Why Few Math Students Actually Understand the Meaning of Means

Did you know there is more than one way to calculate the mean? Mathematics has many means! Arithmetic mean, geometric mean, harmonic mean, least squares mean, and many many more. Why so many means? We tackle this question in this video to better understand why students too often misunderstand the meaning of a mean.

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Created by Doug Corey

Script: Doug Corey and Jennifer Canizales
Audio: Doug Corey
Animation: Jade Gregg
Editing: Jennifer Canizales
Music: Coma Media

© 2023 BYU

[BriTheMathGuy]

We're all kind of average if you know what I mean.

[davidnaranja]

I kinda already knew this implicitly and even manged to solve the MPG problem (didnt try the other) BUT I never explicitly thought of a mean as the value that can be replaced to get the same result. This is a very useful way of thinking of it.

[thing-]

In electrical engineering, the harmonic mean is used to find the aggregate value of resistors in parallel.
The RMS mean is used to find the DC equivalent of a varying voltage waveform for equal power.

[bobthecomputerguy]

I typed this out before the end of the video: The next time I see someone try to argue that mean is completely synonymous with "average, " I'm going to ask them "harmonic, arithmetic or geometric?"

[simihellsten]

Nice explanation (from another mathematician)! That said, I think the reason no-one got your answer for Q1 is that it's too ambiguous. I was quite surprised when you said that the mean you wanted was a s.t. (1+a)^3 = (1+x)(1+y)(1+z) - that's not at all obvious from the phrasing.
The arithmetic mean would be completely appropriate if, for example, you took all of your profits at the end of each year, starting each financial year with a fresh $1000 investment. Equally, you could read it (as I first did) that the three percentages each refer to different investments, each of which you held for three years. Again, the arithmetic mean would be appropriate there.

[diribigal]

The mathematician in me knows you explained very clearly how to think about what mean you need. The nitpicker points out that the spelling is "reciprocal", where the "re" is the same back-meaning prefix as in "return" and the "pro" is the same forward-meaning prefix as in "promote".

[brinleypowell]

I think it’s important to state that if you phrased the car questions as: ‘the two vehicles use about the same amount of fuel’ then the answer would be 25, great video

[kaykwanu]

I cannot thank you enough for helping folks concretize maths. This is a big deal. Please don't stop. You're appreciated.

[aleksey]

I’m a CS student (minoring in maths). I practically memorised a formulae sheet of different averages, and I agree with your thesis. From experience, I have a bad habit of defaulting to the arithmetic mean when faced with averaging problems. Thank you for reminding us that understanding the problem before crunching numbers ❤️🙏

[thetaomegatheta]

The issue with problem 1 is not people not understanding of what an 'average' is. It's with the unclear statement of the problem. It is not explained what a return on an investment is, or how those returns work. Without any further explanation, I could assume that you invest 1000 units of money, then get 133.1% of 1000 over year 1, get 72.3% of 1000 over year 2, and 126.6% of 1000 over year 3.

[JohnPalmoreJr]

I was also recently discussing means in the classroom, for me in the context of heat transfer. For heat transfer in a pipe, the logarithmic mean of temperature is an important quantity. It's not the type of quantity that people would normally think of as a mean, even compared to things like the geometric and harmonic means. This actually inspired a series of posts in my blog about the subject, where I ultimately discus how to use principles from calculus to derive a fairly general class of means. I don't think I can link it here, but my blog is in my bio.

[foozlebagel]

OOOh but there's even more! You didn't even touch the generalized power mean, where you get an infinite set of different "mean" values based on a power number, p.

All you do is raise each value in your data set to the power p, add them together, divide by the number of data points, and raise everything to the power of 1/p.

Actually, all of the means you mentioned in this video are generalized power means:
Harmonic mean -> p = -1
Geometric mean -> p = 0 (this does involve raising something to the power of 1/0, but the limit as p->0 gives you the geometric mean, so usually p=0 is set as the geometric mean)
Arithmetic mean -> p = 1
Root mean square -> p = 2

[somecreeep]

I remember a few months ago I saw a website explaining the abstract definition of a "mean" by fitting some set of criteria. I regret I haven't been able to find it since then even though I search for it regularly, but I remember two of the defining properties of a mean is that it must generalize to work for any number of elements of a set (that is, it can't just work for 2 values and not for more) and that in the special case of 2 values, it must return a number that is between (inclusive) them (assuming ordering is possible), which the three most popular means obviously fit. Since I've only found one place that even suggested there was a textbook definition for a mean, I'm guessing it's not a very popular idea, but I still find it interesting that someone somewhere did define it fairly succinctly

[jaimeduncan]

This is the best explanation I have found. I normally just peek at the correct one, but was never able to explain it with such clarity. An important detail if you are solving an exam, most of the time, mean means arithmetic mean, even if it does not work, so be sure to confirm.

[atmosphericSkull]

I appreciate that you made a point of using mean and average quite interchangeably. Some people insist that "average" must only refer to the arithmetic mean, which never sat right with me...

[michaelwangCH]

Thank you to point out. This problem can be very harmful in reality and those mistakes happen a lot in real work environment with grade students in stats, data science, data analysts and comp. science, also happens in scientific publications as well. This problem is most underestimated in real life and in academia - it leads to wrong decision and strategies in business and data manipulation in academic publications.

[BrettDalton]

So I've been doing engineering and product management for 20 years and this is the best explanation I have ever seen.. I also realise some really bad "means" I've applied in the past

[Chomusuke]

I only solved the first problem. I intuitively had the idea down but you put it into words beautifully.

[gljames]

Root Mean Square is probably used more by electricians as it tells you how much equivalent dc voltage a given ac system would convert to.

[tandemdwarf]

Glad to find a better explanation of this! I encountered all of these in AP stats, but I only memorized where to use them, rather than properly understanding why they were different and where they should be used