Four Kinds of “Mean”

Last week, we looked at exactly what the mean is, referring specifically to the arithmetic mean, the one we first learn as the “average”. But just as we previously saw that there are several things called “average” (mean, median, mode), there are in fact several different kinds of “mean”. We’ll look here at the arithmetic, geometric, harmonic, and quadratic means, focusing on how they are the same, how they differ, and how to choose one.

Definitions of four means

We’ll start with a question from 1996:

Definitions: Average, Mean, Mode

I see that the dictionary says the average is the arithmetical mean and that the geometrical mean is different, but I would like to find a simple definition comparing the meaning of the 

  Mean
  Average
  Mode

Everything I have found is either too complex and over my head, or not complete.

Thanks very much.

Margaret

I’ll ignore the mode here, which we’ve thoroughly covered.

Doctor Pete answered, including a bonus:

"Mean" is a general term, but is most commonly used as an abbreviation for "arithmetical mean."  Now, say you have a set of numbers, say,

  { 5, 6, 1, 1, 1, 7, 4, 3 } .

In this case, there are 8 numbers, but we can have as many as we want.  Then the arithmetic mean, or average is

  (5+6+1+1+1+7+4+3)/8 = 28/8 = 3.5 ;

that is, we add all the elements and then divide by the number of elements.

$$\text{arithmetic mean}=\frac{\text{sum of values}}{\text{number of values}}=\frac{\sum\limits_{i=1}^n x_i}{n}$$

The geometric mean is

  (5*6*1*1*1*7*4*3)^(1/8) = 2.6618 ;

so we multiply by all the elements and then take the 8th root.

$$\text{geometric mean}=\sqrt[\text{number of values}]{\text{product of values}}=\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}$$

The harmonic mean is

  8/(1/5+1/6+1/1+1/1+1/1+1/7+1/4+1/3) = 1.9546 ;

which is like the arithmetic mean except everything is "flipped around."

$$\text{harmonic mean}=\frac{\text{number of values}}{\text{sum of reciprocals}}=\frac{n}{\sum\limits_{i=1}^n \frac{1}{x_i}}$$

This is the reciprocal of the arithmetic mean of the reciprocals.

Later we’ll be seeing one more kind of mean, the quadratic mean. I’ll add that in here: $$\text{quadratic mean}=\sqrt{\frac{\text{sum of squares of values}}{\text{number of values}}}=\sqrt{\frac{\sum\limits_{i=1}^n {x_i}^2}{n}}$$

Arithmetic vs geometric mean: when to use each?

Next, from 2001, we have this question about when each mean is most appropriate:

Arithmetic vs. Geometric Mean

When would it be appropriate to use the arithmetic mean over the geometric, or vice versa? I am not trying be vague, but the geometric mean will always be less than the arithmetic. Are there specific circumstances where one is used over the other?

The comment about the geometric mean being less than the arithmetic mean refers to the “AM-GM inequality”, which is stated (and proved, by a relatively advanced method) by Doctor Floor here:

Arithmetic/Geometric Mean Inequality Theorem

This can be extended further to the HM-GM-AM-QM inequality, which says that if the data are all positive, then all four means we are discussing are always in that order: $$0\le HM\le GM\le AM\le QM$$

Doctor Floor answered:

Hi, Mark - thanks for writing.

Let me try to give examples for when you use one, and when you use the other.

The profit of Company A, SYZO Ltd., has grown over the last three years by 10 million, 12 million, and 14 million dollars. It is appropriate to say that it has grown by an average of 12 million dollars yearly, for which we use the arithmetic mean.

Here the growth each year is measured in dollars – an absolute growth. What we are averaging are numbers that have been added to the company’s profit, so that the total growth is their sum.

The profit of Company B, OZYS Ltd., has grown the over last three years by 2.5%, 3%, and 3.5%. Here we cannot use the arithmetic mean and say that the average growth was 3%. Why not?

Suppose that Company B, OZYS Ltd., started with a 100-million-dollar profit. Three years later it will have become:

     $100,000,000 * 1.025 * 1.03 * 1.035 = $109,270,125

This is less than a yearly increase of 3% would yield, since:

     $100,000,000 * 1.03 * 1.03 * 1.03   = $109,272,700

Here we see that we should use the geometric mean of the growth factors 1.025, 1.03, and 1.035 to find the average percentage. That is always less than the arithmetic mean would yield.

