Content, Philosophy, and Goals
Overview
Prerequisites
Design criteria and implementation
Advantages of Javascript over proprietary Statistics Packages
Suggestions for evaluating the materials
Acknowledgments

## Content, Philosophy and Goals

This text was written for a "terminal" introductory class in Statistics suitable students in Business, Communications, Economics, Psychology, Social Science, or liberal arts; that is, this is the first and last class in Statistics for most students who take it. It also covers logic and reasoning at a level suitable for a general education course. Accordingly, the text is not geared toward theory, numerical analysis, or sophisticated formulae; neither does it contain a bestiary of techniques or named probability distributions. Rather, I hope to help students to think logically about quantitative evidence and to translate real-world situations into mathematical questions; and to expose students to a few important statistical and probabilistic concepts and to some of the difficulties, subjective decisions, and pitfalls, in analyzing data and making inferences from numbers. The text develops probability, estimation, and inference using counting arguments: there is no calculus involved.

I hope that students who study from these materials will:

• Read the newspaper with new eyes: become skilled, circumspect consumers of qualitative and quantitative information.
• Know that probability in particular, and numbers in general, can be used to model some features of the physical world and human behavior.
• Improve their skills in critical thinking and logical reasoning.
• Appreciate the role Statistics plays in many fields, from business to economics, law, politics, science and medicine.
• Know that data can be manipulated to tell many inconsistent stories, that data analysis is not clear cut, and that many subjective judgments are involved in analyzing real data.
• Know important questions to ask when faced with a quantitative argument—be able to analyze arguments and find their strengths and weaknesses.
• Understand that untutored intuition tends to produce faulty probability judgments and know how to reason about probability.
• Appreciate some of the philosophical difficulties in ascribing meaning to probability and in inferring causal relationships from data.
• Be prepared for more advanced courses in Statistics—even though they might not take any.

The text starts with reasoning and fallacies, which is perhaps a bit unusual for a Statistics textbook—but logical reasoning is key to both theoretical and empirical work. The text goes further with counting arguments and combinatorics than most elementary textbooks do; it also goes further with logic and with data analysis. The tools incorporated into the materials enable students to analyze real datasets (the largest has 913 observations of 5 variables) without the pedagogical overhead of teaching students to use a proprietary statistics package. Students also reproduce numerical experiments that demonstrate key concepts, such as sampling distributions, confidence intervals, and the Law of Large Numbers. Using JavaScript based tools also eliminates the need to teach students to read arcane tables associated with different distributions; instead, students type the relevant parameters into textboxes, highlight a range of values, and read off the probability.

I have tried to emphasize topics that can be taught most effectively with this sort of interactive online tool. I have sought to provide enough variety in the material that instructors can pick and choose from among the chapters to find material appropriate to the level at which they desire to teach. The most technical material is in footnotes and sidebars, so that it does not interrupt the flow. Many of the examples and datasets for exercises are real—they arose in my consulting work, in experiments I am familiar with, or they are in the public domain (for example, data on GMAT scores, undergraduate GPA, and MBA GPA).

Many of the inference problems are real, too. For example, the Kassel Dowsing Experiment is a real test of the ability of dowsers to determine whether water is running in a buried pipe; the derivation of Fisher's exact test is in the context of determining whether targeted Web advertising works, a problem I have studied for a consulting client; the case studies about employment discrimination and theft of trade secrets derive from my work as an expert witness.

I have tried to motivate many of the computations by inference problems. Probability, hypothesis testing, randomization, and sampling error, are woven into the discussion of experiments and sample surveys. For some introductory courses, the probability in those sections will suffice. For instructors who desire a more quantitative text, there are additional chapters on probability distributions, discrete random variables, and expectation.

The book does not discuss continuous distributions: The normal curve, Student's t-curve, and the chi-square curve appear as approximations to the probability histograms of discrete random variables, not as probability densities of continuous random variables. These curves are motivated by interactive experiments using JavaScript tools that show empirically that the sampling distributions of some random variables converge to the curves. Probability is developed by counting; inference is developed using counting and sampling experiments that illustrate regularities.

