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So we have arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM). Their mathematical formulation is also well known along with their associated stereotypical examples (e.g., Harmonic mean and it's application to 'speed' related problems).

However, a question that has always intrigued me is "how do I decide which mean is the most appropriate to use in a given context?" There must be at least some rule of thumb to help understand the applicability and yet the most common answer I've come across is: "It depends" (but on what?).

This may seem to be a rather trivial question but even high-school texts failed to explain this -- they only provide mathematical definitions!

I prefer an English explanation over a mathematical one -- simple test would be "would your mom/child understand it?"

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This perhaps oversimplifies but I've always used range and observations. If range is the same = AM (compare scores 0-100, to 0-100), if range is different but observation is the same = GM (compare scores 1-5, to 0-10), if range is same but observations are different = HM (speed of a car at different obs, heights of two ladders, other "rates"). –  Brandon Bertelsen Feb 19 '12 at 20:25
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(+1) As usual from you, this is an interesting and well-constructed question. –  cardinal Feb 19 '12 at 21:35
    
@cardinal: As my dad says: Play with the subject, don't let the subject play with you - for playing sometimes it's just necessary to ask very fundamental questions. Thanks for appreciating! –  PhD Feb 19 '12 at 22:37

4 Answers 4

up vote 60 down vote accepted

This answer may have a slightly more mathematical bent than you were looking for.

The important thing to recognize is that all of these means are simply the arithmetic mean in disguise.

The important characteristic in identifying which (if any!) of the three common means (arithmetic, geometric or harmonic) is the "right" mean is to find the "additive structure" in the question at hand.

In other words suppose we're given some abstract quantities $x_1, x_2,\ldots,x_n$, which I will call "measurements", somewhat abusing this term below for the sake of consistency. Each of these three means can be obtained by (1) transforming each $x_i$ into some $y_i$, (2) taking the arithmetic mean and then (3) transforming back to the original scale of measurement.

Arithmetic mean: Obviously, we use the "identity" transformation: $y_i = x_i$. So, steps (1) and (3) are trivial (nothing is done) and $\bar x_{\mathrm{AM}} = \bar y$.

Geometric mean: Here the additive structure is on the logarithms of the original observations. So, we take $y_i = \log x_i$ and then to get the GM in step (3) we convert back via the inverse function of the $\log$, i.e., $\bar x_{\mathrm{GM}} = \exp(\bar{y})$.

Harmonic mean: Here the additive structure is on the reciprocals of our observations. So, $y_i = 1/x_i$, whence $\bar x_{\mathrm{HM}} = 1/\bar{y}$.

In physical problems, these often arise through the following process: We have some quantity $w$ that remains fixed in relation to our measurements $x_1,\ldots,x_n$ and some other quantities, say $z_1,\ldots,z_n$. Now, we play the following game: Keep $w$ and $z_1+\cdots+z_n$ constant and try to find some $\bar x$ such that if we replace each of our individual observations $x_i$ by $\bar x$, then the "total" relationship is still conserved.

The distance–velocity–time example appears to be popular, so let's use it.

Constant distance, varying times

Consider a fixed distance traveled $d$. Now suppose we travel this distance $n$ different times at speeds $v_1,\ldots,v_n$, taking times $t_1,\ldots,t_n$. We now play our game. Suppose we wanted to replace our individual velocities with some fixed velocity $\bar v$ such that the total time remains constant. Note that we have $$ d - v_i t_i = 0 \>, $$ so that $\sum_i (d - v_i t_i) = 0$. We want this total relationship (total time and total distance traveled) conserved when we replace each of the $v_i$ by $\bar v$ in our game. Hence, $$ n d - \bar v \sum_i t_i = 0 \>, $$ and since each $t_i = d / v_i$, we get that $$ \bar v = \frac{n}{\frac{1}{v_1}+\cdots+\frac{1}{v_n}} = \bar v_{\mathrm{HM}} \>. $$

Note that the "additive structure" here is with respect to the individual times, and our measurements are inversely related to them, hence the harmonic mean applies.

