A sensible characterization of mode, median, and mean

Often “types of data” are introduced all together, and then “measures of central tendency” are introduced all together. For “types of data” I mean nominal, ordinal, and numeric (leaving aside interval vs. ratio). For “measures of central tendency” I mean mode, median, and mean.

A common response to this exposition, even if median is justified with reference to skew, is that mode is a stupid thing and its inclusion in the list is almost insulting.

A much nicer exposition would introduce each type of data together with the “measure of central tendency” that is in some sense the best you can do for that type of data.

With nominal data the best you can do is frequency counts. Mode reports the most common thing. The mode of an election is (often) the winner. This is useful.

With ordinal data you can do better by putting everything in order. Now even if there are 8 A’s, 6 B’s, and 7 C’s, still B is more representative for its middle-ness.

Finally with numeric data you can take the mean, and you may want to or you may not want to, but people often do and they may well be right to.

Described this way, the types of data and the ways of measuring them have a pleasant pattern – an organized relationship that gives every part more meaning.

Intro statistics textbook authors, you may update your texts! (Do any already point this out? I don’t recall ever seeing it written.) (Also, I don’t particularly like the term “measure of central tendency” so I keep it in scare quotes throughout.)

5 thoughts on “A sensible characterization of mode, median, and mean”

1. Japheth Wood says:

Very nice! I haven’t thought of it this way, but this organizes “measure of central tendency” with data type in a very pleasing way.

2. Back to Basics | clioviz

3. A sensible characterization of mode, median, and mean | clioviz

4. Reading Bayesian Data Analysis has made me think about the usefulness of mode when dealing with probability distributions, where the peak (of a unimodal distribution, anyway) may not be the median or the mean of the distribution, but it is the mode. It feels natural to talk about the peak of the distribution, and thinking of it as the mode rather than the place where the derivative of the distribution function is equal to zero is very pleasant.