Above all else, I strive to be correct and not silly—I generally avoid presenting techniques I would not use as a consultant or expert witness. There are exceptions, but I have tried to mark them clearly. For example, I find little use for the t-test or Student t confidence intervals for the mean, but as a concession to their popularity, I have included them—isolated in a single chapter that I usually do not cover. (But I present conservative confidence intervals for population percentages based on Chebychev's inequality, confidence intervals for percentiles, and Fisher's exact test.) ANOVA is missing for that reason. So is any use of regression as an inferential tool rather than a descriptive tool. Such uses are generally shaky, if not misleading or simply wrong.

These materials have been used to teach large undergraduate courses at UC Berkeley since 1997. They are the basis of the first fully online course offered by UC Berkeley, Statistics N21, Summer 2007. They have been used at other schools, including Bard College, City University of New York, John Jay College of Criminal Justice, University of California Santa Cruz, San Jose State University, and Laney College.

### Overview of the Online Textbook

If you are reading this in print, rather than in a Web browser, the following does not apply to the version you are reading. The online version of the materials has much more functionality than any print book can have:

• It is interactive: Students perform numerical experiments, analyze data, and manipulate plots to learn fundamental statistical and probabilistic concepts. The text combines guided exploration with the presentation of new material. The format encourages students to practice and to ask themselves and answer "what if?" questions—to learn through inquiry. The practice exercises are graded instantly, and detailed solutions are a click away.
• It is dynamic: Many examples and exercises change every time a page is reloaded, providing a practically unlimited number of practice problems.
• It is individual: Each student can work at his own pace, and can emphasize those subjects he finds most challenging or most interesting by reloading the chapter. If the instructor desires, every student can be administered a different version of the end of chapter quizzes.
• It is efficient and convenient: Quizzes are graded automatically and logged in on online database. Each student can see the solution to his individual version of the quizzes after the due date.
• It is easy for students to use: The text is accessible through standard web browsers. There is none of the start-up cost associated with learning to use a proprietary statistical software package.
• It is easy for instructors to use: Assignment due dates and enrollment lists are controlled over the Web using a browser. Instructors do not have to write quizzes, collect quizzes, record quiz grades, or return quizzes to students. Moreover, instructors do not have to teach students to use specialized software.

The software empowers students to reproduce numerical experiments themselves, without having to learn a statistical language (using instead a standard Web browser), which encourages exploration and inquiry-based learning. The text uses the power of the Internet in many ways, including the following:

• Links to a glossary of terms.
• Dynamic examples and self-test exercises that change every time a student visits a chapter. Some self-test exercises parse student input to determine whether student formulae are correct—a much more sophisticated notion of correctness than multiple choice or numerical responses.
• online, machine-graded assignments, constructed so that each student gets a different version of the assignment. Grades are posted automatically to the class website, and solutions are available online after the due date.
• Reference and practice materials to prepare for exams.

### Prerequisites

These materials do not assume that the reader has any previous knowledge of statistics or probability. However, the reader needs to be comfortable with percentages, exponentiation and square roots, and "scientific notation" (numbers times powers of ten). Assignment 0 is a review and quiz covering the prerequisite material. The ultimate calculations are all simple, but the logical reasoning needed to reduce the problems to those simple calculations are sometimes subtle.

Some of the footnotes and sidebars rely on elementary calculus to find stationary points of convex, continuously differentiable functions. For example, the mean is characterized as the number from which the rms of the residuals is smallest, and the regression line is characterized as the least-squares line. Those derivations can be skipped with impunity.

### Technical Design Criteria and Implementation

These materials are comprised of XHTML, CSS, and JavaScript. As of 19 June 2011, they consisted of 217 XHTML files containing over 136,000 lines of XHTML and JavaScript, 63 Java classes containing about 16,000 lines of code, 27 JavaScript libraries containing about 14,900 lines of code, 34 data files containing about 5,600 records, four cascading style sheets with about 2,200 lines, and a handful of .jpg and .gif files. The choice to use XHTML, CSS, Java and JavaScript was motivated by these design criteria:

