Quick Revision

Statistics is the science of learning from data. Descriptive summarises what you have; inferential generalises from a sample to a population. A descriptive value for a population is a parameter; for a sample it is a statistic.

Quantitative splits into discrete (countable) and continuous (measured). Qualitative splits into nominal (unordered categories) and rank (ordered 1…n). Qualitative gets a bar graph with gaps; quantitative gets a histogram with touching bars.

Frequency f · relative f/n (sums to 1.00) · percent (f/n)×100 (sums to 100). Pie slice angle = relative frequency × 360°.

Ungrouped: μ = Σx/N, x̄ = Σx/n. Grouped: Σfx / Σf, with x the class midpoint. An inclusive series need not be converted for the mean — but must be for median and mode.

Divide by N for a population, n − 1 for a sample. It has nothing to do with how big the dataset is — only with whether it is everything you care about. Dividing by n − 1 makes s² an unbiased estimator of σ².

CV = (σ/x̄) × 100%. Use it whenever you must compare consistency across things with different means. Same mean? Comparing raw SDs is enough.

Mean < Median ≤ Mode → negative skew, long left tail of low values. In a satisfaction survey that is good news: most people are happy, a few are not. Mean > Median ≥ Mode → positive skew, long right tail.

K > 3 leptokurtic, tall and narrow → consistent. K < 3 platykurtic, flat and wide → inconsistent. K = 3 mesokurtic, the normal curve. Low SD + high kurtosis is the signature of a reliable service channel.

Chebyshev holds for any distribution: at least 75% inside μ ± 2σ, at least 89% inside μ ± 3σ. The empirical rule needs normality and buys precision: 68% / 95% / 99% at 1, 2 and 3σ.

Classical (equally likely outcomes, 1/n each) · relative frequency (from data: count ÷ total) · subjective (judgement).

A priori — before new information. Posterior — revised after it (what Bayes gives you). Marginal — unconditional, a single event. Joint — two events together.

"or" → addition: P(A∪B) = P(A) + P(B) − P(A∩B). "and" → multiplication: P(A∩B) = P(B)·P(A|B), or P(A)·P(B) if independent. Mutually exclusive ≠ independent — they are opposites.

Always 1 − P(none) = 1 − (1−p₁)(1−p₂)…(1−pₙ). Never add the individual probabilities.

Define events → list priors and conditionals → total probability P(A) → one branch ÷ P(A) → managerial insight. Watch the wording: "correctly classifies 90% of Low as Low" means the false-positive rate is 1 − 0.90; "incorrectly flags 20%" is the rate already.

Fixed n trials, two outcomes, constant p → binomial. An average rate per interval, no n → Poisson. A continuous measurement with μ and σ → normal. A binomial with large n → normal with the continuity correction.

P(X = r) = ⁿC_r pʳ qⁿ⁻ʳ. Mean np, variance npq, SD √(npq). Define "success" in words before touching the formula.

P(X = x) = e^(−λ)λˣ/x!. Mean and variance both equal λ; SD = √λ. P(X = 0) = e^(−λ). P(X ≥ 1) = 1 − e^(−λ). Rescale λ to the interval the question asks about.

z = (x − μ)/σ. Bell shaped, symmetric, mean = median = mode, total area 1. Draw the curve, shade the region, then decide: one z negative and one positive → add; same sign → subtract.

Valid when np ≥ 5 and nq ≥ 5. Set μ = np, σ = √(npq), then apply the continuity correction: a single value k becomes k ± 0.5; an interval widens outward by 0.5 at each end.

We want the parameter, but the population is too large to measure, so we estimate it with a statistic computed from a sample.

The probability distribution of the means of all possible samples of a given size from a population. Since x̄ varies from sample to sample, it is itself a random variable — with its own mean (μ_x̄ = μ) and its own SD (the standard error).

(1) μ_x̄ = μ always. (2) As n grows, the sampling distribution of x̄ approaches normality regardless of the population's shape. (3) Threshold n ≥ 30. This is what lets you put a z on a sample mean without knowing anything about the population.

σ_x̄ = σ/√n. It is the SD of the sampling distribution, not of the data. If the question is about a sample mean, this — not σ — goes in the denominator of z. That single confusion is the most expensive mistake available in this portion.

Multiply by √[(N−n)/(N−1)] when the sampling fraction n/N ≥ 0.05. If a question hands you N at all, use it and say why.

μ_p̂ = p, and σ_p̂ = √(pq/n). Check np ≥ 5 and nq ≥ 5 before using the normal approximation — write the check down, it is a free method mark. Then z = (p̂ − p)/σ_p̂.

Error ∝ 1/√n. Quadruple n to halve the error; multiply n by 100 to cut it by ten. Past a point the extra precision is not worth the cost — which is why a national poll uses a few thousand people.

Precision = how close the estimate is (small E). Confidence = how sure you are it holds (large z). Raising one narrows the other, so a trade-off is unavoidable. A 99% interval is wider — more certain, less useful.

Mean: n = (zσ/E)². Proportion: n = z²pq/E², with p = 0.5 when unknown (maximum variability, hence the safest n). Always round up. The three drivers are σ, E and z — not the population size.