The **Poisson distribution** and the **normal distribution** are two of the most commonly used probability distributions in statistics.

This tutorial provides a quick explanation of each distribution along with two key differences between the distributions.

**An Overview: The Poisson Distribution**

The **Poisson distribution** describes the probability of obtaining *k* successes during a given time interval.

If a random variable *X* follows a Poisson distribution, then the probability that *X* = *k* successes can be found by the following formula:

**P(X=k) = λ ^{k} * e^{– λ} / k!**

where:

**λ:**mean number of successes that occur during a specific interval**k:**number of successes**e:**a constant equal to approximately 2.71828

For example, suppose a particular hospital experiences an average of 2 births per hour. We can use the formula above to determine the probability of experiencing 3 births in a given hour:

**P(X=3) **= 2^{3} * e^{– 2} / 3! = **0.1805**

The probability of experiencing 3 births in a given hour is **0.1805**.

**An Overview: The Normal Distribution**

The **normal distribution** describes the probability that a random variable takes on a value within a given interval.

The probability density function of a normal distribution can be written as:

**P(X=x) = (1/σ√2π)e ^{-1/2((x-μ)/σ)2}**

where:

**σ:**Standard deviation of the distribution**μ:**Mean of the distribution**x:**Value for the random variable

For example, suppose the weight of a certain species of otters is normally distributed with μ = 40 pounds and σ = 5 pounds.

If we randomly select an otter from this population, we can use the following formula to find the probability that it weighs between 38 and 42 pounds:

P(38 < X < 42) = (1/σ√2π)e^{-1/2((42-40)/5)2} – (1/σ√2π)e^{-1/2((38-40)/5)2} = **0.3108**

The probability that the randomly selected otter weighs between 38 and 42 pounds is **0.3108**.

**Difference #1: Discrete vs. Continuous Data**

The first difference between the Poisson and normal distribution is the type of data that each probability distribution models.

A **Poisson distribution** is used when you’re working with **discrete data** that can only take on integer values equal to or greater than zero. Some examples include:

- Number of calls received per hour at a call center
- Number of customers per day at a restaurant
- Number of car accidents per month

In each scenario, the random variable can only take on a value of 0, 1, 2, 3, etc.

A **normal distribution** is used when you’re working with **continuous data** that can take on *any* value from negative infinity to positive infinity. Some examples include:

- Weight of a certain animal
- Height of a certain plant
- Marathon times of females
- Temperature in Celsius

In these scenarios, the random variables can take on *any* value like -11.3, 21.343435, 85, etc.

**Difference #2: Shape of the Distributions**

The second difference between the Poisson and normal distribution is the shape of the distributions.

A **normal distribution** will always exhibit a bell shape:

However, the shape of the Poisson distribution will vary based on the mean value of the distribution.

For example, a Poisson distribution with a small value for the mean like **μ = 3** will be highly right skewed:

However, a Poisson distribution with a larger value for the mean like **μ = 20** will exhibit a bell shape just like the normal distribution:

Notice that the lower bound for a Poisson distribution will always be zero no matter what the value for the mean is because a Poisson distribution can only be used with integer values that are equal to or greater than zero.

**Additional Resources**

The following tutorials provide additional information about the Poisson distribution:

An Introduction to the Poisson Distribution

The Four Assumptions of the Poisson Distribution

5 Real-Life Examples of the Poisson Distribution

The following tutorials provide additional information about the normal distribution:

An Introduction to the Normal Distribution

6 Real-Life Examples of the Normal Distribution

Normal Distribution Dataset Generator