1. Maximize accessibility and portability. Recent browsers allow this material to be accessed from almost anywhere in the world, without adding "plug-ins" to the browser, and without buying any proprietary software. If a "plug-in" were required, downloading the plug-in would itself present a considerable barrier to some students. The software runs under every major operating system (unix, linux, Windows 9x and NT, Mac OS), because Mozilla, Opera and/or Microsoft have versions of their browsers that run on those operating systems. Browsers come installed on all new personal computers, so this material is immediately accessible to new computer owners. I have made a considerable effort to make the materials function well with screen reader software for visually impaired students—but there is more work to be done. Converting the mathematics to MathML using MathJax is underway.
2. Maximize interactivity and minimize technological barriers. Students should be able to explore data and to ask and answer "what-if" questions, without needing to learn how to use a conventional statistical software package. Tools should have a point-and-click interface whose use was fairly obvious -- no hidden menus, consistent GUI, etc.).
3. Minimize bandwidth and maximize speed. Using JavaScript allows the figures and plots to be generated on the client-side. The code and data download to the client, then the client computes and creates the figures. This is by far the most efficient way to get dynamic interaction with the data. Otherwise, every time the user changed a parameter value, the client would need to send a message to the server, and the server would have to compute the new figure, and send the resulting figure over the Internet to the client. Interactive real-time data exploration would not be possible. There are a small number of figures that are stored as GIF or JPG files; almost all the figures are computed by the client. Sending just the data and the rules (programs) for generating figures from the data substantially reduces the time it takes pages to load. Many of the figures are pure CSS, which not only is very lightweight, but allows the figures to re-size elegantly if the user changes the dimensions of the page, and displays well even on mobile devices.
4. Make it easy to use the materials in lectures. Because the software is free-standing (it does not need a server for computations), it is easy to display the content in the classroom without an Internet connection. That allows the instructor to demonstrate concepts and the use of the materials in class.
5. Make it easy for instructors to set due dates for assignments and manage a course, and for students to track their own progress. Due dates are controlled by the instructor over the Internet; similarly, the instructor can modify grades, compute course scores, enter extra credit assignments, etc., using a browser. Perl-cgi routines update the database when a student submits an assignment, and allow students and instructors to query the database for grades over the Internet. As an alternative to perl cgi, assignment due dates can be controlled by modifying a simple ascii text file that is queried by an AJAX call, and assignment grades can be redirected to an email address instead of a database.
6. Maximize portability on the server side. Rather than use proprietary solutions, such as Microsoft active server pages (.asp pages) or server-side JavaScript, the "back end" is a collection of perl cgi routines that access a standard database file. The server only needs to support perl scripts and serve static XHTML, JavaScript, text, and CSS. This makes the package platform-independent on the server side as well as the client side: there is public-domain software allowing perl cgi to run with every common web server, and there are public-domain perl implementations for all popular operating systems. Installing the materials on a new server is simple.

Using XHTML with JavaScript and CSS allowed me to make the content dynamic: many of the examples and exercises in the text change whenever the page is reloaded, so students can get unlimited practice at certain kinds of problems. Similarly, each student gets a different version of each assignment and exam, but can see the solutions to his/her version after the due date.

### Advantages of JavaScript over standard Statistical Packages

There are a number of advantages to using JavaScript rather than an integrated statistical package:

1. The material can be accessed from any computer with an Internet connection and a web browser. The computer does not need to have any proprietary software installed. Students therefore can access the material from university and public libraries, Internet cafes, home, etc. Students have even submitted homework using WebTV.
2. Some of the demonstrations would be extremely difficult, if not impossible, to code in a standard statistical package. For example, see the Venn Diagram tool.
3. If a standard package were used, the figures/demonstrations/calculations could not be embedded in the text and the assignments. The student would have to navigate among programs to see demonstrations or solve problems.
4. The intellectual start-up cost to the student is lower than it would be for a general-purpose package. Each tool illustrates a single concept, all the controls are visible, and the interface is as intuitive as I have been able to make it. The student does not need to learn much to get started.
5. I find it preferable pedagogically to use tools with a single function, with all the controls visible.
6. The monetary cost to the student is minimized.

### Suggestions for Evaluating the Materials

I would recommend that instructors who wish to evaluate these materials for possible adoption look first at Those chapters illustrate several aspects of the text: dynamic exercises, the use of real data in examples and exercises, the histogram and scatterplot tools, and the gradual introduction of new functionality (buttons and displayed statistics) into the tools as students learn new concepts. For example, when the scatterplot tool arrives in its only controls change the variables plotted, list the data, show univariate statistics of the variables in the dataset (summary statistics covered in the first two chapters), and display the coordinates of the cursor. (Selecting a row in the data listing highlights the corresponding point in the scatterplot.) In the scatterplot tool acquires the correlation coefficient, and a button to show graphically the standard deviations of the two variables plotted; it is also invoked to display randomly generated data that attain a given value of the correlation coefficient. It also starts to allow students to add points by clicking on the plot, to see the effect of additional data on the correlation coefficient. In the same tool gains buttons to show the graph of averages, the SD line, and the regression line.

