statistics

An Election: Heads vs. Tails

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Yesterday during a tangent in my Algebra 2 class I realized how little understanding my students had about the voting process as well as how predictions are made and how, when, and why networks announce winners without all of the votes counted.

Today, I ran an activity that pitted Heads vs. Tails. Each student received a penny and flipped the coin to determine their vote: Heads or Tails.  The class consisted of 19 students.

In our first election, we strictly counted votes- a popular election.  Tails beat heads 10-9.

In our second election (everyone re-flipped their coin), the girls announced first: tails 6 and heads 5.  I asked our students, what do we EXPECT to happen with the remaining 8 votes?  Assuming fair coins, we expect 4 for heads and 4 for tails.  So if we had to bet, would we bet that heads would win or tails? Since tails already had a 1 vote lead, tails was most likely to win.  In fact, the scenario most likely to result in a win for heads was 5 heads and 3 tails.  This was not a probability class, but with some basic concepts we could have given percentages to each outcome.

Back to this specific election:  With only the girls reporting, the vote was 6 for tails and 5 for heads.  I asked one boy to announce his result.  He flipped tails, so the vote count was 7 for tails and 5 for heads.  I declared, “I am ready to call the election for tails!”  The students protested; there were 7 students still left to vote.  Could heads have come back and won?  Yes, but this was unlikely since 5 of the 7 remaining flips would need to be heads.  We counted the votes and tails won 10-9 (4 of the 7 remaining flips went to heads).  I did not have a specific percentage in mind when I “called” the election (the real chance when I declared tails victorious was around 75%, but I wanted to make a point).  We discussed how, when, why (and the implications) newspapers and television stations “call elections.”

In our third election, I told the class that the front row (5 students) always voted heads and were going to vote for heads again.  With 14 students left to flip, I said that I had enough information to call the election and was sure heads would win.  Again, with the likelihood around 75% that heads would win, there was a decent chance tails could come back, but again, I was making a point.  We then discussed how this activity mirrored the Electoral College with states like California consistently voting democrat.

Finally (and I did not have time to do this election), I was going to weight each row differently.  The first row would vote and their winner would receive 4 votes, the winner of the second row would receive 3 votes, the 3rd row two votes and the back row one vote.  I was going to have each row announce in order 1-4, then redo this system with the back row announcing first.

Please leave a comment if you have any ideas on improving this exercise or need any help implementing it in your class.

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The Role of Gender and Tracking in Student Achievement at the Extremes

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The following is the abstract of a recent paper I wrote analyzing student performance data. Here is a link to the entire paper.

http://dl.dropbox.com/u/33186846/Student%20Achievement%20at%20Extremes.pdf

Measurement of student achievement is at the heart of educational policy and standardized testing has been both supported and contested as a genuine representation of student achievement. This study shows that the interpretations of standardized testing may have starkly contrasting meanings for different cohorts of students. By using quantile regression to account for conditional PSAT scores, educational factors such as gender and tracking are shown to effect students in varying ways. For the students in this study, gender was a significant indicator of performance on the PSAT test, with being male accounting for more than a five point “bump.” The conclusion from the quantile regressions is that students with extreme PSAT scores are outliers based on ability or inability, not because of their gender. Meanwhile, students that participated in the “honors” tracking system had more of an increase in their predicted score the higher their conditional PSAT score.

An Explanation of Conditional Expectations… Part Two

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In my first (found here) of a three part series on statistical analysis, I discussed how data can inform decisions within countless industries.  My research in human capital, and specifically education, provides for various usages of data in decision making.  Specifically, available data can be used for predictions of when, where, and which students may have trouble, determinants of parental satisfaction, admission decisions (at both secondary and collegiate levels), and student achievement both academically, on standardized tests, and in future wages.

This post is intended to shed light on the science of data analysis, specifically conditional expectations.  The mathematical approach to conditional expectation is based on heavy statistical concepts, some of which are not appropriate for this venue.  That being said, my intention is to provide an accessible explanation of the techniques used in statistical analysis.  For a mathematical approach, I recommend the seminal text Econometric Analysis, by William Greene, 2002; or Econometric Analysis of Cross Section and Panel Data, by Jeffrey Wooldridge.