This time the growth is measures as a percentage – a relative growth. We are averaging numbers by which the profit has been multiplied each year, and the total percentage growth is their product.

The big idea: One number, same net effect

In 2001 we got a request for more detail, which was added to the same page:

I am attempting to determine why the geometric mean is more appropriate to use when dealing with percentages and rates of change. However, I am not sure why it works better. I know that the result of applying the geometric mean will be the same as or lower than the arithmetic mean, and that the arithmetic mean is used when finding the average of numbers that are added to find the total, and that the geometric mean is used when the items of interest are multiplied to gain the total (found by looking at your Web site). I still do not know why it works this way.

I looked in some older texts that were in print prior to the use of scientific calculators and saw that there is a formula for the geometric mean that uses logs. This leads me to believe that in doing so, the data was being normalized, and maybe this would be the reason that the geometric mean would be better.

Doctor Rick answered:

Hi, Wes.

I think Dr. Floor did a fine job of briefly explaining the benefits of each mean. I will just say the same thing with a little expansion.

The big idea behind means is this. You have a bunch of different numbers. You want to replace each of these different numbers by the *same* number, in such a way that the net effect (the result of combining the numbers) is unchanged.

This is the essential idea behind any mean.

Different means are used depending on what we mean by "combining" the numbers. In Dr. Floor's example of a company growing annually by 10, 12, and 14 million dollars, the "combined" growth is the sum of the three numbers. We want the total growth (the sum of the annual growths) to equal the total if the company grew by N million dollars each year. The mean we seek, therefore, is the number N such that

  10 + 12 + 14 = N + N + N

The number that works is the arithmetic mean:

  N = (10 + 12 + 14)/3

The arithmetic mean is appropriate when the net effect of the numbers is found by addition, because it finds a number that yields the same sum when repeated.

In Dr. Floor's second example, of a company growing annually by 2.5%, 3%, and 3.5%, the growth rate over the three years is computed differently:

  Profit in year 1 = 1.025 times profit in year 0

  Profit in year 2 = 1.03 times profit in year 1
                   = 1.03 * 1.025 times profit in year 0

  Profit in year 3 = 1.035 times profit in year 2
                   = 1.035 * 1.03 * 1.025 times profit in year 0

Thus the ratio of the profit in the third year to the profit in the base year is

  1.025 * 1.03 * 1.035

If the growth rate had been the same each year, the ratio would be

  N * N * N

For the growth over 3 years to be the same in both cases, we must have

  1.025 * 1.03 * 1.035 = N * N * N

The solution is

  N = (1.025 * 1.03 * 1.035)^(1/3)
    = 1.029992

That is, the cube root of the product of the annual growth factors. This is the geometric mean.

The geometric mean is appropriate when the net effect of the numbers is found by multiplication, because it finds a number that yields the same product when repeated.

Your comment about logs prompts me to point out that there is a close connection between the two means. The log of the geometric mean of a set of numbers is the arithmetic mean of the logs of the numbers:

  log((abc)^(1/3)) = (1/3)log(abc)
                   = (1/3)(log(a)+log(b)+log(c))

Thus you can find the geometric mean by taking the logs of your data, finding the arithmetic mean, then taking the antilog (exponential) of the mean. This is presumably the formula to which you refer.

We’ll be seeing this again.

The mean proportional

For a closely related concept, see:

Mean Proportionals and Geometric Means

The mean proportional of two numbers, say 4 and 9, is another name for the geometric mean, thought of as finding the middle number that makes 4, _, 9 a geometric progression (sequence), so that we are multiplying by the same ratio each time. In this case, that would be \(\sqrt{4\cdot 9} = 6\), so that 4, 6, 9 multiplies by \(\frac{3}{2}\) each time.

