N(0,1) · Normal Distribution
Independent sums, a bell shape, and the assumptions behind it.
Drop thousands of balls through a board of pegs — what shape piles up at the bottom?
When do sums of independent effects approach a bell shape?
Open the immersive version →Why does this question matter?
Heights, test scores, measurement errors, machined part dimensions — utterly unrelated things scatter in the same shape. Coincidence?
A ball on this 12-row board has an exact binomial B(12, 1/2) distribution. More balls stabilize its observed frequencies. Increasing the number of independent rows and standardizing the sum is the separate limit behind the normal approximation.
When a normal distribution is assumed, its mean μ and standard deviation σ determine it. They do not determine an arbitrary distribution. Statistical inference, like 'what score marks the top 2%?', starts here.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
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 the idea from another angle
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'.
Now read the definitions with their conditions
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)
For independent, identically distributed variables with finite mean and positive finite variance, the centered, standardized sum converges in distribution to N(0,1). A 12-row board remains discrete and binomial.
Make one more distinction
Bells only come from special randomness like coins and dice.
A standard central limit theorem assumes independent, identically distributed variables with finite mean and positive finite variance. Under these conditions the standardized sum converges to a standard normal distribution; an arbitrary sum need not do so.
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.
Where else can this idea help?
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.
Explain it in your own words
Try another question
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.
What question comes next?
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 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 →