Steve Kass

I’m a mathematics teacher, and I have some arithmetic problems for you to solve.

1. Mitt drives to the beach and back.

Q. Mitt Romney drives 50 miles from his home one of his homes to the beach and back one summer, with or without a dog on the roof. He averages 50 miles an hour on the way out, and he averages 10 miles an hour on the way back. What was Mitt’s average speed on the road?

A. It was 16-2/3 mph¹, but Mitt’s campaign insists he’s no slowpoke and that he averaged 30 mph. (They also say the trip explains his tan.)

2. “Going Dutch” with Mitt.

Q. You and Mitt Romney have lunch together at Chick-fil-A. The bill is $15, and he only has a $10 bill, so he pays 2/3 of the bill. Later, you have dinner together at the country club, and the bill is $300. He puts in $100, and this time you pay 2/3 of the bill. Are you even-Steven?

A. Of course not, but Mitt says you are, because he paid 1/3 once and 2/3 once.

3. Mitt Sees Red.

Q. In the four squares below, about what percentage of the area overall is red?

A. About 10%. Certainly way less than half. Mitt disagrees. He says the percentage of red is over 75%, and he surprises you by providing actual details instead of saying he has a secret calculation that solves the problem and comes out the way he wants it to. He says there are three 100% red squares and one 5% red square, for an average of 305%/4 or about 76%. Ann tells you to stop giving him a hard time.

Earlier today, the Romney campaign website posted a letter from PriceWaterhouseCoopers “PWC” LLP summarizing the Romneys’ tax returns for the years 1990-2009. The carefully-worded letter² states that

The average of the annual “effective federal personal income tax rates” as computed based on the returns as prepared during the period is 20.20%.

PWC computed each year’s “effective federal personal income tax rate” as instructed by the Romneys.

As you requested, we compute each annual “effective federal personal income tax rate” as total taxes owed divided by adjusted gross income as shown on the federal income tax returns as prepared.

First of all, the Romneys’ “adjusted gross income” (AGI) may be less than what most of us think of as simply “what the Romneys made.” For example, the expenses of carrying on a trade or business (such as dressage) can be deducted from total income. The Romneys’ tax rate on their total income is lower than that on their adjusted gross income. Maybe a little, maybe a lot.

More importantly, however, is how Mitt and Ann instructed PWC to compute a number they could call an “average.”

Despite appearances, no one has said that the Romneys paid 20.20% of their adjusted gross income in federal income taxes during the period 1990 – 2009. They probably paid less.

The 20.20% figure is not the Romney’s federal income tax rate for income they earned during the period 1990 – 2009. It’s the average of 20 separate annual rates. Mathematically, it’s a weighted average of their annual rates.

If you think the average of 20 annual rates is the same as the 20-year average annual rate, you’re wrong. By that flawed logic, Mitt averaged 30 mph, you and he are even-Steven for splitting meals, and the colored squares in the picture above are 75% red. Go away.

Further blurring the issue, the campaign today also posted a letter from Brad Malt, the “trustee of the Romney’s [sic] blind trust,” about Mitt’s taxes. Malt tries to say that PWC calculated something that it didn’t calculate.

Regarding the PWC letter covering the Romneys’  tax filings over 20 years, from 1990 – 2009:

Over the entire 20-year period, the average annual effective federal tax rate was 20.20%.

Malt omitted two short and crucial words: of the. PWC did not say that Romneys’ average annual effective federal tax rate was 20.20%. They said that if you average the 20 separate effective rates on their 20 federal tax returns, you get 20.20%. It’s a silly thing to calculate, but it’s what the Romneys told them (and presumably paid them) to do.

The Romneys’ overall tax rate might have been lower than 20.20%. It might have been higher, too, but given the outrage over the relatively low rates they pay compared to working Americans, if it were higher, he’d have said so. And Mitt won’t release his tax returns. We can only assume the worst.

If you value the truth, vote for Barack Obama on November 6, 2012.

¹ Let’s figure it out. Mitt drove a total of 100 miles. It took him 1 hour (50 miles at 50 mph) to drive the first half of the trip and 5 hours (50 miles at 10 mph) to drive the second half. He drove a total of 100 miles in 6 hours, so his average speed was 16-2/3 miles per hour.

² The letter is signed personally, in blue, by PriceWaterhouseCoopers LLP, because why wouldn’t they sign it? Corporations are people.

The Soda Police are getting noisier lately, but their concern for public health is a subterfuge. When it comes down to brass tacks (and I doubt brass’s slight lead content is going to kill you when used judiciously in plumbing, by the way), the S.P. don’t care most about the public health or about overweight kids at risk for diabetes and heart disease. They’re hell-bent on demonizing soda, especially soda made by Big Food and sold by the Big Chain Store and Restaurant Corporation.

Demon or not, it probably won’t hurt Americans to drink less soda on average than we do now. It will definitely help the environment if we drink less of anything that comes in individual single-use containers — even water — if there’s an environmentally friendly alternative already in place.

Here’s a simple two-part proposal to bring back running water.

