For a Poisson distribution, the mean and variance both equal its parameter. If , then and , while the standard deviation is .
For a Poisson distribution, the mean and variance both equal its parameter. If , then and , while the standard deviation is .
A Poisson distribution models the number of independent events occurring in a fixed interval of time, distance, area, or volume. Events must occur at a constant average rate, and the parameter is the expected number of events in that exact interval.
| Quantity | Poisson result |
|---|---|
| Mean | |
| Variance | |
| Standard deviation |
The key mechanism is interval scaling. If the rate is events per unit and the interval has length , use before finding any summary statistic or probability. This matters because a rate per hour cannot be used unchanged for several hours.
For example, suppose calls arrive at a help desk at an average rate of calls per hour. Over hours, the parameter is
Therefore, if is the number of calls received during those hours,
The standard deviation is
These results mean that the expected number of calls is , the variance is , and the typical spread around the mean, measured by the standard deviation, is approximately calls.
A common misconception is that the standard deviation also equals . This is incorrect: the variance equals , while the standard deviation equals . Also, do not forget to rescale when the interval changes.
This is AHL 4.17 content for Mathematics: Applications and Interpretation HL. In an IB response, first scale the rate so that matches the interval, identify , and then state the requested quantity. Use the GDC for Poisson probabilities, but show the distribution and parameter clearly. Formal derivation of the mean and variance is not required, and final numerical answers should normally be exact or correct to three significant figures.