If the Variance Is Not the Mean, It Is Not Poisson

Capacity sized on the average is exceeded routinely

Five on average does not mean five most of the time. Exactly five occurs in about 17 % of intervals, and everything else happens in the other 83 — including none at all in one interval in a hundred and fifty, and ten or more in 3.2 %, which is roughly once every thirty-one intervals. A queue, a switchboard or a worker rota built for the average will therefore be over capacity several times an hour if the interval is a minute. Size against a chosen upper percentile instead, and state which one: designing for the ninety-fifth means accepting that one interval in twenty overflows, and that is a policy decision rather than an arithmetic one.

Constant rate is the assumption that fails first

Almost nothing arrives at a genuinely constant rate. Web traffic has a daily shape and a weekly one; support tickets spike after a release; road accidents cluster around weather. Fitting one rate across a period that contains several regimes produces a distribution wider than the Poisson it claims to be, which is exactly the excess variance described above and is often mistaken for clustering when it is really a moving average. The fix is to split the data into windows within which the rate is plausibly steady — hours of the day, days of the week — and fit each separately, which usually restores the equality of mean and variance inside each window.

Frequently asked questions

How is this different from the binomial distribution?

The binomial counts successes among a fixed number of trials, so it has a ceiling — twenty coin flips cannot give twenty-one heads. Poisson counts events in a window with no upper bound: nothing in the model forbids a hundred arrivals in a minute that averages five, it merely makes it astronomically unlikely. The two converge when the number of trials is large and the chance of each is small, which is why Poisson is the practical choice for rare events among many opportunities — a few defects in a long production run, a handful of clicks among many impressions.

Can the rate be a fraction?

Yes, and it usually is. The parameter is an average over a window, so half an event per hour is a perfectly ordinary description of something that happens roughly twelve times a day; the counts it predicts are still whole numbers. Changing the window rescales the parameter proportionally — a rate of five per minute is three hundred per hour — and the probabilities change accordingly, which is why the window has to be stated with the rate. A model fitted per minute and applied per hour is out by a factor of sixty in the parameter and by very much more in the tail.