dye

MISUNDERSTANDING OF THE NATURE OF STATISTICS

Example: President Eisenhower expressed astonishment and alarm on discovering that fully half of all Americans have below average intelligence.
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THe author is committing the fallacy himself.  President Eisenhower can rightly be astonished because it is average, not median.  It is possible for the median to be above or below the average if the population distribution is not symetric around the average.

dye

STATISTICS OF SMALL NUMBERS

Example: They say that 1 out of every 5 people is Chinese, but I’ve known hundreds of people and none of them is Chinese.
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Really?

Suppose it is true that 1/5 is Chinese.  Suppose you RANDOMLY select people and test if they are Chinese.

What is the probability of not finding even a single Chinese in your sample?
Sample size = 1, 80%
Sample size = 2, 80% x 80%=64%
...
Sample size = 100, 80%^100=0%

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The problem?  It is not the size of the sample (the small-ness) but the RANDOM-NESS of the sample.

dye

STATISTICS OF SMALL NUMBERS

How will it look like? (As I understand it)

Suppose a person randomly select 3 person in the world and they all happened to be male.  He concluded that there is no female in the world.

Why is it wrong?

Suppose the share of male-female is 50-50.  What is the chance the above scenario happening?  50%^3 = 12.5%.  Not very unlikely.

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Now suppose you randomly select 100 person in the world and they all happened to be male.  You concluded that there is no female in the world.

Suppose the share of male-female is 50-50.  What is the chance the above scenario happening?  50%^100 = 0%.  Very unlikely.

Suppose the share of male-female is 95-5.  What is the chance the above scenario happening?  95%^100 = 0.5%.  Very unlikely.

Suppose the share of male-female is 99-1.  What is the chance the above scenario happening?  99%^100 = 36.6%.  Not very unlikely.

You can see that the conclusion have some force in this case.  That, is the difference.

If he wish to give the conclusion a "full force", try randomly select 500 person in the world, see if they are all male.
Suppose the share of male-female is 99-1.  What is the chance they are all male?  99%^500 = 0.6%. Very unlikely.

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THe validity depends on the size sample as much as the conclusion to be made, as much as the sampling method, as much as the error allowed, etc..