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Statistics in the media

Batting .400 and the Law of Large Numbers

 

Rod Carew, one of the few to make a serious run at .400 since Williams, has studied the .406 season and contends that Williams’s absences were a blessing. 

“The fewer at-bats any hitter has over the required number of plate appearances, the better his chance is of hitting over .400,” Carew wrote in an e-mail responding to questions about 1941. “When I hit .388 in 1977, I had 694 plate appearances and 616 at-bats (239 hits). Ted had something like 450 at-bats in 1941 when he hit .406, and I think George Brett and Tony Gwynn had fewer then 450 at-bats when they made their runs at .400. 

“All in all, the less at-bats, the better.”

He’s trying to articulate the Law of Large Numbers.  Anyone hitting near .400 is deviating from the expected proportion of hits.  If you flip a coin 10 times, you just might get 7 tails.  If you flip if 1000 times, there’s no chance you’ll ever get 700 tails.  Many people may bat .400 during a single game (a handful of at-bats), but almost no one does as the number of at-bats increases.  Their average converges to the more realistic season average.

Categories
Computer Programming Puzzles

Quantitative Interview Questions

He returned for another round of Goldman’s grilling, which ended in the office of one of the high-frequency traders, another Russian, named Alexander Davidovich. A managing director, he had just two final questions for Serge, both designed to test his ability to solve problems.

The first: Is 3,599 a prime number?

Serge quickly saw there was something strange about 3,599: it was very close to 3,600. He jotted down the following equations: 3599 = (3600 – 1) = (602 – 12) = (60 – 1) (60 + 1) = 59 times 61. Not a prime number.

The problem wasn’t that difficult, but, as he put it, “it was harder to solve the problem when you are anticipated to solve it quickly.” It might have taken him as long as two minutes to finish. The second question the Goldman managing director asked him was more involved—and involving. He described for Serge a room, a rectangular box, and gave him its three dimensions. “He says there is a spider on the floor and gives me its coordinates. There is also a fly on the ceiling, and he gives me its coordinates as well. Then he asked the question: Calculate the shortest distance the spider can take to reach the fly.” The spider can’t fly or swing; it can only walk on surfaces. The shortest path between two points was a straight line, and so, Serge figured, it was a matter of unfolding the box, turning a three-dimensional object into a one-dimensional surface, then using the Pythagorean theorem to calculate the distances. It took him several minutes to work it all out; when he was done, Davidovich offered him a job at Goldman Sachs. His starting salary plus bonus came to $270,000.

Michael Lewis: Did Goldman Sachs Overstep in Criminally Charging Its Ex-Programmer?
Categories
Education

Confessions of a College Application Reader

  • …while seeming to prize the high-paying out-of-state students who are so attractive during times of a growing budget gap.
  • …questioned why a student who … had only a 3.5 G.P.A. should rank so highly. Could it be because he was a nonresident and had wealthy parents?
  • Income, an optional item on the application, would appear on the very first screen we saw, along with applicant name, address and family information.
  • In personal statements, we had been told to read for the “authentic” voice over students whose writing bragged of volunteer trips to exotic places or anything that “smacks of privilege.”
  •  Many essays lucidly expressed a sense of self and character — no small task in a sea of applicants. Less happily, many betrayed the handiwork of pricey application packagers, whose cloying, pompous style was instantly detectable, as were canny attempts to catch some sympathy with a personal story of generalized misery. The torrent of woe could make a reader numb: not another student suffering from parents’ divorce, a learning difference, a rare disease, even dandruff!

Confessions of an Application Reader
Lifting the Veil on the Holistic Process at the University of California, Berkeley

Categories
Algebra 2

Alzheimer’s and Exponential Growth

The incidence of Alzheimer’s doubles every five years after the age of 65 until, by 85 years, nearly one in three adults is afflicted.

Are you wondering what percent are afflicted at age 65?  Can you work backards from 33% being affected at age 85?

