## In the Middle – Median

In the previous post, I reported on 7th graders exploring the notion of the middle between two numbers.  The method used by almost all seventh graders is one I dubbed the “two-handed approach.”  The same approach also works very well for finding the median of a set of numbers.  We read about medians all the time: median housing prices, median incomes, median scores.

If the red dots represent housing prices in a neighborhood, what is the median price?  Interestingly enough, we can figure this out without knowing what the scale is in the representation.  All we need to do is point at a particular red dot, a particular house, and say “this one’s price is the median price.”  And how do we find this median house?  We can use the two-handed approach, but this time hopping from dot to dot rather than from blue line to blue line.  We start with our left hand on the dot on the extreme left, and with our right hand on the dot on the extreme right, moving one dot at a time towards the center, in unison with both hands.  This center dot is the one where an equal number of dots are to the left as to the right.  It’s in the center in that particular meaning of the word.

If the data set is large enough, we can talk about median in terms of “half”: if your household income is exactly the median household  income in the country, then half of the households in the country make less, and half of the  households in the country make more.  Or, another way of looking at the same phenomenon, if I visited all households in the country, sorted from poorest to richest, by the time I was halfway done, I would be visiting the household with the median income.  (I better do all this traveling very fast, otherwise the median income will surely have changed during my trip.)

For smaller data sets, the notion of center can be a bit rough.  The one shown above can show us some of the effects through comparison with the one following:

The two data sets have an identical median.  This, even though in the new data set all the high values are really high, and all the low values are only moderately low.  For, the same data point is still the one with four values lower and four values higher.  Intuitively, it may feel that the center must have moved up compared to the earlier one.  Well, depending on the situation, the idea of what would make a good center may not be that clear-cut.

Another “gotcha” comes when we look at a data set with an even number of data points, as this one:

Note that one of the high-end data points is missing; and this time, as we do our two-handed travel from the end points towards the center, we end up with two red dots rather than one.  The standard way that this case is treated is by looking for the center between the two remaining red dots.  This is indicated in the picture by the black arrow.  If the data set represents housing prices, this time the median housing price is not the price of a particular house that we could point to – this time the median price is an in-between price.  Note that the median can still be found with our two-handed approach, but once we get to the center two dots, we must switch from hopping from dot to dot to hopping from blue delineation to blue delineation.  Simple enough, but in a sense very strange – probably the only place in the K-12 curriculum where we summarily shift from one kind of a unit (here, houses in a data set) to an entirely different kind of unit (here, dollars or thousands of dollars) without much fanfare.  At least it is only an artifact of the data set being small and discrete – if there are enough data points, you would expect the middle value, the value just below and the value just above to all be smack on top of each other so that it wouldn’t matter particularly which one you picked.  Off by half a house?  Not a big deal if there are enough of them to point to.

Unlike other things you learn in math class (like the mode of a data set), the median is widely useful in the world outside of school.  Interesting that this single notion of a two-handed approach for finding the center, which students understand so deeply, was so useful here.

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