After those chapters, I would recommend looking at the collection of interactive tools to see how various concepts are presented graphically; in particular, be sure to see the tools for Venn diagrams, sampling distributions, confidence intervals, and the Law of Large Numbers. To see how tables of probabilities are eliminated, see the tools for the Normal Distribution, Student's t-Distribution, and the Chi-square Distribution. I would recommend then looking at Instructors with an interest in logic or who teach general education courses might enjoy The first of those has exercises that parse logical expressions the students type in.

Philip B. Stark is Professor of Statistics and Associate Dean of Mathematical and Physical Sciences at the University of California, Berkeley, where he has been on the faculty since 1988. He received his bachelor's degree in Philosophy from Princeton University in 1980, and his PhD in Earth Science from the Scripps Institution of Oceanography in 1986. He received a National Science Foundation Postdoctoral Fellowship in Mathematical Sciences in 1987 and the Presidential Young Investigator Award in 1989. He is a Fellow of the Institute of Physics and of the American Statistical Association, and professionally accredited as a statistician and as a physicist. Philip dropped out of high school and law school. He has served on the editorial boards of journals in applied mathematics, geophysics, and statistics, and has given over 200 invited lectures at conferences and universities in more than 25 countries. He is the author or co-author of over 150 publications. Philip has done research in astrophysics, microwave cosmology, earthquake prediction, gender bias, geomagnetism, geochemistry, seismic tomography, signal recovery, constrained confidence estimation, probability density estimation, spectrum estimation, information retrieval, inverse problems, election auditing, adjusting the U.S. Census, causal inference, and human hearing.

He specializes in problems with very large datasets; software written by him and his students performs part of the routine data reduction for a geomagnetic satellite and a network of solar telescopes. Philip has consulted in IC mask manufacturing, oil exploration, water treatment, predicting e-mail spool fill, electrical activity of the brain, and targeted Internet advertising. He has served as an expert witness in litigation and legislation on topics ranging from natural resources to agricultural import restrictions, disaster relief, fairness in lending, the U.S. Census, the Child Online Protection Act (sampling the Internet and testing content filters, which involved the controversial subpoena of search records and indexed webpages from Google, Yahoo! and MSN), consumer protection, contested elections, employment discrimination, insurance, product liability, property tax assessment, truth in advertising, marketing, equal protection, trade secrets, intellectual property, risk assessment, wage and hour disputes, and anti-trust.

He has testified to the U.S. House of Representatives Subcommittee on the Census, to the California State Senate, the California State Assembly, and the California Department of Fish and Game. He has consulted for the U.S. Department of Justice, the U.S. Department of Agriculture, the U.S. Census Bureau, the U.S. Department of Housing and Urban Development, the U.S. Attorney's Office of the Northern District of California, the U.S. Department of Veterans Affairs, the Federal Trade Commission, the Los Angeles County Superior Court, the National Solar Observatory, the California Secretary of State, the Colorado Secretary of State, public utilities, major corporations, and numerous law firms, including more than half of the 25 largest. He serves on the Board of Advisors of the U.S. Election Assistance Commission. He won the Chancellor's Award for Public Service for Research in the Public Interest in 2011 for his work on election auditing, and he is currently working with the Secretary of State of California and the Secretary of State of Colorado to implement risk-limiting election audits.

Philip was the Faculty Assistant for Educational Technology at The University of California, Berkeley, from 2001–2003 and chaired the U.C. Berkeley Educational Technology Committee from 2001–2005. He taught UC Berkeley's first official online course, in 2007, and developed one of the first courses to be offered through UC Online Education, and co-developed (with Ani Adhikari) the first introductory statistics course offered through edX. He was Chair of the Department of Statistics from 2012–2015.

Philip does not like to be called "Phil." He likes open-source software, dislikes email attachments, runs 100 mile endurance trail races, roasts his own coffee, and thinks this book is proof that obsessive-compulsive disorder is a job qualification. Philip lives in Berkeley, California, with a laptop, an iPhone, a Portaspresso, a Knock hand mill, muddy homemade huaraches, and an embarrassing number of LED flashlights.