The Bell Curve from Randomness

Drop balls through rows of pegs. Every bounce is a coin flip, yet together the balls pile into a bell curve, no matter how random each one is.

balls per binnormal curve

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How it works

Binomial distribution and the Central Limit Theorem

In one sentence

The number of successes in $n$ independent trials with probability $p$ is binomial; for large $n$ it is approximately normal with mean $np$ and variance $np(1 - p)$ .

General formula

{\htmlData{sym=1}{P(k)}} = {\htmlData{sym=0}{\binom{n}{k}}}{\htmlData{sym=4}{p}}^{\htmlData{sym=3}{k}}(1-{\htmlData{sym=4}{p}})^{{\htmlData{sym=2}{n}}-{\htmlData{sym=3}{k}}} \;\approx\; \frac{1}{{\htmlData{sym=6}{\sigma}}\sqrt{2\pi}}\,e^{-\frac{({\htmlData{sym=3}{k}}-{\htmlData{sym=5}{\mu}})^2}{2{\htmlData{sym=6}{\sigma}}^2}}

Each ball's path is a sum of random left and right steps, and sums of many small random steps pile up into a bell.

What the symbols mean

$\binom{n}{k}$
Number of paths
How many different routes through the pegs end in bin k.
$P(k)$
Bin probability
The chance a ball lands in bin k.
$n$
Rows of pegs
How many times each ball bounces left or right.
$k$
Bin
How many of those bounces went right.
$p$
Chance of going right
The probability of bouncing right at each peg.
$\mu$
Mean
The center of the bell curve: np.
$\sigma$
Standard deviation
The width of the bell curve: √(np(1 − p)).

A ball's final bin counts how many times it bounced right: a sum of coin flips.
That count is binomial, with mean $\mu = np$ and variance $\sigma^2 = np(1-p)$ .
As the number of rows grows, sums of independent random steps approach a normal distribution. That is the Central Limit Theorem.

Learn the math

ProbabilityCalc II · Lesson 15, Applications of Integrals Applications of SeriesCalc II · Lesson 40, Power Series and Taylor Series Double Integrals in Polar CoordinatesCalc III · Lesson 33, Multiple Integrals The Binomial DistributionStats · Lesson 15, Discrete Random Variables The Central Limit Theorem for MeansStats · Lesson 23, The Central Limit Theorem The CLT for Sums and ProportionsStats · Lesson 24, The Central Limit Theorem Planning Sample Sizes and PowerEng Stats · Lesson 16, Intervals for Engineering Decisions X̄ and R Control ChartsEng Stats · Lesson 27, Statistical Process Control Acceptance Sampling and OC CurvesEng Stats · Lesson 30, Statistical Process Control

Learn the math

ProbabilityCalc II · Lesson 15, Applications of Integrals

Applications of SeriesCalc II · Lesson 40, Power Series and Taylor Series

Double Integrals in Polar CoordinatesCalc III · Lesson 33, Multiple Integrals

The Binomial DistributionStats · Lesson 15, Discrete Random Variables

The Central Limit Theorem for MeansStats · Lesson 23, The Central Limit Theorem

The CLT for Sums and ProportionsStats · Lesson 24, The Central Limit Theorem

Planning Sample Sizes and PowerEng Stats · Lesson 16, Intervals for Engineering Decisions

X̄ and R Control ChartsEng Stats · Lesson 27, Statistical Process Control

Acceptance Sampling and OC CurvesEng Stats · Lesson 30, Statistical Process Control