The Language of Argument

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S t a t i s t i c a l G e n e r a l i z a t i o n s

then our confidence in his argument should drop to almost nothing. So, for statis-

tical generalizations, it is always appropriate to ask about the size of the sample.

Is the Sample Large Enough?

One reason we should be suspicious of small samples is that they can be

affected by runs of luck. Suppose Harold flips a Canadian quarter four times

and it comes up heads each time. From this, he can hardly conclude that

Canadian quarters always come up heads when flipped. He could not even

reasonably conclude that this Canadian quarter would always come up

heads when flipped. The reason for this is obvious enough: If you spend a

lot of time flipping coins, runs of four heads in a row are not all that unlikely

(the probability is actually one in sixteen), and therefore samples of this size

can easily be distorted by chance. On the other hand, if Harold flipped the

coin twenty times and it continued to come up heads, he would have strong

grounds for saying that this coin, at least, will always come up heads. In fact,

he would have strong grounds for thinking that he has a two-headed coin.

Because an overly small sample can lead to erroneous conclusions, we need

to make sure that our sample includes enough trials.

How many is enough? On the assumption, for the moment, that our sam-

pling has been fair in all other respects, how many samples do we need to

provide the basis for a strong inductive argument? This is not always an easy

question to answer, and sometimes answering it demands subtle mathemati-

cal techniques. Suppose your company is selling 10 million computer chips

to the Department of Defense, and you have guaranteed that no more than

0.2 percent of them will be defective. It would be prohibitively expensive to

test all the chips, and testing only a dozen would hardly be enough to reason-

ably guarantee that the total shipment of chips meets the required specifica-

tions. Because testing chips is expensive, you want to test as few as possible;

but because meeting the specifications is crucial, you want to test enough to

guarantee that you have done so. Answering questions of this kind demands

sophisticated statistical techniques beyond the scope of this text.

Sometimes, then, it is difficult to decide how many instances are needed to give

reasonable support to inductive generalizations; yet many times it is obvious,

without going into technical details, that the sample is too small. Drawing an

inductive conclusion from a sample that is too small can lead to the fallacy of

hasty generalization. It is surprising how common this fallacy is. We see a per-

son two or three times and find him cheerful, and we immediately leap to the

conclusion that he is a cheerful person. That is, from a few instances of cheerful

behavior, we draw a general conclusion about his personality. When we meet

him later and find him sad, morose, or grouchy, we then conclude that he has

changed—thus swapping one hasty generalization for another.

By making our samples sufficiently large, we can guard against distortions

due to “runs of luck,” but even very large samples can give us a poor basis for a

statistical generalization. Suppose that Harold has tried hundreds of times to use

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