212 Fundamentals of Statistics
6 Continue working with the Population Parameters workbook
and close it when you’re finished. You do not have to save your
changes.These values illustrate an important statistical fact. If a sample is com-
posed of n random variables coming from a normal distribution with
mean m and standard deviation s, then the distribution of the sample av-
erage will also be a normal distribution with mean m but with standard
deviation s/!n. For example, the distribution of a sample average of 16
standard normal values is normal with a mean of 0 and a standard devia-tion of (^51) /" 16 514 , or .25.
The Standard Error
The standard deviation of x is also referred to as the standard error of x.
The value of the standard error gives us the information we need to deter-
mine the precision of x in estimating the value of m. For example, suppose
you have a sample of 100 observations that comes from a standard normal
distribution, so that the value of m is 0 and of s is 1. You’ve just learned that
x is distributed normally with a mean of 0 and a standard deviation of 0.1
(because 0.1 51 " 10 0).
Let’s apply this to what you already know about the normal distribution,
namely that about 95% of the values fall within 2 standard deviations of the
mean. This means that we can be 95% confi dent that the value of xwill be
within 0.2 units of the mean. For example, if x 5 5.3, we can be 95% con-
fi dent that the value of m lies somewhere between 5.1 and 5.5. To be even
more precise, we can increase the sample size. If we want xto fall within
0.02 of the value of m 95% of the time, we need a sample of size 10,000, be-
cause if x is 5.3 with a sample size of 10,000, we can be 95% confi dent that
m is between 5.28 and 5.32. Note that we can never discover the exact value
of m, but we can with some high degree of confi dence narrow the band of
possible values to whatever degree of precision we wish.The Central Limit Theorem
The preceding discussion applied only to the normal distribution. What
happens if our data come from some other probability distribution?
Can we say anything about the sampling distribution of the average in