Statistics and Probability
Statistics and Probability in the Pancyprian Core Mathematics paper asks you to summarise a data set correctly and to reason with events rather than to recall formulas blindly. Two ideas dominate: how a linear change of variable moves the mean and standard deviation, and the distinction between mutually exclusive and independent events.
Summarising data
- Mean of raw data x̄ = Σx/n; for a frequency table x̄ = Σfx/Σf.
- Median and quartiles are read from cumulative frequency, never from the frequency column itself.
- Variance σ² = Σ(x − x̄)²/n for a population, or with n − 1 for a sample; standard deviation is its positive square root.
- Under the transformation y = ax + b: ȳ = ax̄ + b, but σ_y = |a|σ_x - adding a constant never changes spread.
Probability
- 0 ≤ P(A) ≤ 1 and P(A′) = 1 − P(A).
- Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
- Conditional probability: P(A | B) = P(A ∩ B)/P(B), provided P(B) > 0.
- Independence means P(A ∩ B) = P(A)P(B), equivalently P(A | B) = P(A).
- Mutually exclusive means P(A ∩ B) = 0; such events are dependent unless one of them has probability zero.
- On a tree diagram, multiply along a branch and add across branches.
Counting
- Permutations nPr = n!/(n − r)! when order matters; combinations nCr = n!/(r!(n − r)!) when it does not.
Worked example
Given P(A) = 0.5, P(B) = 0.4 and P(A ∪ B) = 0.7, find P(A ∩ B) and P(A | B), then test independence. From the addition rule, P(A ∩ B) = 0.5 + 0.4 − 0.7 = 0.2. Hence P(A | B) = 0.2/0.4 = 0.5. Since P(A)P(B) = 0.5 × 0.4 = 0.2 = P(A ∩ B), the events are independent.
Traps that cost marks
- Treating mutually exclusive and independent as synonyms; for events of positive probability they are near opposites.
- Adding P(A) + P(B) for overlapping events without subtracting the intersection.
- Reading a cumulative frequency as a frequency when locating the median.
- Dividing by n − 1 for a population variance, or forgetting the square root when asked for the standard deviation.