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 it’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 can’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

Confidence Interval = [0.5*X

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

where:

- X
^{2}: Chi-Square Critical Value- N: The number of observed events
- α: 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, **N = 15**.

And since we’re calculating a 95% confidence interval, we’ll use **α = .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, α/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-α/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, here’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.

Great job, but one thing is unclear to me.

For lower bound, α/2 is replaced by .975. However, α/2 is not .975 if α=.05. Should it be 1-(α/2) for lower bound and α/2 for upper bound?

Hello, Thank you for the nice explanation. Can you please explain the alpha values used in the formula vs. the example? In the lower bound formula it is mentioned X2, at alpha/2 but in the example it is used as 1-alpha/2 (1-0.05/2=0.975). In the upper bound it is used the reverse, the formula says X2 at (1-alpha/2) but calculation shows (alpha/2=0.05/2=0.025).

Please clarify.

Thank you.