• Skip to primary navigation
  • Skip to main content
  • Skip to primary sidebar
The Analysis Factor

The Analysis Factor

Statistical Consulting, Resources, and Statistics Workshops for Researchers

  • Home
  • Our Programs
    • Membership
    • Online Workshops
    • Free Webinars
    • Consulting Services
  • About
    • Our Team
    • Our Core Values
    • Our Privacy Policy
    • Employment
    • Collaborate with Us
  • Statistical Resources
  • Contact
  • Blog
  • Login

Differences Between the Normal and Poisson Distributions

by Karen Grace-Martin 4 Comments

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.

poisson

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.

Tagged With: continuous variable, discrete, negative binomial, normal distribution, normality, numeric variable, Poisson Regression

Related Posts

  • The Importance of Including an Exposure Variable in Count Models
  • Count Models: Understanding the Log Link Function
  • Count vs. Continuous Variables: Differences Under the Hood
  • The Problem with Linear Regression for Count Data

Reader Interactions

Comments

  1. Vindhya Singh says

    February 8, 2022 at 3:14 pm

    Thank you so much for this explanation! This sort of explanation was precisely what I was looking for!

    Reply
  2. Adam says

    April 10, 2019 at 4:56 pm

    Thanks for the helpful article. There’s a minor error though when you say that “discrete variables can only be whole numbers”. Technically speaking, a discrete variable is one in which its possible values are countable. For example, consider a variable X that can take any value in {0, 0.5, 1, 1.5, 2}. X is discrete, but not necessarily a whole number!

    Reply
  3. David Harris says

    December 23, 2016 at 2:32 pm

    I just wanted to thank you for your daily Linked-in comments. They are a helpful service to the community, even for the highly trained and experienced among us. Sometimes it is refreshing to think about the simple things that may have slipped your mind and which have unexpectedly great depth because the first time you heard them, you yourself did not have great depth of skill or knowledge and so they just passed as facts into the back of your brain.

    Reply
    • Jon Jon says

      June 2, 2018 at 2:02 pm

      Totally agree with David’s comments. Its a day after the conference in where this became in my mind a highlight.

      Reply

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Please note that, due to the large number of comments submitted, any questions on problems related to a personal study/project will not be answered. We suggest joining Statistically Speaking, where you have access to a private forum and more resources 24/7.

Primary Sidebar

This Month’s Statistically Speaking Live Training

  • Member Training: Analyzing Pre-Post Data

Upcoming Free Webinars

Poisson and Negative Binomial Regression Models for Count Data

Upcoming Workshops

  • Analyzing Count Data: Poisson, Negative Binomial, and Other Essential Models (Jul 2022)
  • Introduction to Generalized Linear Mixed Models (Jul 2022)

Copyright © 2008–2022 The Analysis Factor, LLC. All rights reserved.
877-272-8096   Contact Us

The Analysis Factor uses cookies to ensure that we give you the best experience of our website. If you continue we assume that you consent to receive cookies on all websites from The Analysis Factor.
Continue Privacy Policy
Privacy & Cookies Policy

Privacy Overview

This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.
Necessary
Always Enabled
Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.
Non-necessary
Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.
SAVE & ACCEPT