Let’s build up the equation.  When something doubles, you multiply by 2.  When it doubles a bunch of times, you will be multiplying by a bunch of 2’s… or simply \(2^n\)  Since it is doubling every five years, you only want to multiply by 2 when the year hits a multiple of 5.  So, change it to \(2^\frac{n}{5}\)  You also need some sort of initial amount that is getting doubled in the first place.  Our initial equation to represent the percent of people with Alzheimer’s at  65 is \(y=a(2^\frac{n}{5})\)  Next, we use the given (n,y) pair to solve for that constant (a).  At 85 years (ie: 20 years later), the percent is .33  So, plug it in.  \(.33=a(2^\frac{20}{5})\)     (Also, notice how the division by 5 comes into play?  20/5 = 4 doubles in that 20 years.  ie:  Doubles every 5 years)   Solve and get a = .02  Plug that back into a, and you have \(y=.02(2^\frac{n}{5})\)   Well, let’s answer the question.  What percentage of people have Alzheimer’s at age 65 (n=0)?  \(y = .02(2^\frac{0}{5}) = .02(2^0) = .02(1) = .02\)    So, 2% of people at age 65.  A quick calculation verifies this:  2, 4, 8, 16, 32 at ages 65, 70, 75, 80, 85, respectively.

Seeds of Dementia: What Do Alzheimer’s, Parkinson’s and Lou Gehrig’s Have in Common?
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Education Uncategorized

Why do some habits catch on faster than others?

  • In the era of the iPhone, Facebook, and Twitter, we’ve become enamored of ideas that spread as effortlessly as ether. We want frictionless, “turnkey” solutions to the major difficulties of the world—hunger, disease, poverty. We prefer instructional videos to teachers, drones to troops, incentives to institutions. People and institutions can feel messy and anachronistic. They introduce, as the engineers put it, uncontrolled variability.
  • Mass media can introduce a new idea to people. But, Rogers showed, people follow the lead of other people they know and trust when they decide whether to take it up. Every change requires effort, and the decision to make that effort is a social process.
  •  As the (sales) rep had recognized, human interaction is the key force in overcoming resistance and speeding change.
  • “Why did you listen to her?” I asked. “She had only a fraction of your experience.”
    All the nurse could think to say was “She was nice.”
SLOW IDEAS
Categories
Statistics in the media

Incorrectly Comparing Student Test Scores of Various Countries

…97 percent of all Singaporean students, nearly 90 percent of South Korean primary students and about 85 percent of Hong Kong senior secondary students receive tutoring.
Tutoring Spreads Beyond Asia’s Wealthy

When comparing test scores of various countries, are they comparing similar samples?  No.  Just one (of many) confounding variable that needs to be controlled for is the amount of private tutoring each group of students receive.

Categories
Statistics in the media

Evaluating a study: Online learning

Study Gauges Value of Technology in Schools

Mr. Pane conducted a study, financed by the federal Department of Education, of an algebra software program.  He found that high school students who used the program …showed gains on their state-standardized math tests that were nearly double the gains of a typical year’s worth of growth using a more traditional high school math curriculum.

Double!  Well, that sounds impressive.  But, you want to know what exactly these “doubled gains” actually are.  Does it mean a gain of 2 points instead of 1 point?  Or does it mean 30 points instead of 15?

Note the last step in critically evaluating a study or experiment:

  1. The source of the research and of the funding.
  2. The researchers who had contact with the participants.
  3. The individuals or objects studied and how they were selected.
  4. The exact nature of the measurements made or questions asked.
  5. The setting in which the measurements were taken.
  6. Differences in the groups being compared, in addition to the factor of interest.
  7. The extent or size of any claimed effects or differences.

So, in light of #7, you should to read the actual study, and not a summarized interpretation in a newspaper.  Here are some notable excerpts from the actual study:

  • …treatment effect estimates are not significant the first year. The estimates are negative in the high school study and near zero in the middle school study.
  • …the magnitude is sufficient to improve the average student’s performance by approximately eight percentile points.  Consider a student who would score at the 50th percentile in the control group; an effect size of 0.20 is equivalent to having that student score at the 58th percentile if they were in the treatment group.

So, when you read the fine print, you learn that scores actually go down in the first year, and the improvement may not be as large you the article led you to believe.

