“High school mathematics curriculums spend too much time on traditional topics like Euclidean geometry and trigonometry.  For a typical person, these are useful intellectual exercises but have little applicability to daily life.  Students would be better served by learning more about probability and statistics.”

– N. Gregory Mankiw

I’m with him on learning more about stats but not at the expense of less geometry and trig, which come in very handy for anyone pursuing an engineering career.   Also, budding economists could benefit from a little knowledge of period functions such as sine waves.  It seems to me that what goes around comes around.

It seems to me that someone ought to develop a numerical confidence scale along these lines.  For example:

• 100% Certain.
• 99.9% Iron clad.
• 99% Beyond a shadow of a doubt.
• 95% Unequivocal.
• 90% Definitive.
• 80% Clear and convincing evidence.
• 50% On the balance of probabilities.

There are many other words used to convey a level of confidence, such as: clear-cut, definitive, unambiguous, conclusive.  How do these differ in degree?

Of course much depends on how is making such a statement, many of whom are not always right, but never in doubt. ; )  I’m skeptical of any assertion, thus I follow the advice of famed statistician W. Edwards Deming:

“In God we trust, all others bring data.”

Statistics can be very helpful for stating any conclusion because it allows one to never have to say you are certain.  But are you sure enough to say it’s “iron clad” or what?

Nevertheless, based on observation of three daughters and two sons who went off to college, I believe that Kapinos et al are really on to something.  This was sealed in my mind from the observation of my youngest girl, who soon will start her third year in a biochemistry program.  She said it really is very simple – the female students hang around the dorm dining hall for social reasons, during which times they naturally munch on stuff and pack on the pounds.  I think for the good of their students it would be wise of schools not to put cafeterias in the dorms.

“I think perhaps that women are a little more socially oriented at college, and one social thing they engage in might be, ‘Let’s go get a snack,’ Men may not tend to do that as much.”

– Wayne Westcott, Senior Fitness Executive for the YMCA in Quincy, Massachusetts

By the way, putting on weight after going away to college — the proverbial “freshman 15” – is bound to happen, I think; and for both men and women.  I gained 15 pounds after moving into a dorm at Michigan State University.  It had a dining hall in the building.  However, I’ll bet I’d have added weight just the same even if the meals were served elsewhere on campus.

Thank goodness that our marketing director had become accustomed to working with a bunch of engineers, statisticians and programmers who, when one asks “Could I talk with your for a minute?”, immediately set the timer on their digital watches for precisely 60 seconds (the the nearest one-hundredth).

Coincidentally, while vacationing in Wisconsin’s Door County, I enjoyed a fine demonstration of how hard it can be to bring a quantity of water to a boil.  It’s a tradition there to throw a bunch of fish in one kettle and vegetables in another and cook them up with a wood fire.  However, as I learned and experienced from a somewhat dangerous vantage point, a pitcher of kerosene provides the final heat needed to accomplish the boil-over.  My eyebrows needed a bit of burn-back, so that’s OK.

This sent me scurrying over to my office bookshelf for <em>The Lady Tasting Tea – a delightful collection of stories* compiled by David Salsburg.**  Page 100 of this book reports Fisher saying that below p of 0.01 one can declare an effect (that is – significance), above 0.2 not (that is – insignificant), and in-between it might be smart to do another experiment.

So it seems that Fisher did some flip-flopping on the issue of what value of p is needed to declare statistical significance.

PS.  One thing that bothers me in any discussion of p-values is that it is mainly in the context of estimating the risk in a test of the null hypothesis and almost invariably overlooks the vital issue of power.  For example, see this YouTube video on Understanding the p-value.  It’s quite entertaining and helpful so far as it goes, but the decision to accept the null at p > 0.2 is based on a very small sample size.  Perhaps the potential problem (underweight candy bars), which one could scope out by calculating the appropriate statistical interval (confidence, prediction or tolerance), merits further experimentation to increase the power.  What do you think?

*In the title story, originally told by Sir Ronald Fisher, a Lady claims to have the ability to tell which went into her cup first—the tea or the milk.  Fisher devised a test whereupon the Lady is presented eight cups in random order, four of which are made one way (tea first) and four the other (milk first).  He calculates the odds of correct identification as 1 right way out of 70 possible selections, which falls below the standard 5% probability value generally accepted for statistical significance.  Salsburg reveals on good authority (H. Fairfield Smith–a colleague of Fisher) that the Lady identified all eight cups correctly!

**Salsburg, who worked for some years as a statistician at a major pharmaceutical company offers this amusing anecdote from personal experience:

“When I first began to work in the drug industry…one…referred to…uncertainty [as] ‘error.’ One of the senior executives refused to send such a report to the U.S. Food and Drug Administration [FDA]. ‘How can we admit to having error in our data?’ he asked [and]…insisted I find some other way to describe it…I contacted H.F. Smith [who] suggested that I call the line ‘residual’…I mentioned this to other statisticians…and they began to use it…It seems that no one [in the FDA, at least]…will admit to having error.”