This is a useful way to think of most of our means when there are only two numbers:

  • The arithmetic mean fills in the arithmetic progression a, _, c with the middle number \(b = \frac{a+c}{2}\) so that \(b-a = c-b\);
  • the geometric mean fills in the geometric progression a, _, c with the middle number \(b = \sqrt{ac}\) so that \(\frac{b}{a} = \frac{c}{b}\);
  • and the harmonic mean

Harmonic mean

This question comes from 1996:

Harmonic Mean

I have searched to net to find an explanation of why anyone would use the harmonic mean. Most sites just define it or state the formula.  Some state that it is useful for speed and velocity problems.  Your site talks about the mean as a "representative" value for a set of data. Can you give me an example stating why the harmonic mean is "more representative" than the arithmetic mean?

Katie is quoting from Doctor Pete’s explanation at the top. Each kind of mean is “representative” of a different kind of data, as we’ve been seeing. What kind of data is this mean good for?

Doctor Bombelli answered, starting with some history, and focusing on the mean of two numbers:

The Greeks were into means, and the harmonic mean in particular.  Here is the Greek definition from Porphyry in the "Commentary on Ptolemy's Harmonics": The subcontrary mean, which we call harmonic, is such that by whatever part of itself the first term exceeds the second, the middle term exceeds the third by the same part of the third.  

That is, b is the harmonic mean between a and c if (a-b)/a = (b-c)/c.  Can you get b = 2ac/(a+c) out of this?

The Greeks (Pythagoreans specifically) used these means in music: holding strings in certain ratios and plucking them, for example.  Iamblichus says that the harmonic mean "was then called subcontrary, but which was renamed harmonic by the circle of Archytas and Hippasus, because it seemed to furnish harmonious and tuneful ratios."

This approach to the harmonic mean is parallel to the geometric mean thought of as the “mean proportional”: It fills in a gap in a sequence so that not the ratios, but the relative increase, from one number to the next is constant. Such a sequence is called a harmonic progression, which is defined by the fact that their reciprocals are in arithmetic progression; that is, \(a, b, c\) form a harmonic progression if \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\) form an arithmetic progression, so that \(\frac{1}{b} – \frac{1}{a} = \frac{1}{c} – \frac{1}{b}\). Multiplying this equation by b, we have \(1 – \frac{b}{a} = \frac{b}{c} – 1\), which is equivalent to the equation above.

Observe that if \(\frac{a-b}{a} = \frac{b-c}{c}\), then cross-multiplying, \(ac-bc = ab-ac\), so that \(2ac = ab+bc\) and, as we were told above, \(\displaystyle b = \frac{2ac}{a+c}\), which is a formula for the harmonic mean of two numbers. If we instead divide each side of \(2ac = ab+bc\) by abc, we get \(\displaystyle\frac{2}{b} = \frac{1}{c}+\frac{1}{c}\) which yields our definition of the harmonic mean from above, \(\displaystyle b = \frac{1}{\frac{\frac{1}{c}+\frac{1}{c}}{2}}\).

For an example connected to music, if two notes are an octave apart, their frequencies have a ratio of 2:1. The harmonic mean of 1 and 2 is \(\displaystyle b = \frac{2ac}{a+c} = \frac{2(1)(2)}{1+2} = \frac{4}{3}\), which is a perfect fourth and sounds good with the octaves. (The arithmetic mean is a perfect fifth.)

There are lots of other neat properties of means.  Here is a sampling:

1. If b is the harmonic mean between a and c, then 1/c -1/b = 1/b -1/a so that 1/c, 1/b, 1/a form an arithmetical progression.

2. Let s be the side of a square inscribed within a triangle and having one side lying along the base of the triangle.  s is half the harmonic mean of the base of the triangle and the altitude of the triangle on the base.

3. Let s be the side of a square inscribed within a right triangle and having one angle coinciding with the right angle of the triangle.  s is half the harmonic mean of the legs of the triangle.

4. If s, a, b are chords of 1/7, 2/7, and 3/7 of the circumference of a circle, then s is half the harmonic mean of a and b.

5. If a car travels at the rate of r miles per hour from A to B and then returns at the rate of s miles per hour, the average rate for the trip is the harmonic mean of r and s.

This last example is the sort of “speed and velocity problem” Katie referred to.

Means as transformed averages

A 2002 question about the harmonic mean was added to the same page:

Can you give me a layman's definition for harmonic mean? It would also be helpful if you could give me an equation.