Econometrics, and essentially any data analysis, is based on determining a prediction.  In statistical terms, this is called an expectation.  A conditional expectation is a prediction based on available information.

For instance, consider guessing the height of a random human being.  The average human height is 5 foot 6 inches, so, this would be a logical starting point for our prediction.  But if we know more information about this random person, we can improve our expectation.  Specifically, if we knew the person was male, we would want to change our expectation conditional on that fact.  The average height of an adult man is 5 foot 9.5 inches, so that would be our new prediction given some information.

If we also knew that the person weighed 240 pounds, we may want to increase our expected height.  Note here that there is no causal assumption, just a change in our expectation, given some piece of information.  This is correlation: the taller a person is the more, on average, we expect that person to weigh.  We may predict 6 foot 1 inch for our random male weighing 240 pounds.  Data analysis can help us with our predictions, given some imperfect information.

Next, consider that we also knew this random person’s SAT score was 1200.  We probably would not consider changing our expectation of height.  If we had data on a sample of people with the variables height, sex, weight, and SAT score, some variables may be good predictors of height and be statistically significant while others, SAT score in this example, would not be statistically significant and would not persuade us to change our expectation.

Multiple Regression Analysis essentially considers a sample of data and determines the predictive success of each variable.  Using the information from a statistical data program (even excel can do this reasonably well), we can arrive at a predictive equation for the most logical expectation conditional on the information we have at our disposal.

The data will find the coefficients- the “B’s”- and also determine the likelihood that each “B” is a significant predictor.  In this case, I suspect B3 would not be statistically significant.

In part three, I will discuss an example of student achievement data from a nation-wide sample and how conditional expectations can be used to inform decisions in many fields.

The Rise of Big Data… From Moneyball to the Classroom and Back Again

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This is the first of a three-part series of posts discussing the importance of data analysis and introducing readers to the importance and value in statistical and econometric analysis.

Last Sunday, Steve Lohr wrote a great piece in the New York Times explaining the importance of “big data” in today’s society.  He explained that increasingly “businesses make sense of an explosion of data-  Web traffic and social network comments, as well as software and sensors that monitor shipments, suppliers and customers- to guide decisions, trim costs and lift sales.”

The explosion of data Lohr is referring to is accessible to any field- not just a profit maximizing business- and when used properly, that data can be used to enhance and enrich any institution.  Clearly businesses utilize data to inform their decisions, but increasingly, political campaigns, public health officials and advertising agencies are innovating their traditional practices by developing methods and metrics based on data analysis.

The most glorified example is illustrated in the book and recent film “Moneyball,” written by Michael Lewis describing the revolution in baseball by Billy Beane and the Oakland Athletics.  The short story is that the team began to analyze players using complex statistical analyses instead of traditional benchmarks.  Billy Beane is not the only front office executive to develop and exploit new statistical methods; the general manager of the Houston Rockets, Daryl Morey wrote a piece for Grantland.com regarding the “stats movement in sports” and how the success of Moneyball has transcended sports and become impacted countless industries.

Morey briefly describes how statistical analyses have entered the realm of education: the Gates foundation is gathering data to evaluate teachers.  But Morey and the Gates foundation are only scratching the surface.  Education at all levels is ripe for a takeover of objective data analyses.  Statistics currently used within schools to evaluate programs or students rely on static data.  Static data consists of the most basic statistics we remember from high school: averages and percents.  New data- big data- is about how information, records, numbers move over time and how a fact or figure can be broken down to find relationships and meaning behind the numbers.

Consider a static piece of data such as: In a specific district, 28% of parents are unhappy about their child’s school.  This does not tell an administrator much- probably only something that she already knows.  But, a deeper look into the data could reveal important information such as “of the 28% of parents who are unhappy, 70% of their students play a varsity sport.”  This has more value; specifically, there is a trend among unhappy parents.

The next two posts will dig deeper into the “why” and “how” of how data can be used to improve decision making.  At the most basic level, data can help shape expectations, specifically conditional expectations given some sort of observed trend in the data.  I will explain the concept of conditional expectation in part two.