622 Chapter 16 Business Statistics
his budget. An index number for the cost of living in most cities would include the cost
of owning and operating a car. The costs (parking, insurance, etc.) of operating a car in
Manhattan are extremely high, but many people living there don’t own cars. They may
find it easier and more cost-effective to get around by using cabs and public transportation.
So, how much should the cost of owning and operating a car be counted in an index for
Manhattan? The compiler of the index cannot possibly know whether or not someone using
that index would choose to keep her car when moving from Raleigh to Manhattan.
Along the same lines, in some areas the cost of owning real estate is much higher than
renting (or vice versa), whereas in others the costs are much closer aligned. If you are a
renter, you would want the index numbers to reflect the costs of rent, whereas if you are
an owner, you would want the index to reflect the cost of owning. These, and many other
related factors, can make comparative indexes troublesome if taken too literally. Still, they
do provide a way to quickly make rough comparisons.
Cost-of-living index numbers can be readily found at the library or over the Internet.
Some Internet sites can create specialized indexes to reflect your circumstances, by auto-
matically calculating an index based on the answers given to questions on an interactive
quiz. As with any Internet resources, though, you should be careful in assessing just how
credible the source of the information is before relying on it too heavily.Expected Frequency and Expected Value
Suppose that you are managing a company that manufactures mobile phones. You know that
despite your best quality control efforts some of the phones will turn out to be defective. In
order to effectively manage your business, you would want to have some sense of how many
of your phones will turn out to be defective, to help determine what sort of warranty you are
going to offer and how much you should budget for the costs of returns and repairs.
Of course, there is no way to know in advance how many phones will turn out to have
defects, but still you could make a reasonable attempt to put a good estimate on this.
One logical approach would be to look at what your company’s experience has been
with similar phones made in the past. For example, suppose that in the past year your com-
pany has manufactured 184,300 similar phones, and 5,724 have been returned for repair or
replacement. You should be able to use this information as the basis of an educated guess
about our new phones.
The relative frequency (often simply referred to* as the frequency) of an event hap-
pening is the fraction (or percent) of the time that this event actually has happened. In the
example we are considering, we would say that the relative frequency of defective phones
is 5,724 out of 184,300, or 5,724/184,300, or 3.11%.
The expected relative frequency of an event occurring is the fraction (or percent) of the
time that we predict that a future event will happen. The expected relative frequency may
be only a guess, since we cannot know the future, but even so it should be an educated
guess based on the best information available. In our mobile phone example, it might be
reasonable to take 3.11% as our expected relative frequency, since that is what we have
seen with similar products in the past. If you have good reason to believe that our quality
control has improved, using a slightly lower percent might be justified. On the other hand,
if you want to be conservative and make sure that you do not underestimate the defective
phones (and hence not adequately budget for the cost of repairs and replacement) you
might want to use a somewhat higher percent, such as 3.25% or even higher.
The expected frequency of an event occurring is the number of times we expect it to
actually occur. Expected frequency is very similar to expected relative frequency, except
that frequency is a number while relative frequency is a fraction or percent. In our mobile
phone example, the expected frequency would be the number of defective phones that you
expect. Expected frequency can be found by multiplying:Expected frequency (Expected relative frequency)(Total number of items)* incorrectly