The mean is useful when you are interested in the total amount of a quantity. For example, if you need to know whether water reserves are at an adequate level, the mean rainfall over the past five years may be more useful to you than the median rainfall. If there were an outlier value of an extremely high rainfall during just one of those years, that would affect your total, which is what you care about in this case, and the median would likely not reflect this information.

In studying the wealth or poverty of countries, the mean income could be useful if you were interested in total resources available, but generally we care more about what income range is typical or widespread, and the median is a better measure of this, as it indicates the fraction of citizens earning more or less than a certain amount rather than the amount of money in circulation. The mode might not be terribly useful because there could be very few citizens having the exact same salary (in the data set 18.1, 18.2, 18.05, 18.02, 105, 105 the mode is 105, but that is hardly a good measure of the central tendency of the data). However, if you cluster data points into income ranges, then, depending on how you choose those ranges, the mode of the income ranges could be informative.

When we did this cartoon in class, a couple students took the roles of the two characters and read the parts. One of my loudest students acted the angry girl, and thought reading a word in upper case a fine occasion for yelling “MODE!” so loudly that I wondered what anyone passing in the hall would be thinking. It was fun.