I don’t expect but maybe one person out of a 1000 to remember the Quadratic Equation—even though my algebra teacher wrote in my yearbook that if I forgot everything else she wanted me to remember that one thing. I don’t expect but maybe one person out of ten to be able count change without a computer someplace in the process. But there is one “math thing” that is starting to annoy me. I’ve heard this one claim for decades and I have never heard anyone else point out the obvious fallacy. It’s like an urban myth that everyone believes even though nearly everyone with a room temperature I.Q. could demonstrate it is false.
What finally tweaked me enough to do something about it was listening to a podcast by someone who claims to be smart enough that he should know better. I’m withholding the name to protect the guilty, but what was said was something like, “80% of the population thinks they are better than average drivers. That’s mathematically impossible!” Grrr…
Try proving that without resorting to a far less common definition of “average”.
A year or two ago in a much different forum someone else made a similar statement about penis sizes. I politely explained they were full of it and it hasn’t come up again (pun intended).
Now, with a much larger audience, I will now explain the issue using different example to (mostly) save you from thoughts about penis dimensions. I hope I don’t have to be subjected to this myth again, and if I am I will be able to just glare at them and send them a link to this post.
Imagine we have a sample of 50 male/female couples. All the people, except one, had their spouse as their one and only sex partner (I told you to imagine, remember?). It turns out that before the age of government education loans and grants Trixie earned her way through medical school the old fashion way—in bed. She had 1000 sexual partners prior to her spouse.
Lets compute the average (usually understood to be the arithmetic mean) number of sexual partners in this sample.
MeanSexualPartners = TotalSexualPartners/NumberOfPeople
MeanSexualPartners = ((99 x 1) + (1 x 1001))/100
MeanSexualPartners = 1100/100
MeanSexualPartners = 11
In this case 99% had 1 sexual partner and can truthfully and correctly state they have had fewer than the average number of sexual partners. Furthermore, 99% can correctly state they have had less than 10% of the average number of sexual partners.
I will leave the drivers and penis dimension examples as exercises for the reader.
It seems people are confusing “average” with “median.” It is, of course, mathematically impossible for 80% of drivers to be better than the median driver.
Now that I think about it, let me recant that: most drivers are better than the guy who’s driving on the median.
–John.
LOL. Joe, you can call me out directly, I won’t be offended. Remember, unlike the anti-gunners, we can usually have a polite discussion over differing points of view. 😉 The problem I see with your example is it’s not really an adequate comparison. In your example you’re measuring the number of things had, whereas the driving example is a comparison of people’s being. It’s like the difference between averaging how many apples people have vs. averaging how many of them are apple farmers. This is almost more of a grammatical argument than mathematical, because we have to be clear about what we’re averaging.
Without delving into mean, median, and mode, 80% of students cannot all be above average. 80% of men cannot all be above average male height. 80% of a population can all be under the average bank account balance if you throw, say, a George Soros or a Bill Gates in the mix, but now we’re averaging things (dollars) and not people. Right?
And I don’t remember the quadratic equation precisely, but I remember which one it was (is), and can properly reference the Pythagorean theorem! 😉
Eric, how about a real-world example of the students example – actually, a reversal of it? A bio class I taught a while back had twenty-some students by the end of the year (rest had been transferred, dropped out, been expelled, etc). Final grades were two high As, a B, a couple of low Cs, some Ds, and a lot of Fs. Overall point average was about 72%, IIRC. Only 4 of 21 students were above that average. 17/21 = 80.95%, so 81 percent of students in that class WERE below average, when considering just the normal arithmetic mean, for the class (and WAY below where I hope the cohort average is). Sad (well, pathetic, actually) but true. It was a painful class to teach- lots of toxic personality mixes, challenging home life issues (for the students), and a bad overall school situation.
That said, I know what people mean when they say it can’t be true (given the assumed large enough group with a more normal distribution of continuously measurable criteria) but as Joe says, sometimes things just bug a person.
Thanks Rolf,
Eric if you still don’t get it try doing a formal proof of your case. If you still can’t see it I’ll give you the penis size example.
Yeah, that’s both a promise and a threat.
Eric,
Here are the grades of 10 students: 80, 76, 79, 12, 74, 82, 75, 15, 100, 76
Average Grade: 66.9
Percentage of students above average: 80%
Yes, 80% of students can be above average. The same could happen for height depending on population. If you had in your sample a majority of men between 5’10” and 6’4″, but you had enough men of small enough stature (and it would only take a small percentage of men below, say, 3’6″) to drag the average below 5’10”, then you most definitely could have 80% of men above average height, just as you I had 80% of students who were above average.
Finding those members who are outside of one standard deviation is more significant. In my sample population of students the standard deviation is 27.6, making only one student (with a grade above 94.5) extraordinary in non-statistical terms. (And it makes are two dunces with grades in the teens, extraordinary failures.)
Joe,
I think it all falls flat because of a key word ‘thinks’.
You say ‘80% of the population thinks they are better than average drivers. That’s mathematically impossible!” Grrr…’
It’s not. I’m surprised that it wasn’t 100%. Everyone I know who drives THINKS they are above average even though I know every one bar me is below average. 🙂
Now if thinks was missing from the sentence then I would agree with you.
Ok, I see where the miscommunication is coming from- I’m speaking lazy man’s English and you guys are speaking math. 😉 No, seriously, it’s that you’re using mean and I’m thinking of median (or maybe even mode?). Because in real-world situations I don’t know how many laymen let extreme examples skew their perception of what average is.
I’m with Lord T, in that 80% of people could perceive themselves to be above average, but I maintain that 80% of people cannot all actually BE above average. 4/5 is a clear majority, so the average (or at at least it’s definition) needs to be adjusted.
