Suppose that we have data giving the amount of rainfall in a city each day in a

Question

Suppose that we have data giving the amount of rainfall in a city each day in a certain year. We want useful, informative summaries of how rainy the city was that year. On the majority of days in that year, it did not rain at all in the city. Discuss and compare the following six summaries: the mean, median, and mode of the rainfall on a randomly chosen day from that year, and the mean, median, and mode of the rainfall on a randomly chosen rainy day from that year (where by “rainy day” we mean that it did rain that day in the city).

Explanation / Answer

If we observe across all 365 days , then the mode, the most common
occurence, would be zero. The median would also be zero since more than 50% of days were not
rainy. These are not very useful ways to describe how rainy a city is. The overall mean would be
somewhat useful to describe the total amount of rain across all days, and would likely be used to talk
about whether there were drought conditions compared to normal for that city, or it could be used
to compare overall rainfall totals to other cities. A better measure may actually just be to report the
percentage of days that any rain was received.
If we restrict our summaries to only days that it did rain, then we have more meaningful interpretations, especially of the mode and median. It would be expected that the distribution of the amount
of rain on rainy days would be right-skewed (most days it rains there is a little bit of rain, less than
an inch for sure, and there would be only a few days with torrential rain: 2 inches or more). The
mode would not be very useful since the measures would rarely be repeated. The median would be very useful here since it would represent the amount of rain on a ”typical” rainy day. In practice, the median rainfall during rainy days is often what is reported (or total rainfall across all days, which is just the overall mean times 365).

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