Translations

Code	Language		Translator	Run

Credits

Wolfgang Christian; Loo Kang Wee

Source: Excerpts from the Open Educational Resources / Open Source Physics @ Singapore website.

Overview:

This document describes a JavaScript HTML5 simulation of a Galton Board, a physical device used to demonstrate probability, statistics, and the normal distribution. The simulation, hosted on the Open Educational Resources / Open Source Physics @ Singapore website, provides an interactive way to visualize these concepts.

Key Concepts and Functionality:

• Galton Board Simulation: The core of the resource is a simulation of a Galton Board. The description explains: "This is a simulation of a Galton Board, a vertical board with n rows of pegs onto which a ball is dropped. Every time the ball hits a peg, it has a probability p of bouncing to the left, and a probability 1-p of bouncing to the right."
• Probability and Random Variables: The simulation is tied to the concepts of probability and discrete random variables. The document lists these as key topics.
• Normal Distribution: The simulation demonstrates how repeated iterations of the Galton Board experiment, with equal probabilities of bouncing left or right, approximate a binomial distribution and, with enough trials, a normal distribution. "If p and 1-p are of equal values, then over repeated iterations of the Galton Board simulation, the binomial distribution should form."
• Histogram Representation: The simulation visually represents the results by collecting the balls into bins and displaying the frequency of balls in each bin as a histogram. This provides a direct visual link between the simulation and the resulting distribution.
• Embeddable: The simulation is designed to be easily embedded in other webpages using an iframe.

Target Audience & Learning Goals:

The resource is designed for teachers and students interested in exploring probability, statistics, and the normal distribution. The "Sample Learning Goals" section (represented as "[text]" in the source) and "For Teachers" sections suggest that the simulation is intended to facilitate understanding of these concepts.

Technical Details:

• Technology: The simulation is built using JavaScript and HTML5.
• Authors: Wolfgang Christian and Loo Kang Wee are credited as the creators.

Related Resources:

The document also includes a very extensive list of other interactive resources and projects available on the Open Educational Resources / Open Source Physics @ Singapore website. These cover a wide range of topics in physics and mathematics, often using interactive simulations. A few examples from the list:

• Bohr's Theory of the Hydrogen Atom JavaScript Simulation Applet HTML5
• Simple Electric Circuit JavaScript Simulation Applet HTML5
• Projectile Motion with System of Masses and Spring JavaScript HTML5 Applet Simulation Model by Darren Tan

Key Themes:

• Interactive Learning: The resource emphasizes interactive simulations as a method for learning complex scientific and mathematical concepts.
• Open Educational Resources: The document is hosted on a website dedicated to open educational resources, indicating a commitment to free and accessible educational materials.
• Visualization: The Galton Board simulation specifically provides a powerful visual representation of abstract statistical concepts.

Potential Use Cases:

• Classroom Demonstrations: Teachers can use the simulation to illustrate probability, statistics, and the normal distribution in a dynamic and engaging way.
• Student Exploration: Students can use the simulation to experiment with different parameters (though none are mentioned explicitly in this document beyond the left/right probability) and observe the resulting distributions.
• Online Learning: The embeddable nature of the simulation makes it suitable for inclusion in online courses and learning modules.

Galton Board: A Study Guide

I. Key Concepts

• Galton Board (Bean Machine): A physical device demonstrating the binomial distribution and its relationship to the normal distribution. Balls are dropped onto a staggered array of pegs, bouncing left or right with equal probability at each peg.
• Binomial Distribution: A discrete probability distribution that describes the probability of obtaining a number of successes in a sequence of independent trials, each of which yields success or failure. In the Galton Board, each row of pegs represents an independent trial.
• Normal Distribution (Gaussian Distribution): A continuous probability distribution that is symmetrical and bell-shaped. It is often observed as the limiting distribution of the binomial distribution when the number of trials is large.
• Probability (p): The likelihood of a particular event occurring. In the Galton Board, 'p' represents the probability of a ball bouncing to the left at each peg. Typically, for demonstration purposes, the probability is set to 0.5, thus illustrating a bell curve.
• Discrete Random Variable: A variable whose value can only take on a finite number of values or a countably infinite number of values. The number of balls falling into each bin at the bottom of the Galton Board is a discrete random variable.
• Histogram: A graphical representation of the distribution of numerical data. In the context of the Galton Board, the histogram displays the frequency of balls collected in each bin.
• Simulation: A computer-based representation of a real-world system or process. The Galton Board applet is a simulation of the physical device.
• Open Educational Resource (OER): Freely accessible and openly licensed educational materials, including simulations, that can be used for teaching, learning, and research.

