Distributions

OpenIntro Statistics · Ch. 4 · openintro.org/book/os

Start with an experiment: flip coins and watch a distribution grow.

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 (* 1.0 (z-score score-1 sat-mean sat-sd)))
(newline)

(display "SAT score 900, Z = ")
(display (* 1.0 (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))

A round of coins produces one count. Repeat rounds to see which counts are common. Read the guided introduction →

Make a distributionOne round, one entry

Coins per round

Each coin is independent. Start with a 50% chance of heads.

Start with one round. Its head count will add one entry to the chart.

Now repeat: rounds to run

0 rounds recorded. Bar height shows the percentage of rounds, so you can compare runs of different lengths.

Current experiment · 10 coins per round · 0 rounds

Read current chart as a table

Current experiment: every possible head count
Heads	Proportion	Rounds
0	0%	0
1	10%	0
2	20%	0
3	30%	0
4	40%	0
5	50%	0
6	60%	0
7	70%	0
8	80%	0
9	90%	0
10	100%	0

Another thing to try: change the coin

So far, heads and tails have had equal chances. What shape would you expect if heads were more likely?

Chance of heads

Changing the number of coins or their chance of heads clears the current experiment. A saved comparison stays labeled with its own settings.

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 (* 1.0 (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 (* 1.0 (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 3 5)) (newline)
(display "  P(X=0) = ")
(display (poisson-pmf 3 0)) (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 0.2 4))

Neighbors

Related chapters

🎰 Grinstead Ch.5 — important distributions derived from first principles
🎰 Grinstead Ch.9 — the Central Limit Theorem, why means converge to normal

Foundations (Wikipedia)

Translation notes

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.

Want the full treatment? Read OpenIntro Statistics, Ch. 4.

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