Page 188 - Statistics for Dummies

172 Part III: Distributions and the Central Limit Theorem

Say that one person, Bob, is doing 50 rolls. What will the distribution of Bob’s
outcomes look like? Bob is more likely to get low outcomes (like 1 and 2) and
less likely to get high outcomes (like 5 and 6) — the distribution of Bob’s out-
comes will be skewed right as well.

In fact, because Bob rolled his die a large number of times (50), the distribu-
tion of his individual outcomes has a good chance of matching the distribu-
tion of X (the outcomes from millions of rolls). However, if Bob had only
rolled his die a few times (say, 6 times), he would be unlikely to even get
the higher numbers like 5 and 6, and hence his distribution wouldn’t look
as much like the distribution of X.
If you run through the results of each of a million people like Bob who rolled
this unfair die 50 times, each of their million distributions will look very simi-
lar to each other and very similar to the distribution of X. The more rolls they
make each time, the closer their distributions get to the distribution of X and
to each other. And here is the key: If their distributions of outcomes have a
similar shape, no matter what that similar shape is, then their averages will
be similar as well. Some people will get higher averages than 2 by chance,
and some will get lower averages by chance, but these types of averages get
less and less likely the farther you get from 2. This means you’re getting an
approximate normal distribution centered at 2.
The big deal is, it doesn’t matter if you started out with a skewed distribu-
tion, or some totally wacky distribution for X. Because each of them had
a large sample size (number of rolls), the distributions of each person’s
sample results end up looking similar, so their averages will be similar, close
together, and close to a normal distribution. In fancy lingo, the distribution
of is approximately normal as long as n is large enough. This is all due to the
Central Limit Theorem.
In order for the CLT to work when X does not have a normal distribution, each
person needs to roll their die enough times (that is, n must be large enough)
so they have a good chance of getting all possible values of X, especially those
outcomes that won’t occur as often. If n is too small, some folks will not get
the outcomes that have low probabilities and their means will differ from
the rest by more than they should. As a result, when you put all the means
together, they may not congregate around a single value. In the end, the
approximate normal distribution may not show up.

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