# Statistical Demonstrations

## Measures of Center

Plop It! - Shodor
Click in the graph region and see how the mean, median, and mode are affected by adding data.

## Simulation

Birthday Simulator - Holmes
What is the chance that in a group of people, 2 or more will share the same birthday? Repeat the process several times for groups of n = 10, 20, 25, and 50.
Monty Hall Problem -
A prize lies behind one of three doors and a pig behind the other two. You pick one door and are shown a pig behind another door. You're then asked whether you want to stick with your original door or switch to the other unopened door. What should you do? Run the simulation several times by sticking with your original choice and several times with switching and compare the results. You can also change the number of doors and run the simulation several times with the switch or stay policy.
Chutes & Ladders - Jones
How many moves does an average game of Chutes & Ladders take? Perform this simulation several times, recording the number of moves for each time. Then find the average number of moves per game.
Simulation Project - Jones
This is a project that I wrote for a course I took in 2002. It explains the whole process of simulation and has five different simulations.

## Law of Large Numbers

Binomial Coin Experiment - Siegrist
Set the update size to 10 to speed up the process and then click the fast forward icon. Notice how the observed probabilities (red) approach the theoretical probabilities (blue) as the sample size increases. You can also change the number of coins or the probability of flipping a head. Almost any applet at the Virtual Laboratory in Probability and Statistics can be used to demonstrate the same concept.

## Sampling Distributions

Sampling Distribution - Lane
Sample from a normal, uniform, skewed, or custom distribution and see how the sample mean, median, variance, standard deviation, and range behave. According to the central limit theorem, no matter what the original distribution looks like, the sampling distribution of the means should become more normal as the sample size increases.

## Confidence Intervals

Confidence Interval Simulation - Lane
Although this simulation is written for means, it does demonstrate what the confidence level really means.

## Correlation

Guessing Correlations - Marden
You are given four different scatterplots and correlation coefficients and asked to match them.
Scatter, Correlation, and Regression - Stark
Turn on the regression line and then use the slider for r. Move it towards 1 and then towards -1. Notice how the closer the correlation coefficient is to 1 or -1, the closer the data is to the regression line. The little red square in the middle is the centroid, notice how the regression line always passes through the centroid. Check the "use added points" box and then add some points to the graph. Notice how points close to the line don't influence the graph much, but those far away and on the edges have a lot of leverage.
Regression by Eye - Lane (also under Regression)
Try to guess the correlation (five choices are given). Click "Show r" to find out if you're right. Then try to draw the line that best fits the data by clicking and dragging the mouse to create the line. The Mean Square Error (MSE) is shown and is a measure of how well the line fits the data. The smaller the mean square error, the better your line. You can ask it to show you the minimum MSE as a target and when you think you've gotten as good as you can get, click the box and see what the actual regression line is.

## Regression

Regression - West & Ogden
This applet allows you to see the influence of points with high leverage on the regression equation. It is limited to four points and then one that you add. The regression equation and value of r is given in black for the original four points and then the new equation and correlation coefficient are given in red for the five points once you click and add one. Click on different spots on the graph and see how points close to the line don't make much difference, but values away from the line, especially those towards the maximum or minimum x values, exert a lot of leverage.
Linear Regression - Stanton
Start clicking on the graph on the left and it will add a regression line for you. The green lines drawn between the points and the lines are the residuals and a graph of residuals vs x values are given on the graph to the right. Try making several patterns and notice how the residuals look. If the data has a nice linear pattern, then the residuals should look random. Now try making an exponential or quadratic curve and notice the patterns that appear in the residuals.
Regression by Eye - Lane (also under Correlation)
Try to guess the correlation (five choices are given). Click "Show r" to find out if you're right. Then try to draw the line that best fits the data by clicking and dragging the mouse to create the line. The Mean Square Error (MSE) is shown and is a measure of how well the line fits the data. The smaller the mean square error, the better your line. You can ask it to show you the minimum MSE as a target and when you think you've gotten as good as you can get, click the box and see what the actual regression line is.

## Additional Resources and Demonstrations

Virtual Laboratory in Probability and Statistics - Siegrist
This site at the University of Alabama, Huntsville, has integrated explanations and applications to many topics in statistics.
Project InterActivate - Shodor Educational Foundation
This site has more than just statistical applications and simulations. There are activities and tools for the students and most contain explanations and questions to ask. Many of these are geared for younger ages.
MathTools - Math Forum
This site contains a database of applications, tools, lesson plans, discussions, and stories for most areas of mathematics.