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# How to Calculate a Poisson Confidence Interval (Step-by-Step)

The Poisson distribution is a probability distribution that is used to model the probability that a certain number of events occur during a fixed time interval when the events are known to occur independently and with a constant mean rate.

While itâ€™s helpful to know the mean number of occurrences of some Poisson process, it can be even more helpful to have a confidence interval around the mean number of occurrences.

For example, suppose we collect data at a call center on a random day and find that the mean number of calls per hour is 15.

Since we only collected data on one single day, we canâ€™t be certain that the call center receives 15 calls per hour, on average, throughout the entire year.

However, we can use the following formula to calculate a confidence interval for the mean number of calls per hour:

Poisson Confidence Interval Formula

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Confidence Interval = [0.5*X22N, Î±/2,Â  0.5*X22(N+1), 1-Î±/2]

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where:

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• X2: Chi-Square Critical Value
• N: The number of observed events
• Î±: The significance level

The following step-by-step example illustrates how to calculate a 95% Poisson confidence interval in practice.

### Step 1: Count the Observed Events

Suppose we calculate the mean number of calls per hour at a call center to be 15. Thus,Â N = 15.

And since weâ€™re calculating a 95% confidence interval, weâ€™ll use Î± = .05 in the following calculations.

### Step 2: Find the Lower Confidence Interval Bound

The lower confidence interval bound is calculated as:

• Lower bound = 0.5*X22N, Î±/2
• Lower bound = 0.5*X22(15), .975
• Lower bound = 0.5*X230, .975
• Lower bound = 0.5*16.791
• Lower bound = 8.40

Note: We used the Chi-Square Critical Value Calculator to compute X230, .975.

### Step 3: Find the Upper Confidence Interval Bound

The upper confidence interval bound is calculated as:

• Upper bound = 0.5*X22(N+1), 1-Î±/2
• Upper bound = 0.5*X22(15+1), .025
• Upper bound = 0.5*X232, .025
• Upper bound = 0.5*49.48
• Upper bound = 24.74

Note: We used the Chi-Square Critical Value Calculator to compute X232, .025.

### Step 4: Find the Confidence Interval

Using the lower and upper bounds previously computed, our 95% Poisson confidence interval turns out to be:

• 95% C.I. = [8.40, 24.74]

This means we are 95% confident that the true mean number of calls per hour that the call center receives is between 8.40 calls and 24.74 calls.

### Bonus: Poisson Confidence Interval Calculator

Feel free to use this Poisson Confidence Interval Calculator to automatically compute a Poisson confidence interval.

For example, hereâ€™s how to use this calculator to find the Poisson confidence interval we just computed manually:

Notice that the results match the confidence interval that we computed manually.