*38*

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 FormulaÂ

Confidence Interval = [0.5*X

^{2}_{2N, Î±/2},Â 0.5*X^{2}_{2(N+1), 1-Î±/2}]Â

where:

Â

- X
^{2}: 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*X
^{2}_{2N, Î±/2} - Lower bound = 0.5*X
^{2}_{2(15), .975} - Lower bound = 0.5*X
^{2}_{30, .975} - Lower bound = 0.5*16.791
- Lower bound =
**8.40**

**Note:** We used the Chi-Square Critical Value Calculator to compute X^{2}_{30, .975}.

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

The upper confidence interval bound is calculated as:

- Upper bound = 0.5*X
^{2}_{2(N+1), 1-Î±/2} - Upper bound = 0.5*X
^{2}_{2(15+1), .025} - Upper bound = 0.5*X
^{2}_{32, .025} - Upper bound = 0.5*49.48
- Upper bound =
**24.74**

**Note:** We used the Chi-Square Critical Value Calculator to compute X^{2}_{32, .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.