Lying With Statistics – Stay Off The Median?

There are two types of numbers.  There are numbers that are near the middle and there are numbers that are on the ends.  Being a person with a last name that starts with the letter ‘L’, I can identify with the numbers in the middle.  Whenever we lined up in alphabetical order in school, that is where I was.  Today, we are going to learn the value of that.

Median is another fancy name for average.  Usually you hear it on the new when people are talking about median income, median house price, or median household net worth.  And that is because, as we will see shortly, that gives a different result than the mean, which we discussed yesterday.

The Median is just the middle number in any group.  In the group 3,4,5,11,12, the Median is 5.  There are two values below it and two above.  The Mean, incidentally, is 7.  The Median is handy to use in groups of numbers that have extreme outliers, like incomes, home prices, or net worth.  For example, let’s say in your neighborhood, people have a median income of $51,500, their homes are worth a median price of $189,000, and the Median household net worth is $70,000.  These numbers are important because, among other uses, they are used to calculate taxes.

Then a billionaire builds a multi-million dollar house in your neighborhood.   How much does the Median price move?  Very little, if at all.  All that happened is that there is one more number at the top of the list.

But how would this affect the Mean?  Let’s look at some concrete examples.  For the state of Maine, in 2011, the top 1% of income tax filers had an income of over $800,000.   As you can see on this site, that makes the Mean income, $61,081, substantially higher than the Median, $47,849.  If your whole town is valued that way, that is going to make a big difference in tax calculations.  That is going to be the same with home prices and net worth.  Do you think all of those big, fancy houses on the coast are going to make the Mean higher or lower?  (Hint: higher)

Let’s Check In With Yesterday’s Examples

Let’s look at our examples and see where the Median makes sense and where we can use it to get some false conclusions:

  1. A ‘C’ in class is average.  The median grade still does not help a lot.  Ruling: TBA
  2. The average balance for the deposits at your bank is $5000.  Using the population of  10 customers with a balance of $40,000 and 90 customers with a balance of about $1100, the Median is going to be $1100.  Because that is how much the 50th customer in order of balance has.  If the balances follow a power-law distribution, the Median will still be below the Mean.  If you are running this branch, this is not going to look good.  Ruling: Bogus
  3. The average height of a woman in the US is about 5’4″.  Remember from yesterday that this group fits a normal distribution (or bell-curve) very nicely.  The Median will give you about the same result as the Mean.  Ruling: Winner!  But use the Mean because it is easier to explain.
  4. The average patient stay at a given hospital is 2.5 days. This one is similar to the average deposit balance.  The vast majority of your patients can be staying 2 days or less, but one 90 patient day does not wrecks all of your numbers.  Ruling: Winner! But you still have to note the outliers to be honest.

The Median is very useful in populations that do not follow a Bell Curve.  But don’t go crazy with it.  It is another tool that can be used for good or evil.  In many cases, a little more evil than the Mean because the Median is not as well understood.

The image on this post is from http://imgarcade.com/median.html.

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