Statistics and probability
What this domain covers
This domain covers summarising data (mean, median, quartiles, interquartile range, standard deviation, histograms with unequal class widths), counting with permutations and combinations, probability rules including conditional probability and independence, discrete random variables with expectation and variance, and the normal distribution.
Core rules
- Histogram: frequency density = frequency ÷ class width, so the area of a bar represents frequency.
- Counting: nPr = n!/(n − r)! when order matters; nCr = n!/(r!(n − r)!) when it does not.
- Probability: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); P(A|B) = P(A ∩ B)/P(B); A and B are independent when P(A ∩ B) = P(A)P(B).
- 'At least one' is usually quickest as 1 − P(none).
- Discrete random variable: E(X) = ΣxP(X = x); Var(X) = E(X²) − [E(X)]²; the probabilities sum to 1.
- Normal: standardise with z = (x − μ)/σ, and use symmetry, since P(X > μ + k) = P(X < μ − k).
Worked example
X is normally distributed with mean 20 and standard deviation 3. Find P(X > 24). Standardise: z = (24 − 20)/3 = 1.333. From tables Φ(1.33) = 0.9082, so P(X > 24) = 1 − 0.9082 = 0.0918, roughly 9.2%.
Traps
- Plotting frequency instead of frequency density on a histogram with unequal widths.
- Using nPr where order is irrelevant, or double counting arrangements.
- Substituting the variance σ² instead of the standard deviation σ when standardising.
- Forgetting to square E(X) in Var(X) = E(X²) − [E(X)]².
- Assuming independent events are mutually exclusive, so P(A ∩ B) is set to zero.