Demonstration of the Central Limit Tendency

 The user picks a probability distribution, and the program repeatedly computes the average of n outcomes with this distribution. Each time an average is computed, it is added to a histogram of values that grows over time. Because of the Central Limit Theorem, when n is taken to be large, the histogram that evolves comes to resemble the graph of a normal distribution.

 The program is initially configured to work with a distribution that has 50% probability of outcome "0" and 50% probability of outcome "1" (e.g. flipping a coin). Also, the avarage is initially over only a single trial, so that, as the program starts, it is plotting the histogram of outcomes of repeatedly performing a fair "0-1" distribution. Click "Go" to start the program, and observe how, as the results of trials accumulate, the histogram shows the number of "0" outcomes is roughly equal to the number of "1" outcomes. The graph will automatically rescale itself to accommodate the histogram as the number of trials grows. Use the "Settings..." button to bring up a pop-up panel that controls all the other settings of the applet. Changes only take effect after you press the green check button at the bottom of the panel. The rest of the instructions describe the features you can set on this pop-up panel. The "Distribution" tab: You can experiment with other probability distributions than 50-50. Click on the "Distribution" tab in the pop-up, and edit the probabilities in the right hand column. You can add more outcomes than just "0" and "1", by pressing the "+" button (or delete outcomes with the "-" button). The "View" tab: You can alter the horizontal scale of the graph by changing xMin and xMax to get a closer view of the histogram. The vertical scale adjusts itself automatically to accommodate the histogram. The "Sampling" tab: Initially the histogram displays the distribution of outcomes of repeatedly performing a single trial. Experiment with entering "2" into the "Number of samples to average" field. With the underlying probability still 50-50, this now shows the outcomes of repeatedly preforming two trials (flipping a coin twice, say) and averaging the results (where a head is a 0 and a tail is a 1). Of course the possible averages are 0, 1/2 and 1, and the three columns that you see in the histogram reflect that. Try setting the number of samples to be 5, 10, 20 or 100. In those cases there will be 6, 11, 21, or 101 strips between 0.0 and 1.0, corresponding to all the possible values of the averages. Observe how when n is large, the histogram evolves to a shape close to the normal curve. (It may help to restrict the x range by changing xMin and xMax on the "View" tab.) Observe how even if you start with a very skew distribution, when you make the number of samples being averaged big, the distribution of average values always ends up looking normal. (Remember you set the distribution in the "Distribution" panel, and you can always get a picture of the underlying distribution by running the program with the number of averages set equal to 1.)