Joe, let’s not get into a penis size example/comparison. I understand an extreme example skews the data set. No reason to throw another one into the mix. 😉
IF: y=ax^2+bx+c, THEN x=(-b +- sqrt(b^2-4ac))/2a.
If you have a 2, a 3 and a 10 your average is 5. So 2/3’s of your sample is below average.
“I don’t know how many laymen let extreme examples skew their perception of what average is.”
Most “studies” that folks quote do not drop “extreme examples” from the calculations. It is called “pencil-whipping” when they do and it is usually very bad. So yes, laymen do often let the extremes skew their view unless they personally double check the math.
And then there’s the average folks who confuse “average” and “median” the way some people switch “magazines” and “clips”.
Clint,
“Most “studies” that folks quote do not drop “extreme examples” from the calculations.”
True, but that may actually be legitimate. To properly exclude outliers from a data set, one must perform a Q-test, which considers the ratio of (the difference between the outlying point and the nearest other data point) to (the difference between the outlying point and the mean). If this is larger than some value, which is dependent on the sample size and the confidence interval (90% certainty vs. 95% certainty vs. 99% certainty), the data point can be considered an outlier and discarded with whatever certainty you decided was sufficient (it always leaves a chance that the data point should have been included–i.e. no discarding to 100% confidence). It’s a pretty stringent test, and you can’t usually do so unless your ratio is near unity or your data set is enormous. Most studies that folks go around quoting (the typical newspaper “studies show that…[insert sensationalist, counterintuitive statement]”) suffer from small data sets and probably can’t test out the outliers so well.
If I recall correctly, something like 84.2% of all statistics are made up on the spot.
And Joe, I wholeheartedly agree that most people couldn’t recite the quadratic equation. My latest application I’m writing is full of angles, deviations, averages, and complex maths that I simply had to relearn. And I was in Advanced Calculus in high school and was an electrical technician in the Marine Corp (for a demo of the app as I’m writing it, please give a look at http://www.robballen.com/TargetApp/).
Interestingly the same thing happens when grouping your shots. MOST of your shots can be well below the ‘average’ and very much for the same reason Sally the Slut hoses up the equation for the monogamous.
Lord T beat me to the punch. Stated another way: You ask 1000 people if they think that they are better at driving than the average person. Eight hundred of them respond with “Yes.” Thus the results that 80% of people think that they are better than average drivers.
However, I really like your explanation about the number of sex partners. I’ll have to save that for explanatory purposes.
But in the case of the drivers those 80% could be correct.
I guess I’m going to have to explain it better…
Suppose we measure the proficiency of a driver by the number of vehicle accidents they cause during a 10 year period. 800,000 drivers cause zero accidents during that ten year period. 100,000 cause one accident per ten years. 100,000 cause 10 accidents in ten years. TotalAccidents/TotalDrivers is the average number of accidents per driver.
AverageAccidents = (1,000,000 + 10,000,000)/1,000,000 per ten years per driver
AverageAccidents = 11 per ten years per driver
AverageAccidentsPerDriverPerYear = 1.1
All 800,000 drivers with zero accidents in the last ten years are far better than the average driver.
No matter how you measure it (and use arithmetic mean as your definition of “average”) I can easily construct an example where the 80% could be accurately stating they are “better than average”.
Yeah; I figured it was all about how people perceived themselves, so the number could be anything and still be correct, assuming no one is lying (or do a significant number of respondents lie when answering poll questions?).
I’ve never given that alleged statistic another thought, but it is interesting to see Y’all’s examples.
Erik might also be thinking of a population with a distribution approaching normal. If your population looks like a bell curve, then the mean should be quite similar to the median, and it would be very unlikely for 80% of the population to be greater than the mean.
Certainly people could be thinking that but that does not justify anyone claiming it is “mathematically impossible” for the statement to be true without stating their assumption of a normal distribution.
If the total population of penises is X and the the average penis volume is Y then X/Y is the “average volume per penis”.
If in population X you have subpopulation Z, which is equal to 0.40X and the penis volume for population Z is contained in the smallest volume of Y, say 0.20Y, then you have 40% of the population has a penis volume in the bottom 20%. This means that the remaining 60% of population X has a penis volume in the top 80%. This means that “more than half of population X has a larger than average penis”.
Eric’s initial assumption is an evenly distributed curve (IE the “Bell Curve) with the mean at 50%. The problem is that the Curve has to reflect reality, not reality reflecting the curve. Remember, math is the language of describing reality, that is why it is so important.
Remember, words mean something, being precise with language is important. I was recently reminded that when a Conservative says “compromise” they mean find an equitable middle ground, but when a Liberal says “compromise” they mean “endanger, place in a vulnerable position”. That is why Liberals love to compromise….
“…one must perform a Q-test,…”
I knew I was gonna catch flak for that! Well, I did use “most” and “usually” and I waited to keep it short. Oh well. The point was: people cannot arbitrary drop extremes “just because” you don’t think it fits.
And not to put words in Joe’s month, but I see the root argument here as being the phrase: “That’s mathematically impossible!” -as regards to 80% of the population- is categorically false. It doesn’t matter if they “think” correctly or not.
The point being that “average” is ,not always a halfway point. Especially as most people use the word even if they think it does.
I’ve always been fond of throwing out “90% of the population is below average intelligence” when you see examples of large groups of people being stupid. What I really enjoyed were people that would insist that is not possible. 90% being below the median is impossible. 90% being below the mean is unlikely. There’s a HUGE difference between being mathematically impossible, and just being unlikely.
So Joe, I’ve felt you pain on this one for a long time. I don’t understand why basic math is so hard to grasp for some people.