This collection of Java applets allows users to explore important ideas in statistics and probability via games and simulation. The eight simulations available are Interactive Scatterplot, Interactive Histogram, Interactive Histogram with Error Graph, Galton Board Game, Monty Hall Game, Red and Black Game, Fourier Game, and Wavelet Game. Each simulation has an About button and an i button which explain the assumptions and directions associated with that applet.
Interactive Scatterplot allows users to create a scatterplot by clicking on the graph to add data points; the applet displays the coordinates of each data point, the regression line and equation, the mean and standard deviation of X and Y, the correlation, and the covariance. Both Interactive Histogram and Interactive Histogram with Error Graph allow users to create a histogram by clicking under the x-axis and to set the number of bins by moving the n slider. The first displays either the mean and standard deviation, median, or boxplot of the data, while the latter displays either the mean square error function or the mean absolute error function.
The Galton Board simulation allows users to choose which way a ball will fall through a triangular array of pegs by clicking the right or left arrow buttons. Monty Hall simulates a game where one of three doors conceals a car; users pick a door, and one of the other doors hiding a goat is opened. Users can then switch to the other unopened door; the applet keeps track of the users decision and the outcome. Red and Black simulates Gambler's Ruin, where the user plays a game until he runs out of money or reaches a target fortune. Users set the parameters, where x is initial wealth, a is the target fortune, v is the amount wagered, and p is the probability of winning a play of the game. The applet keeps track of wins, losses, and number of games played.
The Fourier and Wavelet games allow users to visualize the analysis and synthesis steps in the discrete Fourier transform and the discrete wavelet transform, respectively. This site is a part of the larger Statistics Online Computer Resource (SOCR) available at UCLA.
Type of Material:
Java applet simulations.
These simulations could be used as part of classroom demonstrations or in combination with teacher created worksheets or activities in an inquiry-based lesson.
Internet connection with a Java enabled browser.
Identify Major Learning Goals:
The goal of these applets is for students to explore and understand probability and statistical concepts including histograms, scatterplots, regression, and the binomial distribution and to experiment with game simulations in specific contexts.
Target Student Population:
This material reaches a wide population of interest. Several applets such as Monty Hall and Interactive Scatterplot are appropriate for high school AP statistics, others such as the Red and Black game are appropriate for undergraduate introductory statistics, while the Fourier and Wavelet games are better suited for more advanced students.
Prerequisite Knowledge or Skills:
1) Interactive Scatterplot basic knowledge of the least-squares line, correlation, and scatterplot;
2) Interactive Histogram & Interactive Histogram with Error Graph basic knowledge of histograms, mean, standard deviation, mean square error, and mean absolute error;
3) Galton Board, Monty Hall, & Red and Black basic probability knowledge;
4) Fourier & Wavelet games discrete Fourier and discrete wavelet transforms, respectively.
Evaluation and Observation
These demonstrations of some of the classic problems in statistics and probability give users the flexibility to explore different cases. For example, the scatterplot applet allows users to create a large variety of data points and see the least-squares line change with every added point. Similarly, the histogram applet lets users immediately see the associated effects of adding data. The Fourier and Wavelet applets have an impressive number of things users can alter and compare, and the Monty Hall game allows for two methods of playing. In the standard method, Monty knows where the car is and always opens a door with a goat, while the blind method allows Monty to pick a door at random. This comparison of the games gets at the heart of the paradoxical behavior of the game. Additionally, the applets are simple in their presentation, without an overwhelming number of buttons and sliders. These simulations offer tools for teachers to include in their lessons.
1) Minimal instructions are given for the applets under each applets About and i buttons. The notation is defined in mouse-overs, but could be misleading for some. For example, the histogram applet uses n for the number of interval classes (bins), while standard notation would reserve n for the number of data points. The Red and Black game can be difficult to decipher if users are unaware of the mouse-over definitions; I is the outcome of a particular play, X is the wealth at the end of a particular play, J is the outcome of the game (ruin vs. fortune), and N is the number of plays per game.
2) The Galton Board applet seems to function incorrectly. Although a large Galton board is shown, the ball does not fall past row 15, and once row 15 is reached the reset button above the applet does not work; users must use the reset button to the side. The instructions imply that the subset column is meant to represent the value of the binomial coefficient, but it gives the row number when the ball falls to the right and produces a square when the ball falls to the left.
3) The Play and Clear buttons on the Fourier and Wavelet games are not visible when the applet is first opened; users must slide the applet window to the left to see them. Also, it is unclear what the play button does, and therefore, unclear if it is functioning correctly.
4) The applets which represent game simulations (Galton Board, Red and Black, Monty Hall) only let the user run one simulation at a time. Users cannot automate multiple runs of the games to see what happens in the long run. For example, depending on the parameters, the Red and Black game may require a large number of clicks to reach a single outcome (ruin or fortune).
Potential Effectiveness as a Teaching Tool
The ease of use and simplicity of presentation of these applets provide great potential for instruction as demonstrations or as part of an assignment that the instructor creates. The scatterplot, histogram, and Monty Hall applets seem especially useful. The Fourier and Wavelet applets appear to be powerful visualization tools for an engineering or time series class.
In order to use these applets in an introductory statistics course, instructors would have to give a thorough introduction to the problems and notation as well as create an assessment activity to guide student thinking. There is concern over limited functionality on many of the applications which seem to be written for a specific class looking at one or two ideas.
Ease of Use for Both Students and Faculty
The applets load quickly and have quite a bit of flexibility, with an uncluttered, easy to use display and set up. All the applets have a great snapshot feature which allows the user to create a jpeg image of the output in the window. Brief explanations of the applets are provided under the About and i buttons for each applet. This collection is great for instructors who are interested in creating their own educational materials for technological applications.
There did not seem to be a way to copy the data into other software, and the notation was not always easy to decipher. The "Interactive Histogram" applet without the Error Graph has no instructions. If the applications are done in order, then users will see the "Interactive Histogram with Error Graph" first, which does have instructions.
In spite of the simplicity of the applet itself, the interface is not particularly intuitive and will take time to learn due to limited instructions.
Other Issues and Comments:
This collection of applets is useful for demonstrating basic statistical concepts; however, teachers will have to introduce some topics before assigning these applets.