Three Kinds of Averages - How to Calculate

The Arithmetic Mean:

You know by now that the average or mean of a list of numbers is the sum of the numbers divided by the number of numbers.  It is a useful way to get a sense of the typical number in a list.  Quick! What's the average of 725, 726 and 727?  If you reached for a calculator, slap that hand.  "You mean I should add 725+726+727 and divide by 3 in my head?!" Heaven forbid.  There is a time and a place for calculators, and this isn't it.  In this case, if you mentally move 1 from the 727 to the 725 then you have three 726's.  Now if I asked you the average of 726, 726, and 726, please don't tell me that you would add them up and divide by 3!   The average is simply 726.

This trick can be used to mentally calculate the average for other more complicated lists.  For example, you take ten quizzes each scored out of 100, and your grades are: 71, 82, 95, 100, 68, 86, 91, 93, 81, 83.  At a quick glance, it looks like your average is about 80.  Maybe it is, maybe it isn't.  But that's a decent guess in any case.  Now let's see what the real average is by seeing how much over/under 80 each score really is.  71 is 9 under, 82 is 2 over etc.  Here are the under/over values:  -9, 2, 15, 20, -12, 6, 11, 13, 1, 3.  All together these add up to 50 over.  That means all the numbers together are 50 points higher than an 80 average.  If we take that 50 and distribute evenly to all ten quiz scores, we get 5 points per quiz, and that makes for 80+5 real average.  This isn't simple, but it is probably faster than adding up all ten numbers and dividing by 10.

This mean is usually called the arithmetic mean because there are two other kinds of mean that are commonly used: the geometric mean, and the harmonic mean. 

The Geometric Mean:

I drop into Mr. Waldman's office to brag about my investments.  I say that in the last three years my stocks have gone up 5%, 10%, and 15%.  He says that according to his investment statement, his portfolio has averaged 10% a year for the last three years.  If you take the arithmetic average of 5, 10, and 15, it seems that my average gain has been 10% a year, and that we are doing equally well with our investments.  But this time the arithmetic mean is a terrible indicator of typical, and it fails to accurately estimate the average gain.  I am actually making slightly less than 10% a year!  Huh?  Let's try an example.

Say I invest $1000, and I make 5%, 10%, and 15% for the next three years respectively.  That means after one year I have $1000 + .05($1000) = $1050.  After two years, I have $1050 + .1($1050) = $1155.  And after three years I have a grand total of $1155 + .15($1155) = $1328.25.

How does this compare to making 10% a year for each year?  After one year, I have $1100.  After two years, $1210.  And after three years, $1331.  This means that 10% a year is better than the sequence of 5%, 10%, and 15%!

What percent per year for three years would be the same as the sequence of 5%, 10%, 15%?  That's what we really want to know!  This will take some thinking.

When you gain 10% on your money, it's the same as multiplying your money by 1.10.  When you gain 5%, it's the same as multiplying by 1.05.  The idea is that you keep the money you have plus you add the extra percent.  The 1 is for keeping the money you had.  So if you started with 1000 and make 15%, you have 1000(1 + .15) = 1000(1.15).

So if you start with $1000 and make 5%, 10%, and 15% for the next three years, you will have $1000(1.05)(1.10)(1.15) = $1328.25.  Another way to look at it, is that (1.05)(1.10)(1.15) = 1.3825.  To find the real average, we want to find a number that when you multiply it by itself three times will give you 1.3825.  That number is called the cube root of 1.3825.  Now get your calculators out!  The cube root of 1.3825 is about 1.0992, which means that my average earnings per year is about 9.92%, slightly less than Mr. Waldman's.

One last example:  Some slick insurance salesman comes to your home and says "Hey sir, please buy my SLICKO investment, although it lost 40% last year, it went up 50% this year, and that gives you a total gain of 10% for two years, which is a solid average of 5% per year!"  You know this sounds fishy, but he is dressed well, and after all it makes sense that (50-40)/2 is 5.  But somewhere in your heart you remember the math class you had in middle school, and you say. "Hey let's try some numbers".  The salesman looks confused and tells you how pretty your furniture is, but it's too late for flattery.  You know that if you start with $1000, then after a loss of 40% you have $1000(1 - .40) = $600.  Then after a gain of 50% you have $600(1 + .50) = $900.  So after two years, you point out to him that you have lost 10% of your money, not gained!  He recovers to say "oh yes you are right of course, slip of the tongue, sorry, but it's still only a 5% loss each year".  Well now he has confirmed that he is an idiot and you can throw him out the front door.  

What is the real average loss per year?  In this example your money is multiplied by .6 and then by 1.5.  This is like multiplying your money by .9.  What number multiplied by itself makes .9?  Get out your calculator and calculate the square root of .9.  It's about .9486 and that is how much your money gets multiplied by each year on the average.  That corresponds to a 5.14% loss per year.

This kind of average is called the geometric mean, and it is used in situations where the numbers in the list are multiplied (rather than added) in order to calculate what you are interested in,  For bowling scores, we like to add up the scores for a total.  For investments, we like to multiply them to see our total gain or loss.

In general, given a list of n numbers, the geometric mean is calculated by multiplying the numbers together and taking the nth root.  It is definitely a task where a calculator will come in handy.

