Repeat the experiment “flip N coins” M times

Each experiment produces a single number: the fraction of heads ν. The histogram collects the ν of all the experiments and compares it with the predicted Gaussian.

histogram of the experiments (density) Gaussian exact Binomial

Press “Flip” to start.

Flips per experiment, N

Experiments, M

Probability of heads, μ

Variable on the axis

Show the exact Binomial

Experiments done: 0

	from data	theory
mean of ν	–	–
std. dev. of ν	–	–
std. dev. of y	–	–

Try this. Keep M = 10,000 and increase N: in the frequency ν the histogram narrows like 1/√N towards a delta; in the zoom y = μ + √N(ν − μ) it keeps the width σ = √(μ(1−μ)) and approaches the Gaussian. Then keep N fixed and increase M: the histogram does not change width, it only gets less noisy and settles onto the Binomial.

With small N, or with μ close to 0 or 1, the exact Binomial looks discrete and skewed: there the Gaussian is only an approximation.