# Differences Between the Normal and Poisson Distributions

The normal distribution is so ubiquitous in statistics that those of us who use a lot of statistics tend to forget it’s not always so common in actual data.

And since the normal distribution is continuous, many people describe all numerical variables as continuous. I get it: I’m guilty of using those terms interchangeably, too, but they’re not exactly the same.

Numerical variables can be either continuous or discrete.

The difference? Continuous variables can take any number within a range. Discrete variables can only be whole numbers.

So 3.04873658 is a possible value of a continuous variable, but not discrete.

Count variables, as the name implies, are frequencies of some event or state. Number of arrests, fish in a trap, wetlands in a forest are all counts. They’re numerical and discrete, not continuous.

Not only are they discrete, they can’t be negative. You can have 0 or 4 fish in the trap, but not -8.

This point is extremely important for statistical modeling. Count variables have a lower bound at 0 but no upper bound.

A normal distribution, on the other hand, has no bounds. Theoretically, any value from -∞  to  ∞ is possible in a normal distribution.

Count variables tend to follow distributions like the Poisson or negative binomial, which can be derived as an extension of the Poisson. Both are discrete and bounded at 0.

Unlike a normal distribution, which is always symmetric, the basic shape of a Poisson distribution changes.

For example, a Poisson distribution with a low mean is highly skewed, with 0 as the mode. All the data are “pushed” up against 0, with a tail extending to the right. You can see an example in the upper left quadrant above.

But if the mean is larger, the distribution spreads out and becomes more symmetric. In fact, with a mean as high as 12, the distribution looks downright normal.

A Poisson distribution with a high enough mean approximates a normal distribution, even though technically, it is not.

One difference is that in the Poisson distribution the variance = the mean. In a normal distribution, these are two separate parameters. The value of one tells you nothing about the other.

So a Poisson distributed variable may look normal, but it won’t quite behave the same.

Can you treat it as normal?

In some cases, yes. You’ll still get reasonable parameter estimates and standard errors.

But don’t do it blindly. Check your assumptions. (You always do, right?)

If the distribution is too skewed or residual variance too heteroskedastic to assume normality, then no. Stick with a model that takes the true distribution into account.

Poisson and Negative Binomial Regression for Count Data
Learn when you need to use Poisson or Negative Binomial Regression in your analysis, how to interpret the results, and how they differ from similar models.

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1. Maria says

Right away: discrete doesn’t mean “a whole number”. Discrete means: takes particular values, changes abruptly as a function of the parameter set. The particular value that one associates with the steps in this change depends on the normalization and has nothing to do with the value being defined on a set of rational/natural numbers.

E.g.: a set {0.1, 2.0, 4.4} is a set of numbers that can correspond to a discrete variable, say, energy spectrum of an electron in an EM system. Why would one expect a by def discrete (!) variable like this to be defined on a set of whole numbers?? The same with, say, weight of humans in a data set, age… All those can be variables… It depends on your model.

2. David Eisenberg says

I am trying to find mathematical formulas for the Poisson and Normal statistics to compare them side by side. I suspect they are similar with simple to explain differences. I will keep looking.

With thanks.