II. Short-Answer Quiz

1. What does the Galton Board demonstrate? The Galton Board demonstrates the binomial distribution and how, with a large number of trials (balls dropped), it approximates the normal distribution. The device visually represents how randomness at a micro level can create predictable patterns at a macro level.
2. What is the significance of the probability p in the Galton Board simulation? The probability p represents the likelihood of a ball bouncing to the left at each peg. When p = 0.5, the binomial distribution is symmetrical, and the resulting histogram approximates a normal distribution centered in the middle.
3. How does the Galton Board relate the binomial distribution to the normal distribution? As the number of rows of pegs (number of trials) in the Galton Board increases, the shape of the resulting histogram approximates the normal distribution, even though each individual bounce is governed by a binomial probability. The normal distribution can, therefore, be considered an approximation of the binomial distribution.
4. What is the purpose of collecting the balls into bins at the bottom of the Galton Board? Collecting the balls into bins allows for the visualization of the frequency of balls falling into each bin. The distribution of balls across the bins forms a histogram that approximates the binomial distribution.
5. What type of variable is represented by the number of balls collected in each bin? The number of balls collected in each bin represents a discrete random variable. The number can only be an integer value (i.e., a whole number).
6. Explain the role of the histogram in the Galton Board simulation. The histogram displays the distribution of the balls collected in each bin. By doing so, it allows a visual assessment of how the overall distribution approximates the binomial and/or the normal distributions.
7. What is an Open Educational Resource (OER)? An OER is a freely accessible, openly licensed educational material, available for use and adaptation, including for teaching, learning, and research. The Galton Board applet is an example of an OER.
8. Describe one way teachers can utilize the Galton Board simulation in the classroom. Teachers can use the Galton Board simulation to visually demonstrate probability, statistics, the binomial distribution, and its relationship to the normal distribution in an engaging and interactive way. Students can experiment with parameters (such as p) and observe the effects on the distribution.
9. What is the purpose of the "Embed" code provided with the Galton Board simulation? The "Embed" code allows users to integrate the Galton Board simulation directly into other web pages or learning management systems. By doing so, it allows for easy incorporation of the interactive tool into online educational resources.
10. Name one real-world phenomenon that can be modeled using the principles demonstrated by the Galton Board. The distribution of human traits, like height or weight, in a large population can be modeled using the normal distribution, which is approximated by the Galton Board. The Galton Board principle can also be applied to modeling the spread of disease, or genetic inheritance.

III. Answer Key

1. The Galton Board demonstrates the binomial distribution and how, with a large number of trials (balls dropped), it approximates the normal distribution. The device visually represents how randomness at a micro level can create predictable patterns at a macro level.
2. The probability p represents the likelihood of a ball bouncing to the left at each peg. When p = 0.5, the binomial distribution is symmetrical, and the resulting histogram approximates a normal distribution centered in the middle.
3. As the number of rows of pegs (number of trials) in the Galton Board increases, the shape of the resulting histogram approximates the normal distribution, even though each individual bounce is governed by a binomial probability. The normal distribution can, therefore, be considered an approximation of the binomial distribution.
4. Collecting the balls into bins allows for the visualization of the frequency of balls falling into each bin. The distribution of balls across the bins forms a histogram that approximates the binomial distribution.
5. The number of balls collected in each bin represents a discrete random variable. The number can only be an integer value (i.e., a whole number).
6. The histogram displays the distribution of the balls collected in each bin. By doing so, it allows a visual assessment of how the overall distribution approximates the binomial and/or the normal distributions.
7. An OER is a freely accessible, openly licensed educational material, available for use and adaptation, including for teaching, learning, and research. The Galton Board applet is an example of an OER.
8. Teachers can use the Galton Board simulation to visually demonstrate probability, statistics, the binomial distribution, and its relationship to the normal distribution in an engaging and interactive way. Students can experiment with parameters (such as p) and observe the effects on the distribution.
9. The "Embed" code allows users to integrate the Galton Board simulation directly into other web pages or learning management systems. By doing so, it allows for easy incorporation of the interactive tool into online educational resources.
10. The distribution of human traits, like height or weight, in a large population can be modeled using the normal distribution, which is approximated by the Galton Board. The Galton Board principle can also be applied to modeling the spread of disease, or genetic inheritance.

IV. Essay Questions

1. Discuss the limitations of using the Galton Board as a model for real-world phenomena. What assumptions does it make, and how might these assumptions not hold in practice?
2. Explain how varying the probability p (of a ball bouncing left or right) affects the resulting distribution in the Galton Board simulation. What happens when p is significantly different from 0.5, and what does this demonstrate?
3. How does the Galton Board simulation contribute to the understanding of statistical concepts compared to traditional textbook-based instruction? Consider aspects of visual learning, interactivity, and exploration.
4. Analyze the impact of Open Educational Resources like the Galton Board simulation on accessibility and equity in education. Discuss the potential benefits and challenges associated with their widespread adoption.
5. Design a lesson plan that incorporates the Galton Board simulation to teach students about probability, distributions, and the central limit theorem. What activities would you include, and what learning outcomes would you target?

V. Glossary of Key Terms

• Bin: A container or category used to collect and group data. In the Galton Board, bins collect the balls after they pass through the pegs.
• Central Limit Theorem: States that the distribution of sample means approximates a normal distribution as the sample size gets larger, regardless of the population's distribution. The Galton Board provides a visual representation of the Central Limit Theorem because the histogram approximates the normal distribution.
• Equal Values: Having identical or equivalent numerical value. In the Galton Board simulation, an equal probability (p = 0.5) of bouncing left or right at each peg means equal values.
• Frequency: The number of times a particular value or event occurs. The histogram of the Galton board simulation displays the frequency with which the balls fall into each bin.
• Iteration: One single pass through a procedure or algorithm. In the Galton Board simulation, each time a ball is dropped, it is one iteration.
• Peg: A short cylindrical piece of wood, metal, or plastic used to fasten things together or to hang things on. The balls bounce off the pegs in the Galton Board.
• Random Variable: A variable whose value is a numerical outcome of a random phenomenon.
• Vertical Board: A board that is positioned so that it stands on its end or side rather than lying flat. The Galton Board has a vertical board.

Sample Learning Goals

[text]

For Teachers

This is a simulation of a Galton Board, a vertical board with n rows of pegs onto which a ball is dropped. Every time the ball hits a peg, it has a probability p of bouncing to the left, and a probability 1-p of bouncing to the right.

Research

[text]

Video

 The Galton Board by D!NG

 Version:

1. http://weelookang.blogspot.sg/2016/02/vector-addition-b-c-model-with.html improved version with joseph chua's inputs
2. http://weelookang.blogspot.sg/2014/10/vector-addition-model.html original simulation by lookang

Other Resources

[text]