Doctor Schwa answered, again going far beyond the question:

Hi Katie,

To find the mean, or the "arithmetic" mean, you add up all the numbers and divide by how many numbers there are.

The other means, like "quadratic" or "harmonic," are similar.

For the quadratic mean, you square all the numbers, then take the usual average (add them up and divide by how many numbers there are), and then take the square root (un-squaring).

For the harmonic mean, which was your question, you first take the reciprocal of each number, then take the usual average, then take the reciprocal again (because reciprocal is the same as "un-reciprocal" ... the operation is its own inverse).

Observe that in each case, we do some transformation, take the average, and then undo the transformation. For the geometric mean, as we saw above, that transformation is a logarithm:

  • GM = antilog of average of logs
  • QM = square root of average of squares
  • HM = reciprocal of average of reciprocals
For example, if you want the harmonic mean of 10 and 20, you first take 1/10 and 1/20, find their average, which is 3/40, and then take the reciprocal of that, 40/3.

In algebra, the harmonic mean h of two numbers a and b is 

   1 / ( (1/a + 1/b) / 2),

or in other words 

   1/m = 1/2 (1/a + 1/b).

I hope that helps clear things up!

The quick formula for two, which was mentioned earlier, is $$HM = \frac{2ab}{a+b}$$

When is each used?

Let’s finish up with a 2006 question about the application of all these means:

Applications of Arithmetic, Geometric, Harmonic, and Quadratic Means

I'm wondering when to apply the arithmetic, geometric and harmonic means for a certain data set.  I have data that is widely spread. What technique will give me the correct results?

So. how do you decide which to use? Widely spread data may make it important to make the right choice, as means are likely to differ more in this case.

I answered, repeating ideas we’ve seen:

Hi Irfan -

The choice depends not so much on the spread of the data, but on their meaning, and specifically how the numbers naturally combine.

The basic idea is that a mean is a number that can be used in place of each number in a set, for which the NET EFFECT will be the same as that of the original set of numbers.  What determines which mean to use is the way in which the numbers act together to produce that net effect.

For example, if you are looking for a mean amount of rainfall, you note that the total amount of rain, which affects crop growth, etc., is found by ADDING the daily numbers; so if you add them up and divide by the number of days, the resulting ARITHMETIC mean is the amount of rain you could have had on EACH of those days, to get the same total.

If it makes sense to add the numbers, use the arithmetic mean.

If you have several successive price markups, say by 5% and then by 6%, and want to know the mean markup, you note that the net effect is to first MULTIPLY by 1.05 and then by 1.06, equivalent to a single markup of 1.05*1.06 = 1.113; taking the square root of this, if you had TWO markups of 5.499% each, you would get the same result.  This is the GEOMETRIC mean.  In general, you use it where the product is an appropriate "total"; another example is when you combine several enlargements of a picture.

If it makes sense to multiply, use the geometric mean.

If you want the mean speed of a car that goes the same distance (not time!) at each of several speeds, then the net effect of all the driving (the total time taken) is found by dividing the common distance by each speed to get the time for that leg of the trip, and then adding up those times.  The constant speed that would take the same total time for the whole trip is the HARMONIC mean of the speeds.  This amounts to the reciprocal of the arithmetic mean of the RECIPROCALS of the individual speeds.  In general, we use the harmonic mean when the numbers naturally combine via their reciprocals.  Another example is combining resistances in a parallel electrical circuit.

We saw last week how average speed, for speeds each driven for the same time, call for the arithmetic mean. But when the distances are the same, time equals distance divided by rate, so rates combine by adding their reciprocals, and we need the harmonic mean. In a parallel circuit, the combined effect is due to addition of currents, each of which is proportional to the reciprocal of a resistance (I = E/R).

The quadratic mean (which you didn't mention--also called "root-mean-square") is used in situations where it is the SQUARE of the values that matters; for example, electrical current squared is proportional to power, so where it is the total power (energy) that matters, you need the quadratic mean.

Here it’s the square that matters.

In summary, you use the

  ... arithmetic mean when numbers just add up
  ... geometric mean when numbers multiply together
  ... harmonic mean when the reciprocals of the numbers add up
  ... quadratic mean when the squares of the numbers add up

to produce the net effect you are interested in.

And if it’s something else that matters … you can invent your own mean!

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