Application of AM, GM, HM, Median and Mode
Lecturer
Note: Application of AM, GM, HM, Median and Mode
Application
of AM-1
The arithmetic mean
is best used when the sum of the values is significant. For example, your grade
in your statistics class. If you were to get 85 on the first test, 95 on the
second test, and 90 on the third test, your average grade would be 90.
Why don't we use the
geometric mean here?
What about the
harmonic mean?
What if you got a 0
on your first test and 100 on the other two? The arithmetic mean would give you
a grade of 66.6. The geometric mean would give you a grade of 0!!! That's just
mean!
This is why we
generally require all values to be positive to use the geometric mean.
The harmonic mean
can't even be applied at all because 10 is undefined.
Let's face it, people
do get zeros on tests sometimes, so the geometric and harmonic means just
aren't very practical for calculating things like grades.
Could you give the
formula for the geometric mean for a series of numbers if I am trying to get
the compound annual growth rate for a series of number that include negative
numbers?
In general, you can
only take the geometric mean of positive numbers. The geometric mean of numbers
(IMAGE) is the nth root of the product (IMAGE) .
The
GM-1
The geometric mean is
best used when the product of the values is significant.
Suppose you have an
initial investment of $100. It yields a %20 return the first year, %40 return
the second year, and %60 return the third year. After three years, you have $100.00∗1.2∗1.4∗1.6=$268.80. The
arithmetic mean would suggest an average return of %40 per year. So after 3
years you'd expect to have $100∗(1.4)3=$274.40
. Uh-oh. That's not right.
The geometric mean,
however, indicates an approximate yearly return of 1.2∗1.4∗1.6−−−−−−−−−−−√3=. So after 3 years,
you'd expect to have $100∗(1.39)3=$268.80
, which is exactly right!
Note: the arithmetic
mean is strictly larger that the geometric mean for non-negative real numbers,
so be careful when a company tells you their investments average a %30 yearly
return, as they may be using an arithmetic mean to boost their numbers.
Finally, the harmonic
mean suggests an average yearly return of %38.08, so after 3 years, you'd
expect to have $100∗(1.38)3=$263.27
, which is less than you actually have.
So in a case like an
investment, you want to use the geometric mean.
It's worth mentioning
that the above examples give such similar results because I'm only using 3
numbers and they are very close to each other. The closer your numbers are to
each other, the closer the results of the different means will be. All means
will be equal if all the values are the same.
Application
of GM-2
In your example, you
are taking the mean of positive numbers. For example, if you're looking at an
investment that increases by 10% one year and decreases by 20% the next, the
simple rates of change are 10% and -20%, but that's not what you're taking the
geometric mean of.
At the end of the
first year you have 1.1 times what you started with (the original plus another
tenth of it). At the end of the second year you have 0.8 times what you started
the second year with (the original minus one fifth of it). So, the numbers you
are taking the geometric mean of are 1.1 and 0.8. This mean is approximately
0.938.
This means that, on
average, your investment is being multiplied by 0.938 (= 93.8%) each year, a
6.2% loss.
So, the compound anual growth rate is (approximately)
-6.2%.
Application
of GM-3
In the same way, the
geometric mean is relevant any time several quantities multiply together to produce
a product. The geometric mean answers the question, "if all the quantities
had the same value, what would that value have to be in order to achieve the
same product?"
For example, suppose
you have an investment which earns 10% the first year, 50% the second year, and
30% the third year. What is its average rate of return? It is not the
arithmetic mean, because what these numbers mean is that on the first year your
investment was multiplied (not added to) by 1.10, on the second year it was
multiplied by 1.60, and the third year it was multiplied by 1.20. The relevant
quantity is the geometric mean of these three numbers.
The question about
finding the average rate of return can be rephrased as: "by what constant
factor would your investment need to be multiplied by each year in order to
achieve the same effect as multiplying by 1.10 one year, 1.60 the next, and
1.20 the third?" The answer is the geometric mean (IMAGE) . If you
calculate this geometric mean you get approximately 1.283, so the average rate
of return is about 28% (not 30% which is what the arithmetic mean of 10%, 60%,
and 20% would give you).
Any time you have a
number of factors contributing to a product, and you want to find the
"average" factor, the answer is the geometric mean. The example of
interest rates is probably the application most used in everyday life.
Application
of GM-4
Ø In
Growth Rates
The geometric mean is
used in finance to calculate average growth rates and is referred to as the
compounded annual growth rate. Consider a stock that grows by 10% in year one,
declines by 20% in year two, and then grows by 30% in year three. The geometric
mean of the growth rate is calculated as follows:
((1+0.1)*(1-0.2)*(1+0.3))^(1/3)
- 1 = 0.046 or 4.6% annually.
Ø In
Portfolio Returns
The geometric mean is
commonly used to calculate the annual return on portfolio of securities.
Consider a portfolio of stocks that goes up from $100 to $110 in year one, then
declines to $80 in year two and goes up to $150 in year three. The return on
portfolio is then calculated as ($150/$100)^(1/3) - 1 = 0.1447 or 14.47%.
Ø In
Stock Indexes
The geometric mean is
also occasionally used in constructing stock indexes. Many of the Value Line
indexes maintained by the Financial Times employ the geometric average. 1In this type of
index, all stocks have equal weights, regardless of their market
capitalizations or prices. The index is calculated by taking the geometric
average of the percentage change in price of each of the stocks within the
index.
Application
of GM-5
In economic evaluation work, the geometric mean is often useful.
The large P represents multiplication, analogous to the S representing summation.
