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 arithmeticmean 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.

If you enjoyed this article, subscribe for free to Darwin’s Finance. *Are you evolved?*

*More Fun with Numbers:*

Net Present Value – Why You Should Use it in Everyday Life

UFirst Money Merge Mortgage Accelerator Program Cuts Your Mortgage – And Defies Logic

Lazy Portfolio Claims don’t Add Up

Building Your Own Structured Note

401K Asset Allocation – What’s in Yours?

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I love this subject back in College…now it seems that it is forgotten..thanks for sharing it..

Euroangel’s last blog post..Watching TV Program[Reply]

Thank god I was a math nerd in school. I feel bad for those who get taken advantage of because they did not know the difference.

AJ’s last blog post..Why I considered getting a PhDÃ¢â‚¬Â¦[Reply]

Hi,

I have a website, http://mrlsmath.com/, where I host tools and techniques for math teachers and students.

I’d like to refer people to this post if:

1 – It’s okay with you.

2- The post will remain online indefinitely.

Thanks for your consideration,

Bill, a.k.a. Mr L.

Bill Lombard’s last blog post..Strategy Games and Other Fun Resources from N.C.T.M.[Reply]

Yeah, it’s incredible how many people are completely unaware of this. I beat this drum pretty incessantly, though, so people in my sphere of influence are somewhat aware.

We get some stats from another area within my company. The stats measure (among other things) the “average” response time of a certain website action. Unfortunately, they aren’t sure if this is a mean or median. If it’s a mean, then a couple of runaway transactions could skew the numbers for a million good transactions. If it’s the median, then it’s much more useful.

Kosmo @ The Casual Observer’s last blog post..Bill Gates vs. Hurricanes[Reply]

Good article.

How does the word “richest” have anything to do with mean or median? It seems out of place that you threw that in as an example in this discussion.

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Thanks for all your comments.

Bill,

Sure, please feel free to link to the article. I appreciate your interest and endorsement. The post will remain live indefinitely.

Jim

The example sound bites at the end are indicative of the type of rheotoric that is commonplace today. The point of the article was to make people think critically about just what a person is saying and whether they even know what they’re talking about.

Do you know what the richest 2% means? I don’t. Does it mean by annual income, by net worth, pre-tax, after tax (since the “rich” have the

means to offset taxable income by every tactic under the sun) or some other measure of “richness”?

Hope that helps put things into context.

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This is such basic math but most people do have difficulties with it. In my line of work (HR and compensation) it can burned people bad. This concept was beaten into me during my MBA and I am glad they did that – makes it fun to spot the crappy newspaper articles quoting statistics based on bad math!

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Thank you for addressing this, especially in such as read-friendly posting.

I work with numbers for a living (I am a statistician), and am concerned that more people are armed with knowledge such as this, since ignorance is a vulnerability. Math isn’t just for nerds like myself: math is needed to defend oneself against those, as you say, who have an agenda.

Everyday, people who know more about math take advantage of those who know less about math(think: credit card companies, for instance).Math: How else are you going to understand your world?

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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.The median family income would actual be $80,000 for the town since the affluent make between $80,000 and $120,000 and are slightly more (50% +1) than the poor side of town. The median family for the entire town would be the lowest paid in the affluent side of town.

Overall, it doesn’t change your analysis any. However, it does illustrate how easy it is to get the numbers jumbled.

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WOW. Trying to help my kid with 8th grade math. It’s been many, many, years since I studied this. Glad you were here. Please add info re: outlier (sp) for others.

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Bookmarked for an upcoming lecture. The MPA students *usually* get it by midterms, but there’s always a few stragglers (which makes me fear for the future of our country).

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Yeah!! I’m so glad people are talking about statistics!! This is one of my favorite subjects and I even just posted a blog/video post on this subject.

Everyone should pay attention because this is a good example of math that is real world applicable and honestly, utilized a lot!!

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“25% hit to someone earning $25,000”

And don’t forget, while bending the numbers with “statistics”, that there is still room for typos. $5,000 would be a 20% hit to someone earning $25,000, not 25%, or, perhaps it was supposed to be “25% hit to someone earning $20,000” since the range of the not so affluent was $20-40,000.

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Good stuff … wish the Internet had more of this and less of other garbage.

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Thanks for the information. I was reading a study into mortgage indebtedness and came across median and mean in the study. Your insight helped clear up what these numbers mean. So, just to keep things straight, average (arithmetic mean) is a useful number, but mean will eliminate the large outlying values that can skew your figures, right?

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“25% hit to someone earning $25,000!”

I may be mistaken, but wouldn’t a $5,000 tax on someone earning $25,000 be 1/5 of their income, and thus 20%?

Otherwise, very nice explanation of the difference between mean and median. Each has it’s own particular usage. Mean is sensitive to extremes in value, median to extremes in quantity.

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دردشة

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Good article although your political examples at the end show that you clearly have an agenda of your own. We recognize Obama near-quotes when we see them.

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Can you clarify your last example of the 5% tax. you use $5,000 and then 5% tax, and I’m not sure if its $5,000 on each family or 5% tax based on what you say or the politician. Further, where does the 8% rate calculate into, and the 25% on 25,000. I’m not clear on that. Thank you! I appreciate the information you have offered on your site.

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Very helpful overview with certain math concepts, etc. Too bad follow up comments must include opportune politicized comments from folks with their own agenda, like Mike V. This particular blog was written by Darwin in July 2009, well before taxes were high on anyone’s public ‘agenda’, other than Grover Norquist & co., and long before the pres. was being misquoted and maligned by folks of G.N.’s political bent.

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$5000 is TWENTY PERCENT of $25000, not twenty-five percent.

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Thanks for this article! It is super helpful as I take a glance at my statistics notes before lecture. However, could someone explain how you got that a 5% tax levy would actually be 8% for the mean salary ($65,000) while it would be 25% for the $20,000 salary families?

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