Well begun is half done

Recently as part of my research for the Cambridge Mathematics Framework I have been reading a lot about the median, and it has got me thinking about some of the things that I took for granted when teaching this statistical value. 

You may be familiar with this basic approach to teaching median: start with a list of data or a stem and leaf diagram for which there is an odd number of data points. Cross off the largest and smallest values until you reach the middle value – BOOM! This is the median. Many teachers may do this using concrete objects, or lining the students up in some order such as height, but the basic premise is the same – you are hunting for a number which is THE MEDIAN. Once students have a handle on locating this specific value, a curveball is introduced: “What happens if there is an even number of data?” 

At this point a conceptually tricky idea is thrown in: the number now being sought is the average (mean?) of the two central values, a number no longer in the dataset and perhaps reliant on the use of a different sort of average to find. We tend to gloss over the fact that the median is now in some sense a “virtual” thing, a value that does not exist in the data set, and may even be of the wrong data type, for example in the case of a discrete data set with a median falling between two consecutive values. 

Part of the issue here is of our conception of the median, in which we tend to focus on the precise value rather than its implication in terms of the overall distribution, i.e. the point in the data set at which exactly half of the data is either side. We focus on the position of the measure rather than the pattern in the data, using its “halfway-ness” as a (the) method of identification, rather than a fundamental feature. 

Why not approach median in a different way, from the point of view of distribution rather than centrality? Why not start with a data set with an even number of values and try to divide it into two halves? Can we describe the boundary using a single number? The fact that this number is now no longer in the data set is less problematic as it is simply a boundary point; the location of the line through which the data set can be cut exactly in half. It makes sense for this value to be equidistant from the two central data, but conceptually this is not important in context of the distribution itself. 

Of course, the median is useful both as a proportional indicator of distribution and as a central value that can be used to identify signals in a noisy process, but to have a full conception of median, one aspect ought not to be privileged over the other. Perhaps by introducing median through a proportional lens early on, we could begin to link early conceptions of sharing and dividing with the median as a boundary for dividing a dataset.