Category Archives: Applications



Is one’s health helped or hurt by being in a hot sauna? Researchers from Finland conducted a study to find out, with results recently published in a top-flight medical journal.

In the study, data came from 2,315 men who used saunas. And what were the study’s findings? Those who had saunas more often each week, and those who stayed longer in the sauna, had, over time, fewer fatal heart attacks.

Why did this study get published in a prestigious scientific journal? One of the reasons was this: the researchers used complex statistical procedures to examine the relationship between sauna use and heart attacks while controlling for things such as age, BP, tobacco use, SES, physical activity, etc.

Here is a list of the intermediate and advanced statistical procedures used by the researchers: (1) a chi square test, (2) analysis of variance, (3) 95% confidence intervals, (4) a multivariate Cox model with several covariates (age, BMI, systolic blood pressure, cholesterol levels, smoking, alcohol consumption, previous myocardial infarction, diabetes, cardiorespiratory fitness, resting heart rate, physical activity, and socioeconomic status), (5) sensitivity analyses, (6) survival ratios using the Kaplan-Meier method, (7) hazard ratios and cumulative hazard curves, (8) plots of Schoenfeld residuals to check the proportional hazards assumption, and (9) Martingale residuals to check the linearity assumption.

The research report was published on February 23, 2015, in the Journal of the American Medical Association: Internal Medicine. It had this title: “Association Between Sauna Bathing and Fatal Cardiovascular and All-Cause Mortality Events.” The author/researchers were Tanjaniina Laukkanen, Hassan Khan, Francesco Zaccardi, and Jari A. Laukkanen.



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Normal Curve (Product Adoption)

When a new and good product hits the market, how fast or slow are you to buy it? Some people get it immediately. Others wait for varying lengths of time before making their purchase decision.

According to one Internet website (, consumers can be classified into 5 categories based on how quickly they acquire new items. A picture of the famous bell-shaped curve, like the one shown here, indicated the descriptive labels and sizes of the 5 groups.

By considering the percentage of people in each of the 5 groups (as well as the position of the short, dark “notches” on the bell curve’s baseline), you should be able to discern that the statistical concepts of mean and standard deviation were used to “define” each group. For example, a person would be classified as an Early Adopter if he/she tends to purchase new products with a speed that’s between 1 and 2 SDs faster than average.

It is interesting to note that there are 3 sections on the left side of this bell curve but only 2 on the right. The pink area begins 1 SD from the mean and extends all the way to the right. Thus, the percentage of Laggards is equal to the combined percentages of Innovators and Early Adopters. Some people, if creating this picture anew, might split the pink area into 2 parts (thus forming a total of 6 sections rather than 5), with the percentage of Laggards equal to the percentage of Innovators.

To see the original discussion of what was called the “Product Diffusion Curve,” go to

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The accompanying graph shows the number of countries competing in 2 different sporting events: FIFA’s 2014 World Cup that ended last week, and MLB’s upcoming 2014 World Series.

Regarding the red bar, only 32 counties fielded teams that appeared in this year’s “group play” (from which 16 teams advanced to the “knockout” rounds). However, a total of 195 teams were involved in “qualifying” matches—held in 2011, 2012, & 2013—to determine which 31 countries would join Brazil, the host country, as FIFA’s best teams this year.

The World Cup truly is a world-wide event. But baseball’s World Series? Its world is much smaller!


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Of the 48 games played in the initial stage of the 2014 World Cup, 9 ended in ties. The accompanying chart is based on data from the remaining 39 games, with the focus on the winning teams. Each bar shows the percentage of the 39 games in which the winning team outperformed its opponent in some way or another. For example, the yellow bar reveals that in 48.7% of these contests (19 of 39 games), the winning team had more corner kicks than its opponent.

Surprisingly, the winning teams in these 39 games were not uniformly superior in possession time, corner kicks, shots on goal, or fouls. Perhaps most surprising of all is the fact that on 3 occasions the winning team possessed the ball LESS, had FEWER corner kicks, did NOT have as many shots on goal, and had MORE fouls.

World Cup

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Dr. Stephen Jay Gould, a world-famous scientist who taught at Harvard, was 40 when diagnosed with cancer. He discovered that people with his kind of cancer live for a median of 8 months. Gould’s down-to-earth essay, “The Median Isn’t the Message,” deals with his “survival expectancy.” It’s considered by some to be “the wisest, most humane thing ever written about cancer and statistics.”

In his thoughtful commentary, Gould offered some important advice to those who hear (for themselves or a loved one) grim diagnoses based on median survival rates. As Gould so correctly pointed out,

What does “median mortality of eight months” signify in our vernacular? I suspect that most people, without training in statistics, would read such a statement as “I will probably be dead in eight months”––the very conclusion that must be avoided, since it isn’t so….

Dr. Gould’s essay contains important food-for-thought not just for those concerned about cancer or other life-ending diseases, but also for those who produce or receive statistically-based research claims in any disciple. In a nutshell, his admonition says: Don’t focus so heavily on means and medians that the important underlying variability is totally overlooked. As Gould put it,

We still carry the historical baggage of a Platonic heritage that seeks sharp essences and definite boundaries. … This Platonic heritage, with its emphasis in clear distinctions and separated immutable entities, leads us to view statistical measures of central tendency wrongly, indeed opposite to the appropriate interpretation in our actual world of variation, shadings, and continua. In short, we view means and medians as the hard “realities,” and the variation that permits their calculation as a set of transient and imperfect measurements of this hidden essence.

If the median is the reality and variation around the median just a device for its calculation, the “I will probably be dead in eight months” may pass as a reasonable interpretation. … But all evolutionary biologists know that variation itself is nature’s only irreducible essence. Variation is the hard reality, not a set of imperfect measures for a central tendency. Means and medians are the abstractions.

If you’d like to hear Gould’s essay read while you see a series of photos of him at work and play, click this link:

If you’d prefer to read Gould’s essay yourself, or print it, go here:

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Suppose a rare and fatal disease randomly afflicts 1 out of every 100,000 people in the population. Also suppose you are told, after being screened for the disease with a lab test that’s accurate 99.9% of the time, that you have the disease. Are you doomed?

Probably not, because there’s less than a 1% chance you actually have the disease!!!

This surprisingly low probability is given by Bayes’ theorem which can be proven in any of 4 ways: (1) with a simple 2×2 table containing frequency counts, (2) with a Venn diagram, (3) with a tree diagram, or (4) with a formula. Worth watching are 4 short videos that show these alternative ways to determine the conditional probability of having a disease “given that” the screening test is positive. Here are the links to these clear and informative videos:

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The batting averages of the best batters slowly decrease as the baseball season unfolds. Is this because the pitchers get better? Or, do these batters simply get worse? “No” and “no,” say statistical authorities. Their explanation: “regression toward the mean.”

To see some specific information about baseball batting averages and regression-toward-the-mean, go to

To read about the general phenomenon of regression-toward-the-mean, go to

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