Median vs. Mean: Know the Difference or Risk Being Manipulated

The definition of median and mean and the difference between the two terms is poorly understood by many people and while in many data sets, there is only a marginal difference between the two from a practical standpoint, there is sometimes a stark difference between the two.  As I’ll outline in this article, people toss around terms like “average”, mean, median and other statistical nomenclature without either understanding what they’re talking about themselves, or with the intent to silence critics with ‘fact’. Understand that various entities routinely manipulate how a piece of data is presented in order to beguile you, the recipient, in order to advance their agenda.

Median Mean – What’s the Difference?

By understanding the difference between median and mean, and what the implications are of each, you will be able to challenge such assertions and avoid being duped yourself.  First, a brief explanation of each, then some real life examples of median and mean that we can all relate to.

Let’s take this set of 11 numbers:

2    3    3    4    7    9    11    12    14    17    25

The median and the mean are pretty close together, but different nonetheless.  See the attached Excel-generated output:

Definition of Arithmetic Mean: Simply add N numbers in a data set together and then divide by N.  Most people equate the arithmetic mean to their understanding of the term “average”.  The mean of 5 and 10 is 7.5.  Note that excel has a median and mean function built in, whereby you can select a series of cells and have excel calculate the result for you.  Contrary to my warning about relying on “Average” to explicitly = “Mean”, in Excel, you need to use the “AVERAGE” FUNCTION in order to calculate an arithmetic mean (I know, confusing, but that’s the truth).  Also note that there are other types of means like Harmonic Mean and Geometric Mean, but they are generally reserved for more targeted, advanced scenarios.

Definition of Median: The median of a data set is the middle number when the set is sorted in numerical order.  Since there was an odd number of data points above, going to the center to find 9 was easy.  In an even-numbered data set, you take the mean of the middle two numbers.

Definition of Mode: The mode of a data set is the number that appears most frequently.  In the data set above, it would be 3.  Because mode has little utility in most everyday applications, I didn’t focus a great deal on mode.  I can honestly say that I’ve never heard a politician proclaim that the mode income is such and such.

“Average” does not always equate to “Mean”: Mean is the most oft-cited average, but that does not necessarily mean that the word average is equivalent to mean.  There are various types of averages: mean, median and mode being the most common.  So, if someone says, the average player bats .286, they probably mean that .286 is the arithmetic mean, but technically that is not necessarily true unless they specify further which average they are referring to.  Because of this, many people will use the word “average” knowing full-well that their audience will associate it with arithmetic mean, when in fact, they may be referring to a median or mode and as such, the audience falls prey to statistical chicanery.

A Simple Median vs. Mean Example:

Here’s one I like to use when describing the difference in practical terms.

• You live in a town of 1000 residents who are all earning roughly $80,000 per year.
• To keep it simple, the median and the mean are roughly $80,000 give or take.
• Your town is real nice and reminds Warren Buffet of his childhood.  He decides to move in.
• He takes in $1Billion this year.

While the median income for the town remains at $80,000 (because the middle number is still $80,000 as are the other 999, there is only one outlier making a billion), the arithmetic mean income for the town is 1.08 Million Dollars!

So, Money Magazine (or whatever your favorite finance mag is) runs one of their usual stories about “the country’s richest towns”, etc. and utilizes data mining software to crunch some numbers which reveals that the mean or what they consider “average” income is over $1Million per year.  They conduct no actual investigative research before running with it.  They dub your town, “The Town of Millionaires” and runs this amazing story about how all these millionaires live in typical looking homes. They’re a real frugal bunch, look how much they make compared to the lifestyle they lead!  What an example for America!

If only they looked at the data and considered the median and the mean and tried to understand the difference.

Who Uses Median vs. Mean to Manipulate?

For starters, politicians do all the time.  So do college professors and so does someone in your company that’s looking to advance their agenda. Ever have a debate at a dinner party and someone starts spewing out all kinds of statistics?  Ever listen to the pundits on TV who are not actual experts in the area they’re speaking to by any means, but they’ve rehearsed the same line over and over again, so when the topic comes up on air, they robotically spew out some statistic that sounds real compelling?

Chances are they have no idea what they’re talking about.

They were spoonfed a “statistic”, but they don’t know what it means.

Here’s a more complex, but poignent hypothetical example:

• You live in a town with an affluent side and a not so affluent side.
• The folks on the affluent side have annual incomes in the $80,000 to $120,000 range while the opposing side ranges from $20,000 to $40,000.
• Therefore, the median and the mean of the affluent demographic is roughly $100,000 while that of the opposing side is roughly $30,000.
• The number of families in the affluent demographic is slightly larger than the number of families on the opposing side.  As such, the median family income is ~$100,000 for the entire town while the mean income is closer to $65,000.

Due to mismanagement of funds and a complete lack of fiscal oversight (sound familiar?), a significant tax levy is necessary to close a budge shortfall and prevent default and several municipal loans.  The local government seeks to impose a universal, one-time levy on all residents to fix their mess.  They propose a one-time levy of $5,000 on each family.  The politicians state, “The Median Family Income in this town is $100,000.  Surely, a 5% tax hit on a one time basis is a small price to pay to shore up our town’s funds and fix things once and for all”.  While it may actually be a 5% hit in pre-tax income to the affluent group, it’s more like an 8% hit to “mean” population (which is meaningless because there is nobody that actually earns $65,000 so the mean is completely inconsequential) and it is a 25% hit to someone earning $25,000! Now, the debate over whether someone who earns less should pay an equivalent amount of the tax on a net basis vs. a % of income basis is one that occurs in all camps, but there’s no doubt that the politician here was manipulating the audience to advance his agenda.  For all pratical purposes, it’s clear that after tax and with the cost of living, a family earning $20,000 clearly does not have the means to forego $5,000 on a one time basis to support the initiative.

Why is This Important?

The next time your hear a politician say,

“The average family in America will only see their taxes increase by 1%”

“I promise you today…that the average tax deduction you’ll see from my plan is $2,200 per family”

“The average family in America will see No tax increase”

“Only the richest 2% of Americans will see their taxes increase”

“This health plan is only going to cost the average person $500 per year…and inaction is not an option”

The next time you hear this, you know you’re being manipulated.

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