As the title says. This seems to be a little obscure, and I’m curious how many people have run into it.
For the curious, here’s the Wiki entry.
Yes, while studying for a financial certification.
My first guess was that it’s okay to be mean, as long as you can sing well. After looking at the Wiki page, I know that’s not it. I also know it’s unlikely I’ll ever understand the harmonic mean. When you start throwing sigmas at me, I get dizzy. Sacre bleu!
It’s not that hard to understand, AskNott.
Suppose I wanted to take the average of a bunch of, let’s say, gallons of milk from different providers. Maybe one charges $3 per gallon, one costs $4 per gallon, one costs $8 per gallon (I don’t know, it’s super-fancy). One way I would take the average price is by adding them all up and dividing by the total number of items; that’d get me an average of $5 per gallon. I assume you’re familiar with this kind of averaging, which we call the “arithmetic mean” for whatever historical reasons.
Well, now, I might want to carry out the same operation on the same data, only presented differently. Instead of being quoted in terms of dollars per gallon, instead, I’m told the amount of gallons I can get for a dollar; one provider gives me 1/3 gallons per dollar, another 1/4 of a gallon per dollar, and the last gives me 1/8 of a gallon per dollar.
If I want, I can take the arithmetic mean of these, and end up with (1/3 + 1/4 + 1/8)/3 = 17/72 gallons per dollar. But this does not correspond at all to $5 per gallon.
On the other hand, if I want to calculate what would’ve been the arithmetic mean of the dollars per gallon, I can take the so-called harmonic mean of the gallons per dollar. And all “harmonic mean” means is “Take the reciprocal of all your items, then take the arithmetic mean of those, then take the reciprocal again.” If what you’re really interested in is arithmetic mean of dollars per gallon, but you have to speak about everything in gallon per dollar terms for some reason, then you can always take the data you’re given, convert into the form you want to work with, do your work there, and then finally convert it back. In this particular case, we give that the fancy name “harmonic mean”. That’s all.
Er, missing words reinstated in bold.
Also, semi-related question: why do we use the word “arithmetic” for things that have to do with addition and “geometric” for things that have to do with multiplication?
I was not familiar with the term, but I was familiar with the concept, so I clicked “yes”.
Yes, though it has been 20 years or more since I would have heard the term.
I’d be surprised if you could graduate school in a developed country without ever hearing of the harmonic mean. Understanding and remembering is quite another thing, though.
I don’t remember ever learning about it in school, but years later I ran across it while browsing through my copy of the CRC Standard Mathematical Tables.
I knew that the term “harmonic mean” existed and was math-related, but I had no idea what concept it referred to. Basically the opposite of Strinka. I didn’t vote, because I wasn’t sure which answer was correct for me.
I’ve heard of it, but never came across the term in my studies, so far as I know. It’s possible my psychological statistics class mentioned it, but I don’t recall ever using it for any of the typical tests (t-test, ANOVA, chi square, etc.)
Wife’s work involves complicated statistics, and I have a technical background, so yep, I’ve heard of it.
I know what a harmonic mean is, but only through my own reading. It certainly was never mentioned when I went to high school, and I’m pretty sure it was never mentioned when I was in university (unless I was asleep at the time).
I’m only guessing, but it seems to me that after the plain ordinary arithmetic mean, the geometric mean is the next most likely to be encountered.
I definitely learned about the concept in school, though I could not have written out the generic expanded out expression without a lot of pencil chewing (or what is far more likely, a Wikipedia search).
Its proper use underlies a common gotcha on certain kinds of thinking tests and brain teasers. For example:
Let’s say I live 10 miles from my office. Today, I got up and went to work and traffic was good, I was able to achieve an average end-to-end speed of 40 MPH (including stopping at red lights and all that). But, coming home, I hit some traffic and can only go half as fast - 20 MPH. What was my average driving speed for the entire day?
The “obvious” answer is 30 MPH - the (arithmetic) average of 40 MPH and 20 MPH. But that’s not the right answer: in actuality, I traveled a total of 20 miles (10 to get to work and 10 to get back home) in 45 minutes (15 minutes to work, 30 minutes to get back home). 20 miles in 45 minutes, times 60 minutes per hour = 26.67 mph.
A harmonic average gives you that right away. As the Wikipedia page notes, the simplified two-value formula for a harmonic average for two values x1 and x2 is:
(2 * x1 * x2) / (x1 + x2)
where x1 and x2 are in the same units (mph). This makes the harmonic average (2x40x20) / (40+20) = 26.67 mph for the earlier example problem.
Starting from there, you can work out a general expression for any number of values. Or just use the Excel formula builder.
The key thing to grasp is when the harmonic average is the right average to use (it comes up a lot for aggregating Price/Earnings ratios for valuing stocks), versus the arithmetic or geometric average (yet another topic - related to rates of growth over time where the denominator, or the basis for the rate, keeps changing).
Read the wiki link and I STILL don’t know. I thought this was going to be about music.
Now that I know what it is, having perused the Wikipedia article, I can definitely say that neither the harmonic mean nor the geometric mean were ever mentioned in high school or college math. I completed two semesters of calculus, and I would definitely remember learning about those.
I wish they had taught us that; it’s both interesting and useful. I’m frankly surprised I hadn’t encountered it in my reading before now.
To elaborate on my example from my last post: the key concept to understand is that the denominator in the quoted rate, which is the time traveled, was different across the two values (15 minutes in the one case, 30 minutes in the other). It was the distance traveled - the 10 miles between my house and my job - that was fixed.
Therefore it’s not the apples-to-apples comparison necessary for using the straight arithmetic average. If the rates were given in the reverse, as minutes-per-mile, then yes you could use the arithmetic average because the distance was the same.
40 mph = 1.5 minutes / mile
20 mph = 3.0 minutes / mile
Average PACE of the trip = (1.5 + 3.0)/2 = 2.25 minutes / mile
2.25 minutes / mile = 2.667 miles/hour
I vaguely remembered the term (University was a long time ago) but not what it meant, so I didn’t vote.
You misplaced a decimal there: 26.67 miles/hour.
I checked the wiki entry and it does not explain why it is called the harmonic mean. It does have to do with music and it goes back to Pythagoreas or to the Pythagoreans. Suppose I ask what is the average tone between 256 hz (a C, middle C, IIRC) and 1024 hz (two octaves higher). The ordinary mean is 640 hz, which is several tones above C. But musicians would say it was 512 hz, the C in the octave above middle C. And that is harmonic mean of 256 and 1024. The formula, which takes the reciprocal of the arithmetic mean of the reciprocals is easy to derive, but the meaning (no pun intended) behind is clearer.
If you are interested in, say, gas consumption, the harmonic mean between 20 mpg and 40 mpg is 26.7 mpg and that the effect of averaging the consumption, which is the quantity that matters.