Demonstrating the central limit theorem using a TI-83 calculator

Georgia Journal of Science, 2002 by Lazari, Andreas, Goel, Sudhir, Kicey, Charles

ABSTRACT

In this article we show how the graphing calculator can be used to introduce the Central Limit Theorem of Probability and Statistics to the students. We start by giving the history and motivation behind the Central Limit Theorem. We give several examples related to the CLT and finally a TI-83 program demonstrates the CLT. The TI-83 program selects 30 integers between one and six computes the mean and stores it in a list. It repeats itself N times. At the end it gives the histogram plot for the N means to demonstrate the CLT.

INTRODUCTION

In its simplest form, the Central Limit Theorem (CLT) states that for random samples taken from a population that is not normal, the sampling distribution of the sample mean x(overscored) is approximately normal when the sample size n is large enough (n >= 30). A slightly more sophisticated way of stating the theorem is: The mean of n independent, identically distributed random variables approaches the normal distribution as n increases. This is one of the most famous theorems of probability and statistics. Before the discovery of this theorem, mathematicians treated Statistics and Probability as two unrelated disciplines. Our two statements of this theorem illustrate that the Central Limit Theorem serves as a result to unify these two disciplines.

Before introducing the CLT, a brief discussion of some of the rudiments of probability theory is in order. For example, suppose that a "fair" coin is flipped 100 times. What does it mean for the coin to be fair? Does it mean that we should get 50 heads and 50 tails? In our experience, we have seen many students quickly, without reflection, jump to this type of conclusion, when actually the probability of obtaining exactly 50 heads and 50 tails is quite small. Probability and chance, by their very nature, are elusive to us. All that we can say above is that if the coin is fair, then there is a high probability that there will be about 50 heads and 50 tails. Tossing a coin is an experiment, that is, a procedure that leads to results or outcomes that can be observed and recorded.

The law of large numbers tells us that we must repeat an experiment many times in order for probabilistic results to appear. For example, if we roll a fair die 6000 times, we expect to see each number appear approximately 1/6 == 16.7% of the time. However, if we only roll the die 12 times, then it is very likely that some of the outcomes will deviate significantly from the theoretical 16.7% frequency of occurrence.

Here, we have calculated the mean using the MATH submenu from the LIST menu.

Perhaps the simplest way of analyzing this experiment is by using a frequency distribution or a histogram. The power of a histogram relies on the adage "A picture is worth a thousand words." It gives us an instant visualization and a feel for the data list. For this particular list of data, we have set Xmin = -1, Xmax = 8, Ymin = -1, Ymax = 20, and Yscl = 1.

We have turned on STAT PLOT 1, using Histogram Type: (the symbol for Histogram) and Xlist L1. Of particular importance will be the Xscale. For this set of data, the natural choice is Xscale = 1, but later we will want to experiment with various settings.

CONCLUSION

A graphing calculator is one approach to help the students discover the Central Limit Theorem. This approach can be used in the classroom as an activity with discussion to help the students discover and have a much better understanding of this crucial theorem in probability and statistics.

BIBLIOGRAPHY

1. Deborah JB: Randomness. London: Harvard University Press, 1998.

2. Malcolm WB: "Coin-Tossing Computers Found to Show Subtle Bias." New York: The New York Times, 1993.

3. Calzada ME and Scariono SM: "What is Normal, Anyway?" Journal of Mathematics Teacher - Focus on Statistics 92: 682-89, 1999.

*Andreas Lazari Sudhir Goel and Charles Kicey

Department of Mathematics and Computer Science

Valdosta State University

Valdosta, GA 31698

(229) 333-7154 or (229) 333-5778

alazari@valdosta.edu