Law of Averages

There are several choices to make:

1. Are the draws with or without replacement? Choose "w rep" for drawing with replacement, and choose "w/o rep" for drawing without replacement.
2. How quickly should the tickets be drawn? The choices are
1. Fast: Which may not be very fast, depending on the environment
2. Slow: Which is useful for seeing each draw and its effect
3. Batch: This is the fastest. The draws are performed without updating the graphics after each draw, but just after all draws have been made.
3. How many draws to perform? The choices are 1, 10, 25, 100, and 1000. Click on the button that you wish. You can click on the buttons more than once to request more draws. For example, if you wish to have 250 draws, click on "100" twice and "25" twice. (Don't click on them too quickly or they won't all register.) Right below the numbered buttons, there is a number showing how many draws have been requested so far.
4. How many more draws to perform? Even after the draws have been completed, you can ask for more draws by clicking on the numbered buttons again.
5. Start Over: This button sets everything back to 0. You can then make your choices again.
6. Pause: This button stops the drawing. You can restart the drawing from where it left off by pressing "Resume" (which is the same button), or you can hit "Start Over" to start over.

There are several results to watch:

• The running averages: In the middle, there is a graph named "Average." After each draw, the average of the draws is calculated (see the statistics table), and this average is plotted on the graph, where the horizontal axis gives which draw it is, and the vertical axis gives the average. The light blue line on the graph indicates the average in the box.
• The running sums: At the botton, there is a graph named "Sum - Expected Value." After each draw, the sum of the draws, the expected value of the sum of the draws, and the difference between the sum and the expected value of the sum (the error in the statistics table) are calculated, and this error is plotted on the graph, where the horizontal axis gives which draw it is, and the vertical axis gives the error. The light blue line on the graph indicates 0.
• A record of the draws: At the lower right, the draws are recorded as they are being made.
• Histograms: In the middle of the screen, there are two histograms. The one on the left is the histogram for the tickets in the box. It remains the same as the tickets are being drawn. The histogram on the right is that for the tickets that have been drawn so far.
• The statistics: At the upper right, some basic statistics, about the box and about the draws, are collected. The columns:
• Numbers: The "Draws" entry is the number of tickets drawn so far. The "Box" entry is the number of tickets in the box.
• Average: The "Draws" entry gives the average of the draws so far, and the "Box" entry has the average of the tickets in the box.

The "Error" entry is the difference between the average of the draws and average of the box.
The "SE" entry is the standard error of the average of the draws, which gives an idea of the magnitude of the "Error".
• Sum: The "Draws" entry gives the sum of the draws so far, while the "Box" entry gives the Expected Value of the sum of the draws so far, which is (# of Draws)x(Average of Box).

The "Error" entry is the difference between the sum of the draws and the expected value of the sum of the draws.
The "SE" entry is the standard error of the sum of the draws, which gives an idea of the magnitude of the "Error".
• SD: The "Draws"  entry is the standard deviation of the draws so far, and the "Box" entry is the standard deviation of the tickets in the box.