The GM is for situations we want the average used in a multiplicative
situation, such as the "average" dimension of a box that would have
the same volume as length x width x height. For example, suppose a box
has dimensions 50 x 80 x 100 cm.
A cubic box with sides of the GM=73.7 cm would enclose the same 400,000
cm3 or 0.4 m3 volume.
The customary economic evaluation
application is in determining "average" inflation or rate
of return across several time periods. In calculating the GM,
the numbers must all be positive. So, add 1 to each value before
calculating the GM and subtract 1 from the answer.
Suppose that your portfolio has these
five annual returns: .10, -.20, 0, -.10, .20. The
order does not matter if the portfolio has no contributions or withdrawals
during the five years. The return arithmetic average is 0. However,
a portfolio across five years with these annual returns would lose about 5% of
value. The geometric mean:
In checking:
(1.1)(0.8)(1.0)(0.9)(1.2) = 0.95 = (1-.010)5.
That is, a portfolio with this a GM = -.01 would lose .05 in
five years.
Application
of HM-1
The harmonic mean
isn't used as often as the other two, but can be very useful in certain cases.
For example:
Suppose you are
driving in manhattan. All the blocks there going in the same direction are
roughly the same length. You drive North 20 blocks at a speed of simph
on the ith block. Your average speed, S, is the harmonic
mean of your speeds:
S=201s1+1s2+...+1s20 .
The harmonic mean is
particularly sensitive to a single lower-than average value. So your average
speed usually won't be too much larger than your slowest speed.
If you were to use
the Arithmetic or Geometric means for this, you would get very different
results, far from your actual average speed.
Application
of HM-2
In
finance
The weighted harmonic
mean is the preferable method for averaging multiples, such as the
price–earnings ratio (P/E), in which price is in the numerator. If these ratios
are averaged using a weighted arithmetic mean (a common error), high data
points are given greater weights than low data points. The weighted harmonic
mean, on the other hand, gives equal weight to each data point.[7] The simple
weighted arithmetic mean when applied to non-price normalized ratios such as
the P/E is biased upwards and cannot be numerically justified, since it is
based on equalized earnings; just as vehicles speeds cannot be averaged for a
roundtrip journey.[8]
For example, consider
two firms, one with a market capitalization of $150 billion and earnings of $5
billion (P/E of 30) and one with a market capitalization of $1 billion and
earnings of $1 million (P/E of 1000). Consider an index made of the two stocks,
with 30% invested in the first and 70% invested in the second. We want to
calculate the P/E ratio of this index.
Using the weighted
arithmetic mean (incorrect):
P/E = 0.3*30 + 0.7*
1000 = 709
Using the weighted
harmonic mean (correct):
P/E = 
= 93.46
Thus, the correct P/E
of 93.46 of this index can only be found using the weighted harmonic mean,
while the weighted arithmetic mean will significantly overestimate it.
Application of HM-3
The harmonic mean is a better
"average" when the numbers are defined in relation to some
unit. The common example is averaging speed.
For example, suppose that you have four 10 km segments to your
automobile trip. You drive your car:
- 100 km/hr for the first 10
km
- 110 km/hr for the second 10
km
- 90 km/hr for the third 10 km
- 120 km/hr for the fourth 10
km.
What is your average speed? Here is a spreadsheet solution:
|
The harmonic mean formula is:
Application
of AM-GM-HM
A practical answer is
that it depends on what your numbers are measuring. Do some unit analysis and
consider the relationship between consecutive numbers in the series you’re
averaging.
If you’re measuring
units that add up linearly in a sequence (such as lengths, distances, weights),
then an arithmetic mean will give you a meaningful average. For example, the
arithmetic mean of the height or weight of students in a class represents the
average height or weight of students in the class.
If you’re measuring
units that add up as reciprocals in a sequence (such as speed or distance /
time over a constant distance, capacitance in series, resistance in parallel),
then a harmonic mean will give you a meaningful average. For example, the
harmonic mean of capacitors in series represents the capacitance that a single
capacitor would have if only one capacitor was used instead of the set of
capacitors in series.
If you’re measuring
units that multiply in a sequence (such as growth rates or percentages), then a
geometric mean will give you a meaningful average. For example, the geometric
mean of a sequence of different annual interest rates over 10 years represents
an interest rate that, if applied constantly for ten years, would produce the
same amount growth in principal as the sequence of different annual interest
rates over ten years did.
Does an arithmetic
mean of interest rates have any significance? As a number, sure. But as an
“average” interest rate it seems less intuitive because the principal it
produces at the end of ten years is much larger than the geometric mean.
Similarly, the harmonic mean of interest rates produces a smaller principal,
and so is less intuitive.
Now consider areas
and volumes as a test of understanding. What mean should we use to report the
“average” area or volume in a sequence of areas or volumes? Area is measured in
units of length squared. Volume is measured in units of length cubed. In a
sequence of areas or volumes, we could either add them up linearly and divide
or multiply them and take the roots — which is correct? It depends on what
we’re measuring. If these areas or volumes are dependent upon each other (e.g.,
the size of the same microbe at different times), then a geometric mean
probably makes more sense. If these areas or volumes are independent of each other
(e.g., the size of a house or pool), then an arithmetic mean probably makes
more sense.
Application
of AM-GM-HM-2
We have reviewed three different ways of
calculating the average or mean of a variable or dataset.
Each mean is appropriate for different types of
data; for example:
If values have the same units: Use the arithmetic
mean.
If values have differing units: Use the geometric
mean.
If values are rates: Use the harmonic mean.
The exceptions are if the data contains negative
or zero values, then the geometric and harmonic means cannot be used directly.
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