Varying distances, constant time

Now, let's change the situation. Suppose that for $n$ instances we travel a fixed time $t$ at velocities $v_1,\ldots,v_n$ over distances $d_1,\ldots,d_n$. Now, we want the total distance conserved. We have $$ d_i - v_i t = 0 \>, $$ and the total system is conserved if $\sum_i (d_i - v_i t) = 0$. Playing our game again, we seek a $\bar v$ such that $$ \sum_i (d_i - \bar v t) = 0 \>, $$ but, since $d_i = v_i t$, we get that $$ \bar v = \frac{1}{n} \sum_i v_i = \bar v_{\mathrm{AM}} \>. $$

Here the additive structure we are trying to maintain is proportional to the measurements we have, so the arithmetic mean applies.

Equal volume cube

Suppose we have constructed an $n$-dimensional box with a given volume $V$ and our measurements are the side-lengths of the box. Then $$ V = x_1 \cdot x_2 \cdots x_n \>, $$ and suppose we wanted to construct an $n$-dimensional (hyper)cube with the same volume. That is, we want to replace our individual side-lengths $x_i$ by a common side-length $\bar x$. Then $$ V = \bar x \cdot \bar x \cdots \bar x = \bar x^n \>. $$

This easily indicates that we should take $\bar x = (x_i \cdots x_n)^{1/n} = \bar x_{\mathrm{GM}}$.

Note that the additive structure is in the logarithms, that is, $\log V = \sum_i \log x_i$ and we are trying to conserve the left-hand quantity.

New means from old

As an exercise, think about what the "natural" mean is in the situation where you let both the distances and times vary in the first example. That is, we have distances $d_i$, velocities $v_i$ and times $t_i$. We want to conserve the total distance and time traveled and find a constant $\bar v$ to achieve this.

Exercise: What is the "natural" mean in this situation?

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+1, I'm happy to be the first to upvote this. I had been thinking along these lines (particularly the top bolded line), but wouldn't have done anything so comprehensive, or clear, or well laid out. Bravo! –  gung Feb 20 '12 at 2:47
    
+1, this great response is no surprise coming from you. I would like to borrow your intuition if you don't mind :) –  Mike Wierzbicki Feb 20 '12 at 4:59
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+1 This is a great answer. However, I think it's incomplete in an important way: in many cases the right mean to use is determined by the question we are trying to answer rather than by any mathematical structure in the data. A good example of this occurs in environmental risk assessment: regulatory authorities want to estimate a population's total exposure to contaminants over time. This requires an appropriately weighted arithmetic mean, even though environmental concentration data usually have a multiplicative structure. The geometric mean would be the wrong estimator or estimand. –  whuber Feb 20 '12 at 15:03
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@whuber: (+1) This is an excellent comment. On my path to constructing an answer, I took a decidedly nonstatistical fork, so I'm glad you mentioned this. It's a topic worthy of a complete answer (hint). –  cardinal Feb 20 '12 at 17:20
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@whuber: It also brings up the fact (perhaps unintentionally), that statistical analysis can oftentimes be subject to the oversight of domain experts (or, perhaps in your example, even nonexperts), who want to estimate something meaningful to their domain but almost wholely unnatural statistically. The issue I've run into there in the past is that they sometimes want to also dictate the way that statistical estimation is carried out! :) –  cardinal Feb 20 '12 at 17:20

Expanding on @Brandon 's excellent comment (which I think should be promoted to answer):

The geometric mean should be used when you are interested in multiplicative differences. Brandon notes that geometric mean should be used when the ranges are different. This is usually correct. The reason is that we want to equalize the ranges. For example, suppose college applicants are rated on SAT score (0 to 800), grade point average in HS (0 to 4) and extracurricular activities (1 to 10). If a college wanted to average these and equalize the ranges (that is, weight increases in each quality relative to the range) then geometric mean would be the way to go.

But this isn't always true when we have scales with different ranges. If we were comparing income in different countries (including poor and rich ones), we would probably not want the geometric mean, but the arithmetic mean (or, more likely, the median or perhaps a trimmed mean).

