The Mathematics of Money

Copyright © 2008, The McGraw-Hill Companies, Inc.

numerical values like averages which are used to summarize a large amount of data with

a single number.^1

Mean and Median

While the term “average” may be a familiar example of a statistical measure, matters are

not quite as simple as we might assume. There are actually several commonly used mea-

sures that may be meant when we use the term average, each of which has its pros and cons

in any given situation. The term measure of central tendency is a general term for any of

these “averages”.

The most familiar measure of central tendency, and what most people usually think of

when they hear the word “average,” is the mean. The mean of a set of numerical values

is found by adding all of the values up and then dividing by the number of values. For

example, suppose that you are considering opening a coffee shop and are doing some mar-

ket research about what other coffee shops in town charge for a large regular coffee. You

investigate five competitors, and find that their prices are:

Competitor Price

A $1.49

B $1.59

C $1.25

D $3.75

E $1.50

The mean price in town then would be:

Mean 

$1.49  $1.59  $1.25  $3.75  $1.50

__ 5
$1.92

While $1.92 is the mathematical “average” in the usual sense, if we look at the prices them-

selves we can see that it is not a particularly good estimate of what is “average” in the sense

of being typical; $1.92 is significantly higher than the prices charged by four of the five com-

peting coffee shops, and so if you based a pricing decision on the assumption that the market

would regard $1.92 as an “average” price you would be mistaken. The mean is being brought

up by one shop that charges a much higher price than the others. This shop may or may not

be successful with its higher pricing, but its pricing certainly is not typical. Its impact on the

mean makes this “average” a misleading measurement of what is typical.

An alternative measure of “average” is the median. The median of a set of values is

calculated by listing the values in increasing (or decreasing) order, and then choosing the

value in the middle of the list (if there are an odd number of values) or the mean of the

middle two values of the list (if there are an even number).

Putting our coffee prices in order,

$1.25, $1.49, $1.50, $1.59, $3.75

we can see that the median price is $1.50. Looking over the prices, we see that $1.50 is a

pretty good measure of a typical, middle-of-the-road price. The strength of the median is

that it effectively ignores the size of the highest and lowest values. If Competitor E raised

its price from $3.75 a cup to $5,000 a cup, the mean would shoot up ridiculously high, but

the median would not change at all.

(^1) Technically speaking, if the value is calculated on the basis of all relevant individuals, it is called a parameter. The
average salary mentioned here would be a parameter if it is calculated from the salaries of all pharmacists. If the
value is based on some sampling of individuals, it is called a statistic. The average mentioned would be a statistic if
it is calculated from, say, “500 pharmacists surveyed” or some other collection of pharmacists short of all of them.
The distinction between parameters and statistics is important, but for our purposes here it will not be necessary to
distinguish between the two. We will use the term statistical measure regardless of whether it is based on the whole
group or just a portion of the whole.
16.2 Measures of Average 617