Statistics Essentials For Dummies (19 page)

formula is
, where
is population standard deviation and

n
is sample size. In the next sections you see the effect each of has on the standard error.

Sample size and standard error

Because
n
is in the denominator of its formula, the standard error decreases as
n
increases. It makes sense that having more data gives less variation (and more precision) in your results.

A visual can help you see what's happening here with respect to gaining precision in
as
n
increases. Suppose
X
is the time it takes for a worker to type and send 10 letters of recommendation. Suppose
X
has a normal distribution with mean 5 minutes and standard deviation 2 minutes. Figure 6-2a shows the picture of the distribution of
X
.

Now take a random sample of 10 workers, measure their times, and find the average,
each time. Repeat this process over and over, and graph all of the possible results for all possible samples. Figure 6-2b shows the picture of the distribution of
. Notice that it's still centered at 10 (which we expected) and that its variability is smaller; the standard

error in this case is
. The average times are

closer to 10 than the individual times shown in Figure 6-2a. That's because average times for 10 individuals don't change as much as individual times do.

Now take random samples of 50 workers and find their means. This sampling distribution is shown in Figure 6-2c. The variation is even smaller here than it was for
n
= 10; the standard

error of
in this case is
. The average times here

are even closer to 10 than the ones from Figure 6-2b. Larger sample sizes mean more precision and less change from sample to sample.

Figure 6-2:
Distributions of a) individual times; b) average times for 10 individuals; c) average times for 50 individuals.

Population standard deviation and standard error

In the standard error formula for
,
you see that the