Stats Made Easy

In the latest issue of American Scientist magazine, Howard Wainer, an adjunct professor at Wharton, makes a case for how ignorance of how sample size affects statistical variation has created havoc for nearly a millennium – and continues to do so today. Simply put, the variance of sample means increase as the sample size decreases. Wainer deems Abraham de Moivre’s 1730 discovery of this mathematical relation The Most Dangerous Equation .

The article details a major investment by Bill and Melinda Gates Foundation and others to provide better education via smaller schools. Statistics showed that among high-performing schools were an unrepresentatively large proportion of smaller ones. However, the Wainer found that the same can be said for low-performing schools – smaller ones are over-represented. This finding agrees with De Moivre’s equation – smaller schools display greater variance of test score averages and thus over-populate the extremes. Wainer presents statistical evidence that “overall bigger schools do better…not unexpected, since very small high schools cannot provide as broad a curriculum or as many specialized teachers.” He concludes that “Spending more than a billion dollars on a theory based on ignorance of de Moivre’s equation – in effect serving only to increase variation – suggests just how dangerous that ignorance can be.”

Wainer provides another example showing how U.S. cities rank for automobile safety. Smaller cities come out on top and bottom – again demonstrating how variance increases as sample size decreases. How often have you seen tables like this in the popular media that rank areas by some criterion and seen the same result – smaller cities coming out on top and bottom? Perhaps some of the investments for improving education ought to be directed toward basic statistics!

Subject matter expertise from a technical guru often works wonders. For example, consider Charles Steinmetz of General Electric, who learned his trade from Thomas Edison. Steinmetz invented more than 200 electrical devices. It’s said that after he retired, GE hired Steinmetz to solve a particularly difficult machine problem. He looked here and there, tested various parts, and then marked an “X” on a specific spot. The GE engineers were amazed to find a defect precisely at the Steinmetz mark. Later they got a bill for $1,000. The itemization read: “For making one chalk mark – $1. For knowing where to put it – $999.”

Many years ago (40!*) as a pre-teen I read the story “Eleven Blue Men” by Berton Roueché. This and other true stories about medical detectives fascinated me. Given the popularity of the television show House, I am not the only one who appreciates experts that make astoundingly accurate diagnoses based on a few facts and fantastic leaps of intuition.

Dr. Jerome Groopman in his book on How Doctors Think says that “The mind acts like a magnet, pulling in clues from all directions.” However, Groopman advises experts keep their guard up against these common biases:
– Availability – the tendency to reach for the easiest plausible explanation and reject all others,
– Confirmation – see only evidence that fits preconceived notions,
– Commission – the rush to action when doing nothing would be best.
(If you never want to believe any expert opinion again, see an exhaustive list of cognitive errors in diagnosis compiled by Dr. Pat Croskerry (Dalhousie University Faculty of Medicine, Halifax, Nova Scotia, Canada).)

Groopman suggests that patients who fear they are not getting a good diagnosis ask these simple questions: “What else could it be? Could two things be going on simultaneously?” This strikes a chord with me as a specialist in the field of experiment design. It seems that the toughest problems for those who rely on intuition, or heuristics as Groopman calls it, are those where factors interact to create an important effect.

By the way, Groopman has some interesting caveats about the new approach of “evidence-based medicine,” which you can see in an excerpt from his book posted along with the sound track of an interview by NPR’s “Morning Edition” radio show. He warns, “Statistics embody averages, not individuals.”

*PS. Yes, I do feel old, but age may provide some benefit via the accumulation of experience. For example, the March 22nd issue of Wall Street Journal says that age 53 is best for financial decisions. “The age of reason” they call it. Guess how old I am. It’s funny how one notices stuff like this that confirms a prior notion.

Hmmm, the French philosopher Camus would not approve, but I have some odds and ends left over from my previous blog on happiness and well-being.

1. The New York Times published a much fancier graphic on happiness by country than mine but with out-dated stats (’95) on per capita income.

2. In their article titled “Reversal of Fortune,” this month’s issue of Mother Jones magazine (March/April 2007) refutes the axiom “Make money, get happy.” They report that money consistently buys happiness right up to about $10,000 income per capita – what an average Mexican earned in 2006 while achieving the second ranking in the World Values Survey on happiness and well-being.* Folks at the poverty level get a great lift from even a small reversal of misfortune. The author, Bill Mckibben, calls this the “Laura Ingalls Wilder effect” after the writer made famous by “Little House on the Prairie”. One of my favorite stories is Wilder’s reminiscence of Christmas on the banks of Plum Creek near Walnut Grove, Minnesota, in 1875. Her parents could only afford to give less than a dozen pieces of hard candy to each of their children, but this sufficed to create great happiness in the prairie-dwellers’ dugout. This really puts things in perspective!

