Poisson Distribution Explorer

Drag the average rate and watch the Poisson distribution change shape, from sharply skewed at low rates to nearly bell-shaped at high ones.

Distribution

Average rate λ = 4.0

Mean = variance = 4; standard deviation = 2.

Look up k = 4 Overlay the normal approximation

Poisson(λ = 4.0)

P(X = 4)

0.1954

P(X ≤ 4)

0.6288

P(X ≥ 4)

0.5665

Drag λ upward and turn on the normal overlay: once λ is roughly 10 or more, a normal curve with mean λ and standard deviation √λ tracks the bars closely, an instance of the central limit theorem at work. At small λ the distribution stays visibly right-skewed, since it can never go below zero.

How to use this tool

Drag the lambda slider to set the average number of events per interval.
Drag the k slider to shade outcomes up to k and read the three probabilities below the chart.
Turn on the normal-approximation overlay to see how closely a normal curve with mean and variance both equal to lambda tracks the bars.
Compare a small lambda (skewed) with a large one (closer to bell-shaped).

Formula Reference

Poisson probability

P(X = k) = e^-lambda * lambda^k / k!

Mean and variance

E[X] = Var(X) = lambda

Normal approximation

X is approximately Normal(lambda, sqrt(lambda)) for large lambda

ArticleThe Poisson Distribution ArticleThe Central Limit Theorem, Explained With Worked Examples CalcPoisson Distribution Calculator ToolBinomial Distribution Explorer GlossaryGlossary: Poisson Distribution

Frequently Asked Questions

Why does the distribution look skewed for small lambda?+

A Poisson variable can never be negative, so when the average rate is low, the bulk of the probability is squeezed against zero, leaving a long right tail.

How large does lambda need to be for the normal approximation to work?+

A common rule of thumb is lambda of about 10 or more. Turn on the overlay and drag lambda upward to see the dashed normal curve fit increasingly well.

How is this different from the Poisson Distribution Calculator?+

The calculator gives exact numeric probabilities for one chosen lambda and k. This tool is for building intuition about how the whole shape responds as lambda changes.

Why are the mean and variance always equal for a Poisson distribution?+

It is a defining property of the distribution, derived directly from its probability formula. If real data have a variance much larger than the mean, the counts are overdispersed and a Poisson model may not fit well.

How to use this tool

Drag the lambda slider to set the average number of events per interval.

Drag the k slider to shade outcomes up to k and read the three probabilities below the chart.

Turn on the normal-approximation overlay to see how closely a normal curve with mean and variance both equal to lambda tracks the bars.

Compare a small lambda (skewed) with a large one (closer to bell-shaped).