The only use I've seen for harmonic mean is that of comparing rates. As an example: If you drive from New York to Boston at 40 MPH, and return at 60 MPH, then your overall average is not the arithmetic mean of 50 MPH, but the harmonic mean.

AM = $(40 + 60)/2 = 50$ HM = $2/(1/40 + 1/60) = 48$

to check that this is right for this simple example, imagine it is 120 miles from NYC to Boston. Then the drive there takes 3 hours, the drive home takes 2 hours, the total is 5 hours, and the distance is 240 miles. $240/5 = 48$

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Why would your SAT/GPA/extracurricular example use a geometric mean rather than a weighted or scaled arithmetic mean? Why should an SAT or GPA of zero mean that the other two values become irrelevant (as a geometric mean would imply)? And what if (say) extracurricular-activities tend to cluster in a much narrower band than its theoretical range? It seems like it would make more sense to take an arithmetic mean of percentiles (or other adjusted values) than a geometric mean of raw values. –  ruakh Feb 20 '12 at 2:08
    
@ruakh Interesting. The 0 issue doesn't really matter in this case, as SAT and GPA can't really be 0 (SAT = 0 is almost impossible, and GPA of 0 would not graduate). I think an arithmetic mean of percentiles will be close to the geometric mean in its conclusions (even though not in the actual numbers). –  Peter Flom Feb 20 '12 at 12:04

I'll try to boil it down to 3-4 rules of thumb and provide some more examples of the Pythagorean means.

The relationship between the 3 means is HM < GM < AM for non-negative data with some variation. They will be equal if and only if there's no variation at all in sample data.

For data in levels, use the AM. Prices are a good example. For ratios, use the GM. Investment returns, relative prices like the Bloomberg Billy index (the price of Ikea's Billy bookshelf in various countries compared to the US price) and the UN's Human Development Index are all examples. HM is appropriate when dealing with rates. Here's a non-automotive example courtesy of David Giles:

For instance, consider data on "hours worked per week" (a rate). Suppose that we have four people (sample observations), each of whom work a total of 2,000 hours. However, they work for different numbers of hours per week, as follows:

Person      Total Hours       Hours per Week          Weeks Taken
1                  2,000                  40                   50
2                  2,000                  45                   44.4444
3                  2,000                  35                   57.142857
4                  2,000                  50                   40

Total:           8,000                                       191.587297

The Arithmetic Mean of the values in the third column is AM = 42.5 hours per week. However, notice what this value implies. Dividing the total number of weeks worked by the sample members (8,000) by this average value yields a value of 188.2353 as the total number of weeks worked by all four people.

Now look at the last column in the table above. In fact the correct value for the total number of weeks worked by sample members is 191.5873 weeks. If we compute the Harmonic Mean for the values for Hours per Week in the third column of the table we get HM = 41.75642 hours (< AM), and dividing this number into the 8,000 hours gives us the correct result of 191.5873 for the total number of weeks worked. Here is a case where the Harmonic Mean provides the appropriate measure for the sample average.

David also discusses the weighted version of the 3 means, which come up in price indices used to measure inflation.

A Hijacky Aside:

These ROTs are not perfect. For instance, I often find it hard to figure out if something is a rate or a ratio. Returns on an investment are usually treated as a ratio when calculating means, but they are also a rate since they are usually denominated in "x% per unit of time." Would "use HM when the data is levels per unit of time" be a better heuristic?

If you wanted to summarize the Big Mac Index for Northern European countries, would you use the GM?

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A possible answer to your question ("how do I decide which mean is the most appropriate to use in a given context?") is the definition of mean as given by the Italian mathematician Oscar Chisini.

Here is a paper with a more detailed explanation and some examples (mean travelling speed and others).

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It might be ideal if you could add a few lines about Chisini's definition here in case the link goes dead, &/or to help readers know if they want to click the link to pursue the ideas further. –  gung Jun 2 at 19:54

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