Page 184 - Statistics for Dummies

168 Part III: Distributions and the Central Limit Theorem

Why is having more precision around the mean important? Because some-
times you don’t know the mean but want to determine what it is, or at least get
as close to it as possible. How can you do that? By taking a large random
sample from the population and finding its mean. You know that your sample
mean will be close to the actual population mean if your sample is large, as
Figure 11-2 shows (assuming your data are collected correctly; see Chapter 16
for details on collecting good data).

Population standard deviation

and standard error

The second component of standard error involves the amount of diversity
in the population (measured by standard deviation). In the standard error
formula you see the population standard deviation, , is in the
numerator. That means as the population standard deviation increases,
the standard error of the sample means also increases. Mathematically this
makes sense; how about statistically?

Suppose you have two ponds full of fish (call them pond #1 and pond #2), and
you’re interested in the length of the fish in each pond. Assume the fish lengths
in each pond have a normal distribution (see Chapter 9). You’ve been told that
the fish lengths in pond #1 have a mean of 20 inches and a standard deviation
of 2 inches (see Figure 11-3a). Suppose the fish in pond #2 also average 20
inches but have a larger standard deviation of 5 inches (see Figure 11-3b).

Comparing Figures 11-3a and 11-3b, you see the lengths for the two populations
of fish have the same shape and mean, but the distribution in Figure 11-3b
(for pond #2) has more spread, or variability, than the distribution shown in
Figure 11-3a (for pond #1). This spread confirms that the fish in pond #2 vary
more in length than those in pond #1.

Now suppose you take a random sample of 100 fish from pond #1, find the
mean length of the fish, and repeat this process over and over. Then you do
the same with pond #2. Because the lengths of individual fish in pond #2 have
more variability than the lengths of individual fish in pond #1, you know the
average lengths of samples from pond #2 will have more variability than the
average lengths of samples from pond #1 as well. (In fact, you can calculate
their standard errors using the formula earlier in this section to be 0.20 and
0.50, respectively.)

Estimating the population average is harder when the population varies a lot
to begin with — estimating the population average is much easier when the
population values are more consistent. The bottom line is the standard error
of the sample mean is larger when the population standard deviation is larger.

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