Mean Vs. Median

Categories: Fact Check, Finances
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Published on: 2017.02.12

Not everyone deals with math every day, and that’s fine. Most people don’t need to know the following in their day-to-day lives. However, I write this entry so I can link to it from other entries that mention “mean”, “average”, or “median”. Most of us see these terms only in news stories about the economy: “median home price in Seattle” or “average household income in South Carolina”. It’s obvious from the comments (and sometimes from the articles) that many don’t understand the difference.

“Mean” is what most people mean when they say “average”. You get the mean by adding up figures and then dividing by the number of figures. For example, add the prices of all the homes sold in your neighborhood over the past year, and then divide by the number of homes sold. The result is the mean.

“Median” is something quite different, though in many cases it will be the same as “mean”. The median is obtained by listing the values in ranked order and then selecting the middle value.

Here are a couple of concrete examples.

Think of a group of people of differing ages. Let’s say there are five people, aged 20, 23, 25, 26, and 31. The median age is 25 (the middle value), and the average age is 25 (125 divided by 5). So the median and mean are identical in this case.

Now imagine that two more friends show up. One is 21 years old, but the other is 60 years old. The median remains 25, because that’s still the middle of the seven values. But the average jumps up to 29.4. This is because the outlier age of 60 skews the average upward.

To get more extreme examples, we’ll have to pick something besides age — because everyone’s age will be between 0 and 120 or so.

Let’s take income for our second concrete example. Imagine 10 people, with incomes ranging from $20,000 per year to $110,000 per year.

$20,000
$30,000
$40,000
$50,000
$60,000
        mean/median = $65,000
$70,000
$80,000
$90,000
$100,000
$110,000

In this case, the mean is $65,000, and so is the median (when using an even number of values, the median is the average of the two middle figures — the ones in bold above).

Let’s add one more person, someone who’s looking for work; his income is currently $0. That drops the median to $60,000, but drops the mean a little lower: $59,091. Still pretty close to each other.

Then something improbable happens. Lawrence Ellison walks into the room — perhaps he got lost and is asking for directions. Employed by Oracle, Ellison’s pay is around $40 million per year. The median pops back to where it was: $65,000 — the middle of the 12 values. But the mean — the average of the 12 figures — skyrockets to $3.39 million. Now the list looks like this:

$0
$20,000
$30,000
$40,000
$50,000
$60,000
        mean = $65,000
        median = $3,387,5000
$70,000
$80,000
$90,000
$100,000
$110,000
$40,000,000

So if a survey taker for the Department of Labor took information from the 12 people in that room and then released her findings — “Average salary is $3.39 million!” — many readers of the resulting news story would assume that it was a room full of ultra-high income people. But if instead the headline looked like: “Median salary is $65,000”, readers would assume the room was full of middle-class workers.

Which number is more accurate? Mathematically, both are accurate. The mean, however, only tells you the average of the total amount of income. The median tells you how much the middle person gets paid. Neither tells you that only one person in the room is a multi-millionaire and another person in the room has no income at all.

When reading news stories about income, prices, etc., watch carefully for “median”, “mean”, and “average” (usually means “mean”), and remind yourself of what those figures could indicate.

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