Behavioral Properties of the Sample Average

What percentage of adult men are between 5'6" and 6' tall? Population surveys have shown that men's heights are approximately normally distributed with mean 5'9" and SD 3". Thus the percentage of men between 5'6" and 6' is estimated as 68%, the percentage within 1 SD of the mean. See Figure  6.1

  

Figure 6.1: Percentage of men's heights between 66 and 72 inches

If the population of men are randomly assigned into groups of 9, and the average heights are computed for each group, what percentage of groups average between 5'6" and 6' in height? Is the answer approximately 68%? No. In fact, more than 99% of the groups will average between 5'6" and 6', even though only 68% of individuals do. Why? Because averages tend to include tall, short and medium heights - therefore averages tend to fall closer to middle than individuals. (Think about this: We put in a hat the names of all the men in class. You will win $20 if the names you draw average over 6 feet tall. Would you rather draw 1 or 2 names? What are you chances of winning if you draw 9 names?)

Figure  6.2 shows the histogram of averages ,     superimposed over the histogram for individuals.

  

Figure 6.2: Histogram of individual heights (solid line) and 9-men averages (dashed line)

The main result of this chapter explains the different look of the two histograms. We say it in words first:

  

Now we say it in statistical notation.

The symbol `` $\sim N(\;\cdot\; , \; \cdot \;)$'' is read ``having normal shaped histogram with mean equal to ...(whatever is written before the comma) and SD equal to ...(whatever is written after the comma).''. If we were so inclined to compute average height for as many n-member samples as we can draw from the population, and if we took this big long list of averages and printed a histogram in Excel, the Main Result says we will observe 3 things: (i) this histogram of averages will look like a bell curve, (ii) with mean equal to the mean for individuals, (iii) and much smaller SD than the SD for individuals.

When we first talked about the population    mean and SD, we did not mention (because it seems obvious) that these are the mean and SD for individuals. If male heights average 5'9'' with SD 3'', then approximately 68% of individuals fall between 5'9'' $\pm$ 3''. At this point, it may be helpful to think of individual heights as a 1-member average. If we took 9-member averages instead, approximately 68% will fall between 5'9'' $\pm$ 1.0'', because the SD of 9-member averages equals the SD of individuals (or 1-member averages) divided by $\sqrt{9}$. Furthermore, approximately 68% of 25-member averages will fall between 5'9'' plus or minus ________ inches?. Answer: $3/\sqrt{25}=.6$ inches.

Exercise 1
Recall that men's heights are normally distributed with mean 5'9" and SD 3".
a. What percentage of men are over 5'11" tall?
b. Select a man at random. What is the probability that he is over 5'11" tall?
c. If the average height were calculated for all possible samples of size 9 that can be taken, what percentage of averages will exceed 5'11"?
d. Given one randomly selected sample of size 9, what is the probability that the average height will exceed 5'11"?
e. Given a sample of size 25, what is the probability that the average height will exceed 5'11"?
f. Given a sample of size 9, the average height of the sample will exceed ______ with probability .90.
g. 90% of samples of size 9 will have average height exceeding ______.