, dice and playing cards.
Statistics
- Mean (direct): x̄ = Σfx/Σf. Step-deviation: x̄ = a + h(Σfu/Σf) with u = (x − a)/h.
- Median (grouped): 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 (grouped): mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, using the modal class and its neighbouring frequencies.
- Empirical relation: 3 Median = Mode + 2 Mean.
Probability
- P(E) = number of favourable outcomes / total number of equally likely outcomes, with 0 ≤ P(E) ≤ 1.
- P(not E) = 1 − P(E).
- Standard deck: 52 cards, 4 suits of 13, 26 red, 12 face cards (J, Q, K), 4 aces.
Worked example
One card is drawn from a well-shuffled deck of 52. Find the probability that it is a king or a heart. Kings: 4. Hearts: 13. The king of hearts is counted in both, so favourable outcomes = 4 + 13 − 1 = 16. P = 16/52 = 4/13 ≈ 0.31, which lies between 0 and 1 as required.
Traps that cost marks
- In the median formula, using the frequency of the median class where the cumulative frequency of the preceding class belongs.
- Double-counting overlapping events (king of hearts) instead of subtracting the intersection.
- Writing a probability greater than 1 or negative after an arithmetic slip - always sanity-check.
- Mixing class limits and class marks when choosing the median or modal class.