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Drawing from a Nonnormal Population

The histogram of n-member averages looks like the normal curve when the histogram of individuals (i.e. the population histogram) looks like the normal curve. What if the population histogram is not normal shaped? Answer: The histogram of averages may still look like the normal curve depending on the sample size. Consider rolling a regular six-sided die. Rolling a die is like drawing from a large population consisting of the numbers 1, 2, 3, 4, 5 and 6 in equal proportion. Then the histogram of individuals looks rectangular (Figure 6.3.).

**Figure 6.3:** Histogram of the outcome of rolls of a die
$\begin{figure} \begin{center} \epsfig{file=Uniform.ps, height=4.5in, width=2.5in, angle=-90}\end{center}\end{figure}$

Randomly draw groups of 9 members each, as before, and take the average of each group. This is the same as taking the average of 9 rolls of the die. What will the histogram of the averages look like? The averages will tend to include as many 6's and 5's as there are 1's and 2's, wouldn't it? Hence, the averages will tend to fall close to center, with central values more likely than outlying values. Check out the histogram of averages of 9 rolls, superimposed over the histogram for individuals in Figure 6.4.

**Figure 6.4:** individuals (n=1) versus averages (n=9). Note the `centering' or `central tendency' of averages.
$\begin{figure} \begin{center} \epsfig{file=unifnormal.ps, height=4.8in, width=2.7in, angle=-90}\end{center}\end{figure}$

Compare the histogram for averages of 9 rolls versus 25 rolls in Figure 6.5. The histogram for averages looks more and more like the normal curve as the number of rolls increase.

**Figure 6.5:** Histogram for averages (n=9 versus n=25). Note the smaller range for n=25.
$\begin{figure} \begin{center} \epsfig{file=annieside1.ps, width=2.70in, angle=-90}\epsfig{file=annieside2.ps, width=2.70in, angle=-90}\end{center}\end{figure}$

The behavior of the sample average in Figures 6.4 and 6.5 may be summarized in one of the better known theorems in statistics.

$\fbox{ \parbox{5.5in}{ \vspace*{1ex} {\bf The Central Limit Theorem:}\\ The his... ... averages equals the SD of individuals divided by $\sqrt{n}$ . \vspace*{1ex} } }$

The Central Limit Theorem is one theorem with three components. It is helpful to see how each component is reflected in the histograms in Figure 6.4. (i) the shape of the histogram of averages is approximately normal (even if the histogram of individuals is rectangular), (ii) both histograms are centered over the same value (i.e. the population mean ), and (iii) the histogram of averages gets narrower as sample size increases.

Example 2. Suppose that gross weekly rentals for Campus Video historically average $1200 with a standard deviation of $200.
a. There are 16 weeks this current semester. The weekly rental for this semester should average around $________ give or take $________ or so.
b. Estimate the chance that the average weekly rental for this semester will fall below $1000.
c. Estimate the chance that this week's rental will fall below $1000.
d. With 90% chance, the average weekly rental for this semester will exceed $________.

Next: Estimating the Population Mean Up: Sampling Distribution of the Previous: Behavioral Properties of the

2003-09-08