Probability and statistics formulas

Statistics summarises data you have; probability describes data you might get. This section covers the averages and spreads you calculate from a list of numbers, the normal and binomial distributions behind most exam questions and opinion polls, the counting rules for arrangements and selections, and the rules for combining probabilities, up to Bayes' theorem.

Printable cheat sheet Formula finder

Mean (average)

x̄ = (∑[i = 1..n] xᵢ)/n

The arithmetic mean: add up the values and divide by how many there are.

Standard deviation

σ = √((∑(xᵢ − μ)²)/N)

How spread out data are: roughly the typical distance of a value from the mean. Shows both the population (σ) and sample (s) versions.

Z-score (standard score)

z = (x − μ)/σ

How many standard deviations a value is above or below the mean, which lets you compare values from different scales.

Normal distribution probability

P(a < X < b) = Φ((b − μ)/σ) − Φ((a − μ)/σ)

The probability that a normally distributed value falls below, above or between given values, from the mean and standard deviation.

Permutations

P(n, r) = n!/((n − r)!)

The number of ways to choose r items from n when the order matters, such as podium places or PIN-style codes without repeats.

Combinations

C(n, r) = n!/(r! (n − r)!)

The number of ways to choose r items from n when the order does not matter: hands of cards, committees, lottery tickets.

Binomial distribution

P(X = k) = C(n, k)p^k(1 − p)^(n − k)

The probability of exactly k successes in n independent trials that each succeed with probability p; also gives "at most k".

Addition rule (or)

P(A∪B) = P(A) + P(B) − P(A∩B)

The probability that A or B (or both) happens: add the two, then subtract the overlap so it isn't counted twice.

Multiplication rule for independent events (and)

P(A∩B) = P(A)·P(B)

When two events don't affect each other, the probability that both happen is the product of their probabilities.

Conditional probability

P(A∣B) = P(A∩B)/(P(B))

The probability of A once you know B has happened: the share of B's outcomes that are also in A.

Bayes' theorem

P(A∣B) = (P(B∣A) P(A))/(P(B))

Reverses a conditional probability: from how likely the evidence is if A is true, to how likely A is given the evidence.

Expected value

E(X) = ∑xᵢ pᵢ

The long-run average outcome of a random variable: each value weighted by its probability.

Margin of error (proportion)

E = z^∗√((p̂(1 − p̂))/n)

How far a poll's percentage could be from the true value at a given confidence level, and the sample size you need.