* Evidenced in Figure 1 presented by Ed Diener (University of Illinois, Gallup Organization) and Martin E.P. Seligman (University of Pennsylvania) in their article Beyond Money Toward an Economy of Well-Being. It shows that “Over the past 50 years, income has climbed steadily in the United States, with the gross domestic product (GDP) per capita tripling, and yet life satisfaction has been virtually flat.”

3. The San Francisco Chronicle (12/24/00,Science Tracks the Good Life ) offered this list of the most important sources of personal happiness from researcher Michael Hagerty:
— Close ties to friends and family;
— Wide political freedom*
— High income, and
— A narrow gap between rich and poor.
According to Hagerty’s analysis, the United States is the fifth-happiest nation, a fact that baffled him because for the most part, the top-rated countries are small and homogeneous. That may explain Puerto Rico’s positive feelings of well-being.

*(See also Democracy and Happiness: What Causes What? by Ronald Inglehart (University of Michigan).)

4. Here are some other topical references I found on the internet, but after I reached a point of diminishing happiness trying to find the secret to happiness (Camus was right!):

a. The Reliability of Subjective Well-Being Measures by Alan B. Krueger (Princeton University) and David A. Schkade (University of California, San Diego).

b. A Simple Statistical Method For Measuring How Life Events Affect Happiness by Andrew E. Clark (CNRS and DELTA, Paris, France) and Andrew J. Oswald (Department of Economics, University of Warwick, UK).

(Photo of Guatemalan girl by K. M. Anderson)

After enjoying a fun vacation in Puerto Rico, I was not surprised to see this country* topping the list in a worldwide ranking by happiness reported in the April 2007 issue of Minnesota Business. It stems from a study by the World Values Survey that questioned such things as “All things considered, how satisfied are you with your life as a whole these days?” See the combined index of happiness and well-being (positive versus negative) ranked by country (or the original Word doc).

The author of the Minnesota Business article, Tor Dahl, added 2006 per capita income to the mix. This reveals some surprising statistics, such as Mexicans earning less than half the income of Spaniards, yet reporting twice the level of happiness and well-being. However, using Design-Expert® software’s historical regression tools, I find overall a significant (p<0.0001) positive relation of income with happiness and well-being. Luxembourg, which at $65,900 per capita income (’06) exceeds the next richest country (Norway!) by 50 percent, stands out as an extremely high leverage point. If ignored, the non-linear plateauing of happiness and well-being becomes insignificant. In my opinion, though, this makes sense – at some point more money probably cannot buy more happiness. The quest for the secrets of happiness will no doubt continue with little chance of a totally satisfying outcome. Princeton professors Daniel Kahneman and Alan B. Krueger in their Developments in the Measurement of Subjective Well-Being present a list of variables which are correlated with global reports of life satisfaction and happiness, among which is an “unfakeable smile.” Isn’t that wonderful! 🙂

*(I enjoyed a good chuckle calling Earthlink before my vacation in San Juan, Puerto Rico to get their local dial-up accesses for internet. Their representative could not find Puerto Rico on the list. “Is it a United State?” he asked. “Not exactly,” I replied. “Well, then, it must be a country – correct?” “No.” I then tried to briefly explain the status of Puerto Rico as a self-governing commonwealth – possibly even more difficult to explain than why their people seem to be so happy. Anyways, the Earthlink rep pleased me greatly by saying he found two access numbers for San “Chew-on.” Evidently this fellow’s native language was not Spanish, but in a case like this, I do as Bobbie McFerrin says – “Don’t worry, be happy”.)

While vacationing in San Juan, Puerto Rico this week, I came across a tricky triangular staircase. It’s in a bastion of the Spanish fort El Morro. Military engineers, evidently well-versed in geometry by the late 16th century when these fortifications were built, favored polygonal designs because the oblique angles resisted cannon balls. Triangular outworks provided fields of fire along adjoining walls. In the jargon of fortress design these are called “demilunes” if inside the surrounding ditch or moat, and a “ravelin” if outside. I think even the 5th graders who outsmarted their adult contestant in knowledge of a trapezoid would be challenged to identify all the shapes in a fort like El Morro.

“Sometimes things happen in the world that we are not capable of understanding.” “Yeah, like geometry.” 3/23/07 comic strip Baldo

It turns out that the only one to successfully take over El Morro was Sir George Clifford, 3rd Earl of Cumberland. His English predecessor Sir Francis Drake failed to defeat the Spanish in San Juan three years earlier. Perhaps Clifford’s solid education in mathematics helped him formulate a successful strategy. However, although he was fearsome on a horse — being one of the top jousters of the age, Sir Clifford struggled on foot. While suited up in his legendary armor, he slipped while bridging a moat and nearly drowned. After traversing the pictured triangular stairway (termed “ternary” as explained in my previous blog), I cannot see how anyone could manage this in full armor. Perhaps by then Sir Clifford had achieved victory and stripped down to his Bermuda shorts to sip on one of San Juan’s famous pina coladas.

For more on El Morro and its place in the battle against pirates of the Caribbean, see this blog by scholar Jon Beasley-Murray.