3.2 Median

The three measures of central tendency are all measures that tell us where typical cases fall or where cases tend to cluster. After exploring the mode in the previous section, in this section we turn to the second measure of central tendency called the median.

The median lives up to its name: it derives from the Latin root medi, meaning “middle”, and that’s exactly the type of information it provides. Specifically, the median divides the cases of a variable into two equal halves and identifies the case in the middle. As such, it points out the “centre” of the data in a very straightforward way — it simply reports the middle observation.

Consider, however, the following point: even in everyday life, the middle implies a beginning and an end (e.g., “in the middle of the book”); something that is in-between, a gradation from a point A to a point C, as it were. From clothes sizes (“small, medium, large”) to how spicy you like your Thai food (“a little, medium, or hot”), through turning the volume up or down while listening to music (“low, medium, high”), the “centre” category bisects whatever it is applied to into a smaller/larger, less/more, left/right, etc. parts. That is, to speak of the middle of something we need to know where it starts (e.g., the minimum) and where it ends (e.g., the maximum). Simply put, we need an order.

What all this should tell you is that the median is not applicable to nominal variables. Speaking of the middle of gender, or the middle of ethnicity, or religious affiliation, or hair colour, or degree major, or of the middle of any other nominal variable makes no sense. After all, the order the categories of a nominal variable appear is either arbitrary or a matter of preference; nothing precludes rearranging the categories in some other way so that a case belonging to one category that ends up in the middle of one arrangement would not necessarily be in the middle of another arrangement. A case belonging to any category can easily end up being the middle one. A statistic shouldn’t depend on such a chance/preference; as such nominal variables have no median.

On the other hand, as you know by now, ordinal and interval/ratio variables do have an inherent order arranging their categories/values. They have a “beginning” and an “end”, and therefore a “centre”. As such, the median applies (only) to ordinal and interval/ratio variables.

Note that while the mode applies to a category (reflecting the largest number of cases), the median is determined by the case (observation) that falls in the middle of the category-ordered listing of all cases. Thus it’s not the middle category that is the median; depending on the size of the categories, the median case can belong to any category/value. The median category/value is the one to which the middle case belongs. Presented this way, the explanation sounds undoubtedly as clear as mud but do not despair.  It will get better when we establish the manner in which we obtain the median, so trust me and read on.

We could continue the same way as in Example 3.2 (A) above for larger groups too: we could arrange the cases in order of their categories/values, find the middle case (or two middle cases) and report its category/value as the median. However, you can guess that this would quickly become impractical the larger the group size gets. We need some other way of finding the median, one that generalizes across groups of any size.

Consider the following formula:

  “numbered position of the median case in the ordered list of cases”

where, as usual, N is the group size.

Instead of counting, let’s apply this formula to Example 3.2 (A).

It is easy to see that we could substitute a group of any size for the N in the formula. Even when working with hundreds or thousands of cases, we can always use the formula to find the place (or which case number) bisects the variable’s distribution in two haves.

So far I only used an ordinal variable to illustrate the median. How does finding the median work for interval/ratio variables? Would it matter that interval/ratio variables have numerical values rather than qualitative categories? No, not in the least. After all, finding the median doesn’t depend on the category or value of any case in any substantive sense — only on its numbered position in the ordered list of categories/values.

There is something a bit different in the way interval/ratio variables look, however, that some people seem to find a tad more confusing when working with values rather than categories. To illustrate, I’ll give you another example.

What might be confusing for some people is differentiating between the numbered positions of the cases on the list and their values since both are expressed numerically. In this example I have tried to make it easier to distinguish by putting the numbered positions of the cases in brackets and the values next to them (just like the categories in the ordinal example above). Thus you can see that Case #1 has 0 siblings, Case #2 has 1 sibling, etc. Had I chosen different set of values — for example, if Case #1 had 1 sibling, Case #2 had 2 siblings, Case #3 had 3 siblings, etc. — you might have found it a bit harder. For that reason, make a mental note to keep a clear track of what is a case’s value and what is its numbered position.