A distribution describes how values are spread across possible outcomes. The normal distribution is the workhorse of statistics: most sample means converge to it. Other distributions (binomial, Poisson, geometric) model counting, rare events, and waiting times.
The normal distribution and Z-scores
The normal distribution is symmetric, bell-shaped, and fully described by its mean (μ) and standard deviation (σ). A Z-score standardizes any value: Z = (x - μ) / σ. A Z-score of 2 means the value is 2 standard deviations above the mean.
Scheme
; Z-scores: standardize values; Z = (x - mean) / sd
(define (z-score x mean sd)
(/ (- x mean) sd))
; SAT scores: mean=1060, sd=195
(define sat-mean 1060)
(define sat-sd 195)
(define score-1 1250)
(define score-2 900)
(display "SAT score 1250, Z = ")
(display (exact->inexact (z-score score-1 sat-mean sat-sd)))
(newline)
(display "SAT score 900, Z = ")
(display (exact->inexact (z-score score-2 sat-mean sat-sd)))
(newline)
; Reverse: what score is at Z = 2?
(define (from-z z mean sd) (+ mean (* z sd)))
(display "Score at Z=2: ")
(display (from-z 2 sat-mean sat-sd))
Python
# Z-scores in Pythondef z_score(x, mean, sd):
return (x - mean) / sd
sat_mean, sat_sd = 1060, 195print(f"SAT 1250, Z = {z_score(1250, sat_mean, sat_sd):.2f}")
print(f"SAT 900, Z = {z_score(900, sat_mean, sat_sd):.2f}")
print(f"Score at Z=2: {sat_mean + 2 * sat_sd}")
Binomial distribution
The binomial distribution counts successes in n independent trials, each with probability p. P(X = k) = C(n,k) * p^k * (1-p)^(n-k). Mean = np, standard deviation = sqrt(np(1-p)).
Scheme
; Binomial distribution: n trials, probability p; P(X = k) = C(n,k) * p^k * (1-p)^(n-k)
(define (factorial n)
(if (<= n 1) 1 (* n (factorial (- n 1)))))
(define (choose n k)
(/ (factorial n) (* (factorial k) (factorial (- n k)))))
(define (binomial-pmf n p k)
(* (choose n k)
(expt p k)
(expt (- 1 p) (- n k))))
; Example: 10 coin flips, P(exactly 7 heads)
(define n 10)
(define p 0.5)
(display "P(7 heads in 10 flips) = ")
(display (exact->inexact (binomial-pmf n p 7))) (newline)
; Distribution of successes
(display "Full distribution:") (newline)
(let loop ((k 0))
(when (<= k n)
(display " P(X=") (display k) (display ") = ")
(display (exact->inexact (binomial-pmf n p k)))
(newline)
(loop (+ k 1))))
Poisson and geometric
The Poisson distribution models rare events per unit time: P(X = k) = e^(-λ) * λ^k / k!. The geometric distribution models waiting time until first success: P(X = k) = (1-p)^(k-1) * p.
Scheme
; Poisson: events per unit time; P(X = k) = e^(-lambda) * lambda^k / k!
(define (factorial n)
(if (<= n 1) 1 (* n (factorial (- n 1)))))
(define e 2.718281828)
(define (poisson-pmf lam k)
(* (expt e (- lam))
(/ (expt lam k) (factorial k))))
; Average 3 emails per hour. P(exactly 5)?
(display "Poisson(lambda=3):") (newline)
(display " P(X=5) = ")
(display (poisson-pmf 35)) (newline)
(display " P(X=0) = ")
(display (poisson-pmf 30)) (newline)
; Geometric: trials until first success; P(X = k) = (1-p)^(k-1) * p
(define (geometric-pmf p k)
(* (expt (- 1 p) (- k 1)) p))
; P(first success on 4th trial), p=0.2
(newline)
(display "Geometric(p=0.2):") (newline)
(display " P(first success on trial 4) = ")
(display (geometric-pmf 0.24))
Neighbors
Related chapters
🎰 Grinstead Ch.5 — important distributions derived from first principles
🎰 Grinstead Ch.9 — the Central Limit Theorem, why means converge to normal
OpenIntro devotes significant space to normal probability tables and calculator use. We compute Z-scores and probabilities directly. The 68-95-99.7 rule (empirical rule) is the key heuristic for the normal distribution. For the Central Limit Theorem that justifies the normal's dominance, see Grinstead Ch. 9.
① Normal distribution