Collaborative Exercise
Suppose eight of you roll one fair die ten times, seven of you roll two fair dice ten times, nine of you roll five fair dice ten times, and 11 of you roll ten fair dice ten times.
Each time a person rolls more than one die, he or she calculates the sample mean of the faces showing. For example, one person might roll five fair dice and get 2, 2, 3, 4, 6 on one roll.
The mean is [latex]\frac{\text{2 + 2 + 3 + 4 + 6}}{5}=3.4[/latex]. The 3.4 is one mean when five fair dice are rolled. This same person would roll the five dice nine more times and calculate nine more means for a total of ten means.
Your instructor will pass out the dice to several people. Roll your dice ten times. For each roll, record the faces, and find the mean. Round to the nearest 0.5.
Your instructor (and possibly you) will produce one graph (it might be a histogram) for one die, one graph for two dice, one graph for five dice, and one graph for ten dice. Since the “mean” when you roll one die is just the face on the die, what distribution do these means appear to be representing?
Draw the graph for the means using two dice. Do the sample means show any kind of pattern?
Draw the graph for the means using five dice. Do you see any pattern emerging?
Finally, draw the graph for the means using ten dice. Do you see any pattern to the graph? What can you conclude as you increase the number of dice?
As the number of dice rolled increases from one to two to five to ten, the following is happening:
- The mean of the sample means remains approximately the same.
- The spread of the sample means (the standard deviation of the sample means) gets smaller.
- The graph appears steeper and thinner.
You have just demonstrated the central limit theorem (clt).
The central limit theorem tells you that as you increase the number of dice, the sample means tend toward a normal distribution (the sampling distribution).