1. A team of U.S.-British researchers announced earlier this month that they deciphered previously-illegible scrawling by African explorer David Livingstone, which he made 140 years ago under desperate circumstances using the juice of local berries.  See the image enhancement in this article by New Scientist Tech.  Given the depressing content of Livingstone’s laments, it may be just as well he used ephemeral ink.
2. The Dead Sea Scrolls, now on exhibit at the Minnesota Science Museum (see this picture, for example), were written with extremely durable black ink (well over 2000 years old!) comprised of lamp black (soot), gum Arabic and flaxseed oil.  According to this Numerica entry on the chemistry of ink a red version was made by substituting cinnabar (mercury sulfide ? – HgS).  That must have been used by the editor overseeing publication of the Scrolls. ; )
3. Printer.com suggests that we all save ink by favoring certain fonts over others.  For example Century Gothic* uses 30 percent less ink than Arial.  As a general rule the serif fonts do better than the sans serif ones.  An article by Dinesh Ramde of Associated Press on 4/7/10 reported that a school of 6,500 students, such as the University of Wisconsin-Green Bay, can save up to $10,000 per year by switching to an ink-stingy font.  To really make a statement about their support for Earth, UW-GB ought to go with the “holey” ecofont.  However, rather than going to something so ugly, perhaps the best thing for all concerned about going green would be to be prohibited from printing anything and just hand-write what’s absolutely essential to put on paper (or papyrus).

Unfortunately the people funding consultants and researchers do not want to hear any equivocation in stated results.  However, it’s vital that experts convey the possible variability in their findings if we are to gain a true picture of what may, indeed, transpire.

“Error is to be expected and not something to be scorned or obscured.”

– Trevor Butterworth

• A pie chart reporting on a June 1-7 telephone survey by Stanford University of 1000 American adults asking their opinion on belief in global warming.  A pie chart illustrated that about 75% do believe in global warming, 20% do not, and 5% “don’t believe in pie charts”.  I suspect that the author of this editorial, Jon A. Krosnick – a professor of communications at Stanford, meant this last bit of the chart to represent those who are undecided, but the graphic designers (Fogleson-Lubliner) figured they’d have some fun.
• Olivia Judson’s comments on “Galton’s legacy” note that this preeminent British statistician once published a comment in Nature (June 25, 1885 “Measure of Fidget”) that correlated boredom by how the audience squirmed during particularly wearisome presentations.  I wish I would’ve thought of this “amusing way of passing an otherwise dull” lecture before attending two statistical conferences over the last several weeks.  Based on this 2005 assessment of “Nodding and napping in medical lectures”, the more things change the more they stay the same, at least so far as presentations are concerned.  The only difference is cost.  For example, the authors figure that at a typical 1 hour talk to 100 high-powered professionals, say master statisticians, perhaps as much as $20,000 goes up in snores.

“Nodding was common, but whether in agreement with the speaker or in reverie remains undetermined.”

– Kenneth Rockwood (Dalhousie University), Christopher J. Patterson, McMaster University, David B. Hogan (University of Calgary)

Bonferroni corrected

Uncorrected (random results)

I enjoyed a fine afternoon in the old Citta Alta of Bergamo in northern Italy – a city in the sky that the Venetians, at the height of their power as the “most serene republic,” walled off as their western-most outpost in the 17 century.

In statistical circles this town is most notable for being the birthplace of Carlo Emilio Bonferroni.  You may have heard of the “Bonferroni Correction” – a method that addresses the problem of multiple comparisons.

For example, when I worked for General Mills the head of quality control in Minneapolis would mix up a barrel of flour and split it into 10 samples, carefully sealed in air-tight containers, for each of the mills to test in triplicate for moisture.  At this time I had just learned how to do the t-test for comparing two means.  Fortunately for the various QC supervisors, no one asked me to analyze the results, because I would have simply taken the highest moisture value and compared it to the lowest one.  Given that there are 45 possible pair-wise comparisons (10*9/2), this biased selection (high versus low) is likely to produce a result that tests significant at the 0.05 level (1 out of 20).

This is a sadistical statistical scheme for a Machiavellian manager because of the intimidating false positives (Type I error).  In the simulation pictured, using the random number generator in Design-Expert® software (based on a nominal value of 100), you can see how, with the significance threshold set at 0.05 for the least-significant-difference (LSD) bars (derived from t-testing), the supervisors of Mills 4 and 7 appear to be definitely discrepant.  (Click on the graphic to expand the view.) Shame on them!  Chances are that the next month’s inter-laboratory collaborative testing would cause others to be blamed for random variation.

In the second graph I used a 0.005 significance level – 1/10^th as much per the Bonferroni Correction.  That produces a more sensible picture — all the LSD bars overlap, so no one can be fingered for being out of line.

By the way, the overall F-test on this data set produces a p value of 0.63 – not significant.

Since Bonferroni’s death a half-century ago in 1960, much more sophisticated procedures have been developed to correct for multiple comparisons.  Nevertheless, by any measure of comparative value, Bergamo can consider this native son as one of those who significantly stood above most others in terms of his contributions to the world.