Counting Unequal Things
Suppose a school offered 20 classes with 10 students each and one class
with 200 students. What would be the average class size?
There are a total of 21 classes, and the total class attendance
(counting students multiple times if they attend more than one) is
10x20 + 1x200, totaling 400. So the average would be 400/21 or
about 19 students per class on the average. Now it could be that
there are 200 students, and each attends exactly two classes: one
of the size ten classes, and the size 200 class. Now if you asked
any of these students what her average class size was, she would say
105, since the average of 10 and 200 is 105. That's very
different from the school's average of 19, but the student's view is
more likely what we'd care about.
Why did the first average give such a misleading result? Because
we were counting things that were unequal. When we count the
total number of classes, the huge class only counts as one, as if it
were equal to a small class. When we count things, we are
expecting to get a number that will tell us some useful quantity.
If we count apples, the result is usually useful because it gives an
indication of how much food we have, since apples don't usually vary
much from the size we're used to.
On the other hand, if we count cakes in a bakery, it makes a big
difference if we're counting cupcakes or wedding cakes. It will
still make sense to count cupcakes as long as we know that's what they
are, but there's not much point in knowing the total number of cakes if
we don't know how many are small and how many are large.
This comes up fairly often. A politician might point out that
most of the donors to his campaign gave amounts less that $50.
That may be true, but if a thousand donors gave twenty dollars each and
two donors gave a million each, almost all of his money came from large
donors (twenty thousand compared to two million).
When I was young (in the 50's) my father was a Republican. I
recall him being pleased after reading that most of the newspaper
editors endorsed Republican candidates. It didn't occur to me
until a long while later that small town papers tended to support
Republicans and large city papers were more likely to support
Democrats. Since large city papers have a much larger
circulation, it may well have been that many more people actually read
newspapers that supported Democrats, so my father's optimism may have
been faulty. Again when counting newspapers, counting huge
circulation papers as if they were equal to small ones could be very
misleading.