Categories
Education Learning Skills

Reliance on calculators can backfire

Math Skills Placement Test Practice

Here is a Math Placement test a student of mine was recently given by her prospective college.  Take a look at the questions, then notice the directions; they were not allowed to use any sort of calculator.  40 mins. / 40 questions.  This is a good example of where over-reliance on calculators for basic k-7 math can backfire.

Also, the AP Calculus exam has entire sections that do not allow the use of any type of calculator.

Categories
Algebra

Two examples of “proof” using Algebra (Not Geometry)

In high school, you almost never see a “real” proof.  You only see those Statement/Reason proofs in Geometry.  These types of proofs never really show you what real math proofs are really like.  So, here are 2 for you:

Prove that \(\sqrt{2}\) is irrational

A Geometry proof using Algebra

Categories
Education

Measuring College Prestige vs. Cost of Enrollment

Measuring College Prestige vs. Cost of Enrollment

…. found that equally smart students had about the same earnings whether or not they went to top-tier colleges. The big difference, their studies found, came from minority and low-income students who went to top-tier colleges: They did better later on.

 

 

 

Categories
Education

Critical Thinking Is Best Taught Outside the Classroom

  • Formal education, which is driven by test taking, is increasingly failing to require students to ask the kind of questions that lead to informed decisions.
  • Perhaps many teachers have too little time to allow students to form and pursue their own questions and too much ground to cover in the curriculum and for standardized tests
http://www.scientificamerican.com/article.cfm?id=critical-thinking-best-taught-outside-classroom

 

Categories
Statistics in the media

Intentionally Misleading Graphs (How to Lie With Statistics)

This is a great example of a shamelessly biased graph.  USA’s cost is grossly exaggerated by extending the bars to include 95th percentile, while none of the others do this.   Blatant!

 

Categories
Education

Grouping Students by Ability

Grouping Students by Ability
Categories
Education

Raising Successful Children

Raising Successful Children

  • Praising children’s talents and abilities seems to rattle their confidence. Tackling more difficult puzzles carries the risk of losing one’s status as “smart” and deprives kids of the thrill of choosing to work simply for its own sake, regardless of outcomes.
  • The central task of growing up is to develop a sense of self that is autonomous, confident and generally in accord with reality.
  • Hanging back and allowing children to make mistakes is one of the greatest challenges of parenting.
  • To rush in too quickly, to shield them, to deprive them of those challenges is to deprive them of the tools they will need to handle the inevitable, difficult, challenging and sometimes devastating demands of life.
  •  If pushing, direction, motivation and reward always come from the outside, the child never has the opportunity to craft an inside.

 

 

 

Categories
Education

The Touch-Screen Generation

Young children—even toddlers—are spending more and more time with digital technology. What will it mean for their development?

  • Before hiding Piglet, the researcher effectively engaged the children in a form of media training. She asked them questions about their siblings, pets, and toys. She played Simon Says with them and invited them to sing popular songs with her. She told them to look for a sticker under a chair in their room. She gave them the distinct impression that she—this person on the screen—could interact with them, and that what she had to say was relevant to the world they lived in.
Categories
Education

The Case For Repeating Algebra

The Case For Repeating Algebra
Reader Responses

Categories
Education

College Completion Rates for Full-Time Students

A Closer Look at College Completion Rates for Full-Time Students
Categories
Education

Study: Internet Addicts Suffer Withdrawal Symptoms Like Drug Users

Study: Internet Addicts Suffer Withdrawal Symptoms Like Drug Users

Categories
Education

Helicopter Parents May Breed Depression and Incompetence in Their Children


Helicopter Parents May Breed Depression and Incompetence in Their Children

  • “Parents are sending an unintentional message to their children that they are not competent,”
  • “When adult children don’t get to practice problem-solving skills, they can’t solve these problems in the future.”
  • Helicopter parenting decreased adult children’s feelings of autonomy, competence and connection. 
Categories
Statistics in the media

Harvard researchers challenge results of obesity analysis

Weight and mortality

The studies that Flegal did use included many samples of people who were chronically ill, current smokers and elderly, according to Hu. These factors are associated with weight loss and increased mortality.