# Here’s the difference between averages and medians

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Former Microsoft Chairman Bill Gates plays bridge during the Berkshire Hathaway annual meeting weekend in Omaha, Nebraska May 1, 2011. Gates is on the Berkshire Hathaway board of directors. REUTERS/Rick Wilking
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REUTERS/Rick Wilking

There are a couple different ways to mathematically think about the average or typical member of a group, and they have advantages and disadvantages in different situations.

As you likely recall from math class, the mean or average of a set of numbers is calculated by adding up all the numbers in the set, and then dividing by the count of how many numbers are in the set. So, to find the average of 1,2,3, and 4, we add those up: 1+2+3+4 = 10, and then divide by 4: 10/4 = 2.5.

Averages are great! They are frequently a good way to get an idea of what the middle or typical member of a set looks like. They are used in all kinds of statistical analysis. They are a key part of thinking about what happens in probabilistic situations in the long term.

Looking at something like the average household income in a large metropolitan area also has a handy interpretation: If we were to take the total income produced in that metro area and evenly distribute it out in equal amounts to every household, that amount would be equal to the average.

But, if we’re trying to get a sense of what’s going on with a typical member of a group, averages can sometimes be misleading.

Problems emerge when looking at highly skewed populations, in which a large part of the group is fairly close to each other, but there are some distant outliers. Consider a room with nine fairly normal people with \$10,000 in net worth each, and Bill Gates, with a net worth of about \$78 billion, according to Forbes.

The average net worth of the ten people in our room is \$7,800,009,000 – just over \$7.8 billion. The contribution of nine of our room’s ten denizens amounts to a rounding error. Further, saying that the average net worth of everyone in this room is in the ten figures tells us very little about the majority of our group, and certainly doesn’t seem like a “typical” net worth in the room.

Enter the median. The median tries to get the middle value of the group: the value so that the same number of elements of our set are bigger than the median as are smaller. One way to calculate the median is to begin by lining values up from smallest to largest. Then, if you have an odd number of elements, the median will be the number in the middle of your ordered set. If you have an even number of values, take the average of the two middlemost.

For our nine average Joes plus Bill Gates, we have a median of \$10,000, a much better description of a “typical” person in the room.

The tendency for averages to be distorted by outliers shows up in lots of places, though usually in a less extreme way than in our regular people vs. Bill Gates example. For example, according to the Census Bureau’s recently released 2014 American Community Survey estimates, the median household income in the United States was \$53,657 while the average was \$75,591.

That difference largely comes from the fact that a relatively small number of very high income households are pulling up the average income in a similar way Bill Gates pulled average wealth up in our example. When considering the experiences of a typical American household, or making comparisons between different groups or geographic areas, then, it’s usually ideal to look at medians for these kinds of variables rather than averages.