Rate per n values

Consider an event such as annual rainfall below 900 mm that has occurred in 20 out of the last 100 years in some village. The proportion, 20/100 = 0.2, can be expressed in a few equivalent ways:

This can be generalised to give a rate per n years, where n is usually some small value.

The rate per n years was n  × proportion

Note that:

Percentages

A percentage is simply the rate of occurrences per 100 values — i.e. the proportions times 100.

Annual rainfall in Samaru

The diagram below again shows the cumulative distribution function for the rainfall data in Samaru.

Initially the diagram shows the rate of years with rainfall 900 mm or less per 56 years. This is simply the number of years with that rainfall in the 56 years between 1928 and 1983.

Use the slider to increase n to 100 years. The calculations show the rate of years with rainfall 900 mm or less per 100 years — the percentage of years with rainfall below 900 mm in the data.

The numbers can be easier to interpret if they are expressed as the rate of years with rainfall 900 mm or less in n = 10, or 20 or 25 years. Use the slider to try these values.


Finally select larger values from the pop-up menu and observe that


Expected occurrences in n new values

The cumulative proportions and rates that we discussed earlier have all described aspects of the existing data. In the rainfall data from Samaru, they describe what happened in the years between 1928 and 1983.

If we can assume that rainfall in the future will have the same distribution (no climate change), a rate per n years can also be interpreted as the expected number of occurrences in n future years.

Average time with one occurrence (return period)

A third way to express proportions is the number of values in which there is an average of exactly one occurrence of the event of interest. This is the inverse of its probability. In many application areas of statistics, this is called the return period of the event.


Annual rainfall in Samaru

The diagram below again shows the cumulative distribution function for the Samaru rainfall data.

Drag the vertical line to the right and observe how the return period changes.

Why so many different ways to express a proportion?

When explaining the results of an analysis to others, it often helps to repeat the same information in different words. Many people find it difficult to understand what a proportion tells them about the data, and expressing the proportion in a few different ways can help.

Rainfall below 800 mm in Samaru

In the 56 years of rainfall data that were recorded in Samaru, the proportion of years with rainfall below 800mm was 0.054. The following are equivalent ways to explain this proportion to others.

Note that return periods are easier to interpret, when the period is long, so we would not usually use return periods to describe proportions less than 0.5.

These descriptions are factual summaries of what happened between 1928 and 1983. Assuming that there is no climate change and that future weather patterns and distributions will be as in the past, we could also state that: