Heights, test scores, measurement errors — why do they all trace the same bell?
Drop thousands of balls through a board of pegs — what shape piles up at the bottom?
Stack enough small accidents and you always get the bell.
Experiment
Hands-on experiment
Predict first — drop thousands of balls through 12 rows of pegs. What shape piles up at the bottom?
Drop the balls (each row: 50/50 left or right)
Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
Heights, test scores, measurement errors, machined part dimensions — utterly unrelated things scatter in the same shape. Coincidence?
Not coincidence — a theorem. When an outcome is built as the sum of many small independent factors, the sum's distribution converges to a bell regardless of the ingredients: the central limit theorem.
Thanks to it, just two numbers — mean μ and standard deviation σ — reconstruct the whole distribution. Statistical inference, like 'what score marks the top 2%?', starts here.
Insight
Insights from the video
“The normal distribution is order hiding inside the world of chance.”
Each ball's path is fully random, yet thousands piled together draw the same curve every time. Disorderly individuals, orderly crowds — the founding sentiment of statistics.
“μ is the bell's position, σ its width — two numbers are everything.”
A normal distribution is completely determined by two parameters. That's why 'knowing the mean and SD' equals 'knowing the whole distribution'.
Misconception
Common misconceptions
Bells only come from special randomness like coins and dice.
Whatever the ingredients, a sum of small independent accidents converges to a bell (the central limit theorem). Heights, errors, and averages are all bell-shaped not because of their ingredients but because they are sums.
In a normal distribution any value can occur, so prediction is impossible.
Individual values are unknowable, but the proportions are exact — 68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ. This rule is the skeleton of quality control and statistical inference.
Formula
Writing it as math
The bell you witnessed on the Galton board, written as formulas:
The normal distribution
Completely determined by mean μ and variance σ². The cockpit's two sliders were these two parameters.
The 68-95-99.7 rule
The fractions inside ±1σ, ±2σ, ±3σ are identical for every normal distribution. 'Area under the curve = probability' (integration!) is doing the work.
The central limit theorem (intuition edition)
A sum of independent small accidents converges to normal whatever the ingredients. The Galton board's 12 left-right bounces were exactly this.
In Real Life
Where you meet it in real life
Grading on a curve
'Top 4% gets the highest grade' is computed in σ under a normality assumption. The reason standardized scores exist.
Factory quality control
A part dimension escaping ±3σ has probability 0.3% — control limits are drawn in σ.
Measurement error
Repeated measurements scatter bell-shaped around the true value — the basis for averaging to reduce error.
Finance's (limited) assumption
Many models assume normal returns, but real markets have fatter tails — knowing an assumption's limits is literacy too.
Try Yourself
Test yourself
Q1Heights: mean 170cm, σ 7cm. Between which heights do 95% of people fall?Show answer ▾
The ±2σ rule: 170 ± 14, so roughly 156–184cm. Two numbers pin down 95% of a population.
Q2A test score with z = +3 is roughly what percentile?Show answer ▾
By the 99.7% rule, ±3σ leaves 0.3% outside; the upper side alone is ~0.15% — a 1-or-2-in-a-thousand extreme.
Q3One die is uniform. Why is the sum of 100 dice bell-shaped?Show answer ▾
The central limit theorem — regardless of the ingredient (uniform), a sum of independent accidents converges to the bell. The Galton board's balls were exactly this experiment.
Try answering yourself before revealing it — getting it wrong is where learning starts.
Connection
Concepts connect
Previous concept
Variance & Standard Deviation
With μ and σ in hand, it's time to meet the distribution those two numbers fully determine.
← Variance & Standard Deviation labLeads to next
Sampling & Estimation
The normal distribution's regularity enables the magic of 'estimating 50 million from 1,000' — the mathematics of polling is next.
Go to the Sampling & Estimation lab →Related
Labs worth exploring together
Related lab
Integrals
'Area under the curve = probability' requires integration — the source of the number 68%.
Go to the Integrals lab →Related lab
Variance & Standard Deviation
σ sets the bell's width — master measuring wobble first.
Go to the Variance & Standard Deviation lab →