Mean = variance — reading the shape and spread
For Poisson both the mean and the variance are λ. Learn to separate absolute spread from relative spread (the coefficient of variation).
What changes when λ gets larger
As λ grows, the center of the bar chart moves right and the absolute spread also grows. But the spread relative to the mean (the coefficient of variation) actually gets smaller.
A comparison between λ = 1 and λ = 9. The center of the peak moves to the right and the absolute spread grows.
For Poisson, mean and variance are equal
A characteristic property of Poisson is that both the center of the distribution and the size of the spread can be read from the same λ.
E[X] = λ(the long-run mean count)Var(X) = λ(the size of the spread of the count)SD(X) = √λ(the square root of the variance is the standard deviation)
The fact that mean equals variance is also useful as a quick diagnostic in practice. From your data, compute the sample mean and the sample variance, and look at their ratio (variance ÷ mean, the dispersion index).
- variance / mean ≈ 1: a sign that Poisson is a reasonable fit.
- variance / mean ≫ 1 (overdispersion): for example, mean 3 but variance 10. Poisson alone cannot explain this, so consider the negative binomial distribution or a mixture model that captures clustering or unobserved heterogeneity.
- variance / mean ≪ 1 (underdispersion): a hint of capped counts or strong regularity. A signal to revisit the assumptions of the data-generating process.
Keeping "mean = variance" in mind lets you decide already from a histogram whether Poisson alone is good enough or whether additional structure is needed.
Check 1 — Mean, variance, standard deviation
Use the formulas directly to read off the mean, variance, and standard deviation.
Q1. What is the mean of a Poisson distribution with λ = 4?
For Poisson, E[X] = λ, so the mean is 4.
Q2. What is the variance for the same λ = 4?
For Poisson, the variance is also λ, not just the mean. Here the variance is 4 as well.
Q3. For λ = 9, what is the standard deviation σ = √λ?
σ = √9 = 3.
Absolute spread grows, relative spread (coefficient of variation) shrinks
A distribution with larger λ looks wider in the bar chart. Indeed, the standard deviation √λ grows together with λ.
But the mean grows at the same time, so in the sense of standard deviation / mean = 1 / √λ, the relative spread (the coefficient of variation) actually shrinks. From here on we call this quantity the "relative spread" or the "coefficient of variation." That is why the shape looks smoother as the counts get larger.
Check 2 — Absolute spread vs. relative spread (coefficient of variation)
As λ grows, the absolute spread grows too, but the relative spread (coefficient of variation) 1 / √λ shrinks.
Q1. Comparing λ = 1 with λ = 9, which bar chart has its center further to the right?
λ is itself the mean count, so a larger λ pushes the center further to the right.
Q2. Comparing λ = 2 with λ = 8, which has the smaller relative spread in the sense of standard deviation / mean = 1 / √λ (the coefficient of variation)?
The relative spread (coefficient of variation) is 1 / √λ, so it shrinks as λ grows.
Key takeaways from this chapter
- For Poisson, both the mean and the variance are
λ. - The standard deviation is
√λ, so the absolute spread grows with λ. - On the other hand,
standard deviation / mean = 1 / √λ, so the relative spread (coefficient of variation) shrinks. - If variance / mean drifts far from 1, suspect that Poisson alone is not enough.