Harmonic Average:

I ride my bike to work and back every day.  It’s downhill there and uphill on the way back.  My speedometer says I average 18 mph going there, and 14 mph going back the same route.  What’s my overall average speed?

This is a trick problem.  Without thinking you quickly take the arithmetic mean of the two values and get the wrong answer of 16 mph. 

Intuitively it seems right, but it is wrong because you spend less time going 18 mph and more time going 14 mph, so the actual average speed should be closer to 14 than to 18.

Should we just take the geometric average? That is the square root of 14 times 18, which is about 15.87, and that is closer to 14 than it is to 18!  But why would that be right?  We hope it is right but we have no good reason why it should be, and in fact it is not right!

To correctly calculate the average speed, we first need to calculate the total time traveled.  We could do that if we knew the total distance traveled.  The total time is just the total distance divided by the speed.  For example, if you travel 20 mph for 40 miles, that takes 40/20 = 2 hours.   But how can we do this calculation if we don’t know the distance? 

Why weren't we told the distance?  Maybe the actual distance doesn’t matter, and the final average will come out the same no matter what the distance is! If that's true, we could just pick any distance.  Let's say 10 miles.  The total time going there is the distance 10 miles divided by 18 mph, which is 10/18 hours, and the total time back is 10/14 hours.   The average speed overall is the total distance divided by the total time.  This equals 20/(10/18 + 10/14) = 20/(5/9 + 5/7) = 20/ (80/63) = 1260/80 = 63/4 ≈ 15.75 mph.   

For those of you who are not afraid of algebra, the calculation using a distance of d, is:  2d/(d/18 + d/14).  Adding up the fractions (you can do it) gives 2d/(14d+18d)/(14×18) = (2d ×14×18)/d(14+18) = (2×14×18)/(14+18) ≈ 15.75 mph.   So the distance d doesn't effect the answer!  If we had been given two different speeds A and B, then the answer would have been simply: 2/(1/A + 1/B) =  2AB/(A+B).

Conclusion:

Three averages, each with a different slant.  I do not recommend that you memorize formulas to understand this stuff, because: 

1.  You are better off mastering the ideas with your own thoughts.  

2.  The formulas will not tell you which mean to use, or which numbers to plug in.

3.  There is no formula without understanding.

Nevertheless, if you insist and you understand the ideas underlying the formulas,  here are the formulas for calculating the different means for two numbers A and B:

 

Arithmetic Mean:  (A+B)/2

Geometric Mean:  √AB

Harmonic Mean:  2/(1/A + 1/B) or equivalently 2AB/(A+B)

For example, if A = 10 and B = 40, then the values of these averages are:  25, 20, and 16.

Which is the real average?  There is no such a thing – just the right average for the right job.  These three averages are called respectively, the arithmetic, geometric, and harmonic averages.  Don’t memorize them, just remember how to solve these three problems.

Problems: (Calculators are encouraged - for a change :-) )

1. Mr. Flipflop's  weight went up by 10% last year, and down 10% this year.  What % did his weight go up or down on the average over the last two years?
   
2. You are filling a swimming pool from two hoses.  One hose can fill the pool by itself in 18 hours.  The other can fill the pool in 24 hours by itself.  If you run them both simultaneously, how long does it take to fill the pool? (Hint: it is not 42 hours!  That's silly!)
3. You have a rectangle with length 49 and width 100.  What is the side of the square with the same area as the rectangle?
   
4. You bowl 100 games.  Your average for the first 30 games is 185.  Your average for the next 30 games is 175.  Your average for the last 40 games is 180.  What os your overall average for the 100 games?
5. You put $3000 in a mutual fund for 7 years for college.  The fund had the following performance for each of the seven years:  10%, 18%, 2%, -30%, 2%, 16%, and 26%.  What was the average performance of the fund over the 7 years?
6. You can hike a trail 3 hours and your friend can hike it in 5 hours.  If you start on opposite sides of the trail, how long does it take for you to reach each other?
7. You eat 2 Snickers bars every day except Shabbat and Sunday.  On Shabbat and Sunday you don't have any candy.  How many Snickers bars do you on the average each day of the week?
8. Prove using algebra that the harmonic average times the arithmetic average equals the square of the geometric average.
   

Exploring:

Note that all three of these averages generalize to the case when we have more than just two numbers.  For example, with ten numbers:

1. This is a hard problem because at first glance it seems that you do not have enough information to solve it.  However, it can be solved using the average tools you learned in this unit.  Two people walk toward each other from opposite ends of a hiking trail.   They both start at sunrise, and they meet somewhere in the middle at noon.  They both continue hiking, the slower one finishing at 9:00 PM and the faster one finishing at 4:00 PM.  What time was sunrise?  Hint:  There is a harmonic average problem in disguise which surprisingly turns into a geometric average problem.  It also requires some basic 8th grade alegbra I.
   


2. This problem explores the relationship between the three averages you learned in this unit.  In the example of 10 and 40, the arithmetic average is 25, the geometric average is 20, and the harmonic average is  16.  Do you think it is always true given any two numbers that the arithmetic average is the greatest, the harmonic average the least, and the geometric average in between?   If not, then find an example where it is not the case.  If you believe it is always true, can you prove it?  Hint:  Make a picture of a rectangle with sides 10 and 40.  Try to figure out how to manipulate the picture to construct the three different averages. 
   

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