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Probability and statistics cheat sheet

All 13 formulas on one page. Print it or save it as a PDF; it fits on 3 sides of A4 or Letter.

Mean (average)

x̄ = (∑[i = 1..n] xᵢ)/n

$\bar{x}$ the mean of the data ( $μ$ is used for a whole population)
$x_{i}$ the data values $x_{1}, x_{2}, \dots, x_{n}$
$n$ the number of values
$∑$ "add up": the sum of all the values

Standard deviation

σ = √((∑(xᵢ − μ)²)/N)

Sample standard deviation

s = √((∑(xᵢ − x̄)²)/(n − 1))

Variance

σ² = (∑(xᵢ − μ)²)/N

$σ$ the population standard deviation: the data are the whole group you care about
$s$ the sample standard deviation: the data are a sample from a bigger population
$x_{i}$ the data values $x_{i}$
$μ$ the mean of the data ( $\bar{x}$ for a sample)
$N$ the number of values ( $n$ for a sample)

Z-score (standard score)

z = (x − μ)/σ

$z$ the z-score: positive above the mean, negative below
$x$ the value
$μ$ the mean of the distribution
$σ$ the standard deviation (greater than 0)

Normal distribution probability

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

Probability density

f(x) = 1/(σ√2π)e^( −((x − μ)²)/(2σ²))

Below b

P(X < b) = Φ((b − μ)/σ)

Above a

P(X > a) = 1 − Φ((a − μ)/σ)

$P$ the probability, between 0 and 1
$μ$ the mean
$σ$ the standard deviation (greater than 0)
$a$ the lower bound (leave it empty for "below b")
$b$ the upper bound (leave it empty for "above a")
$Φ$ the standard normal cumulative distribution: $Φ(z)$ is the probability of a value below z

Permutations

P(n, r) = n!/((n − r)!)

$P_{n, r}$ the number of ordered arrangements, also written $_nPᵣ$
$n$ the number of items to choose from
$r$ how many are chosen and placed in order (r ≤ n)
$n!$ n factorial: $n×(n − 1)×⋯×2×1$ , with $0! = 1$

Combinations

C(n, r) = n!/(r! (n − r)!)

$C$ the number of selections, "n choose r", also written $_nCᵣ$
$n$ the number of items to choose from
$r$ how many are chosen (r ≤ n)

Binomial distribution

P(X = k) = C(n, k)p^k(1 − p)^(n − k)

At most k (cumulative)

P(X ≤ k) = ∑[j = 0..k] C(n, j)p^j(1 − p)^(n − j)

Mean and standard deviation

μ = np, σ = √(np(1 − p))

$P (X = k)$ the probability of exactly k successes
$P (X \leq k)$ the cumulative probability of at most k successes, $P(X ≤ k)$
$n$ the number of trials
$k$ the number of successes (0 to n)
$p$ the probability of success on each trial

Addition rule (or)

P(A∪B) = P(A) + P(B) − P(A∩B)

Mutually exclusive events

P(A∪B) = P(A) + P(B)

$P (A \cup B)$ the probability of A or B or both, $P(A∪B)$
$P (A)$ the probability of event A
$P (B)$ the probability of event B
$P (A \cap B)$ the probability that both happen, $P(A∩B)$

Multiplication rule for independent events (and)

P(A∩B) = P(A)·P(B)

General (any events)

P(A∩B) = P(A)·P(B∣A)

$P (A \cap B)$ the probability that both A and B happen
$P (A)$ the probability of A
$P (B)$ the probability of B

Conditional probability

P(A∣B) = P(A∩B)/(P(B))

$P (A ∣ B)$ the probability of A given B, $P(A∣B)$
$P (A \cap B)$ the probability of both A and B
$P (B)$ the probability of B (must be greater than 0)

Bayes' theorem

P(A∣B) = (P(B∣A) P(A))/(P(B))

Total probability of B

P(B) = P(B∣A)P(A) + P(B∣Ā)P(Ā)

$P (A ∣ B)$ the updated (posterior) probability of A after seeing B
$P (A)$ the prior probability of A, before the evidence (for example how common a disease is)
$P (B ∣ A)$ the probability of the evidence when A is true (a test's sensitivity)
$P (B ∣ \bar{A})$ the probability of the evidence when A is false (a test's false-positive rate)
$\bar{A}$ "not A"

Expected value

E(X) = ∑xᵢ pᵢ

Variance

Var(X) = ∑pᵢ(xᵢ − E(X))²

$E (X)$ the expected value (mean) of X
$x_{i}$ the possible values $x_{i}$
$p_{i}$ the probability $p_{i}$ of each value; together they must add up to 1

Margin of error (proportion)

E = z^∗√((p̂(1 − p̂))/n)

Sample size needed

n = ((z^∗)²p̂(1 − p̂))/(E²)

Confidence interval

\hat{p} \pm E

$E$ the margin of error, as a proportion (0.03 means ±3 percentage points)
$z$ the critical value $z^∗$ : 1.645 for 90% confidence, 1.96 for 95%, 2.576 for 99%
$\hat{p}$ the sample proportion $\hat{p}$ (use 0.5 when planning, which gives the largest margin)
$n$ the sample size

letx.io · Probability and statistics cheat sheet · 2026-10-31