The Mathematics of Money

620 Chapter 16 Business Statistics

$2,766.50

__ 105

 $26.34

Notice that when we compare the three “averages” in this example, the weighted average

is lower than the mean or median. Neither the mean nor the median takes into account the

fact that the workers earning the higher wages are working fewer hours; this is taken into

account with the weighted average, and so it is lower. In this case, the weighted average is

the better representation of the average rate per hour that the company pays.

Indexes^2

An index is another type of (usually weighted) average. An index is a numerical value,

calculated according to some set formula, intended to numerically summarize some overall

situation. One familiar example is a market index, used to reflect the value of the stock

market (or some other financial market). The Dow Jones Industrial Average (the Dow) and

the Standard and Poor’s 500 (S&P 500) are two U.S. stock market indexes whose values

are reported in the daily news. The S&P 500 is based on an average of the prices of 500

large companies’ stocks, weighted by the overall market value of each company. The Dow

Jones Industrial Average is based on 30 of the largest publicly traded companies’ stock

prices. It is also a weighted average, though over the many years that this index has been

kept, the formula has become quite complicated; it has been adjusted many times to reflect

changes in which companies are included in the average.

Market indices also exist for foreign markets, or for companies engaged in certain types

of businesses, or for companies that meet some other criterion. There are indexes specifi-

cally for telecommunications companies or retailers, and indexes for companies that meet

certain guidelines for “socially responsibility.”

Indexes differ from averages, though, in that the formulas used to calculate them need not

correspond to any of the familiar average types. For example, suppose you are constructing a

very simple index to track the performance of five stocks. The stocks, and their prices, are:

Company Stock Price

A $25

B $45

C $30

D $10

E$ 5

You could construct an index by taking the straight average of the prices of these stocks,

in which case your index would now stand at 23.00. Or, your index formula could be to

add up all the stock prices and use that number, in which case your index would now stand

at 115.00. Or you could use a formula that weights Company A and Company B twice as

heavily as the others, calculates a weighted average, and then divides the result by 0.0175.

(Never mind asking why you might be using this formula; you could use it if you wanted

to.) In that case your index would be 1510.20. Any formula that calculates a numerical value

based on these stocks’ prices could be used as a formula for an index of these prices.

Because the value of an index depends on the formula used to calculate it, and the formulas

can vary so wildly, the value of a market index looked at in isolation does not necessarily tell you

much. If the GlobalInvestexx CleanTech 75 Index stands at 24,535.72, does that mean that stocks

of the companies it tracks are doing better or worse than the stocks in the Dow Jones Industrial

Average, which stands at 12,000? Without knowing anything about how those indexes are cal-

culated, it is impossible to say. One may be higher than the other because of differences in the

stocks, or one may be higher than the other simply because of how the formula is calculated.

(^2) The technically correct plural of “index” is “indices.” However, “indexes” is now in common use, and so we will follow
modern usage and feel free to use either term.