# Poisson vs. Normal Distribution: What’s the Difference?

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) = λk * 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) = 23 * 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√)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√)e-1/2((42-40)/5)2 – (1/σ√)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.