BBRW Part 1. Require public water fountains everywhere.

Schools, parks, subway stations, airports, shopping centers, offices, stores, and more. We already require a lot of things, sensible and otherwise, so the means is in place. Require enough of them so no one has to wait in line. These water fountains (bubblers in Wisconsin and parts of New England) should have good water pressure, and they should be designed so they can fill up a bottle, too — or there should be some faucets for that. Simply making it possible to fill a personal water bottle in an airport — and yes, you can carry one through security so long as it’s empty — will reduce heart disease.

No flow restrictors, either; use spring-loaded knobs to conserve. (I’m not going to say a word about those infrared hand-wavy travesties.) Restrictors belong in kitchens and showers, if anywhere. It doesn’t need to take ten minutes to deliver half a cup of water. ADA compliant, but otherwise basic and solid. Call me nostalgic, but I like porcelain-coated cast iron.

Room-temperature, pure water is already available from every municipal water system. Only a little effort makes it ubiquitous. (If you’re afraid it will give you cancer, carry your own personal PET-free container full of home-purified water.)

BBRW Part 2. Require water to be available everywhere soda is available, for less.

If a restaurant offers a meal that includes soda, require it to offer the same meal with the same size tap water for less money. Less by at least half the restaurant’s own à la carte price for the included soda. Except during water emergencies, require restaurants to offer tap water when patrons are seated.

Stop the endless debates over soda vs. fruit juice, sugar versus high-fructose corn syrup, artificially-sweetened beverages vs. sugary ones, and aspartame vs. stevia extract. Bring back running water.

I spent a good chunk of the last 24 hours at one of my favorite hangouts, Language Log. My reason for lingering was to pore over (in good company) some interesting graphs Mark Liberman had put up about the ever-controversial adverb “literally.” [Link: Two Breakfast Experiments™: Literally]

The graphs purported to show, inter alia, “a remarkably lawful relationship between the frequency of a verb and the probability of its being modified by literally, as revealed by counts from the 410-million-word COCA corpus.” [Aside: Visit COCA some time. It’s beautiful, it’s open 24/7, and admission is free.]

Sadly (for the researcher whose graphs Mark posted), there was no linguistic revelation; happily (for me and other mathophiles) the graphs highlighted a very interesting statistical artifact. Good stuff was learned.

Instead of rehashing what you can find in the comment thread at Language Log, what I’ll do here is give a non-linguistic example of this statistical artifact. First, a very few general remarks about statistics.

Much of statistics is about making observations or drawing inferences from selected data. In a nutshell, statistical analysis often goes like this: look at some data (such as the COCA corpus), find something interesting (such as an inverse relationship between two measurements), and draw a conclusion (in this case, a general inference about American English, of which COCA is one of the largest samples available in usable form).

Easy as a, b, c. One, two, three. Do, re, mi.

Sometimes. The mathematical underpinnings of statistics often make it possible, given certain assumptions, to make inferences from selected data with some (measurable) measure of confidence. Unfortunately, it’s easy to focus so hard on measuring the confidence (Yay, p < 0.05! I might get tenure!) that you forget the assumptions or you get careless about how you state an inference or calculation.

When bad statistics happens, there’s often a scary headline, but I can’t think up a good one at the moment and I’ll go straight to the (artifactual) graph.

This graph shows that for not-too-small cities, there’s a modest negative relationship between city size and homicide rate: on average, smaller cities tend to have higher homicide rates.

But the truth is that among not-too-small cities, smaller cities don’t tend to have higher homicide rates than larger ones. Here’s a better graph:

This graph shows almost no relationship between city size and homicide rate.

What’s going on, and what’s wrong with the relationship that shows up (and is real) in the first graph? The titles hold a clue (but don’t count on such clear titles when you see or read about graphs in the news). The first graph only shows cities that had at least 10 homicides in 2009. For that scatterplot, cities were selected for analysis according to a criterion related to the variable under investigation, homicide rate. That’s a no-no.

The 10-homicide cutoff biased the selection of cities used in the analysis. Most very large cities show up simply because they’re large enough to have 10 or more homicides, but the smallest cities that appear are only there because they had high enough homicide rates to reach 10-homicide threshold despite their relatively small populations. For the first graph, I (pretendingly) unwittingly chose all large cities together with only some smaller cities, specifically smaller cities with unusually high homicide rates for their size. Then I “discovered” that smaller cities had higher homicide rates.

Oops. It’s an easy mistake to make, and it wouldn’t surprise me if it happens often. I can easily imagine medical studies that compare the rates of some disease among cities and exclude any city that has “too few” cases of the disease to analyze.

Statistics is a powerful tool. Follow the instructions.

A few months ago, I posted a graph and parametric equations for conchiglie rigati. Today I’m sharing a graph and equations for cavatappi. As before, I started with equations from Chris Tiee’s 2006 class notes for vector calculus.

Coming soon, a graph and equations for Möbius pasta.

Wandering the internet today, I stumbled upon pasta and mathematics. At the same time. Chris Tiee, a teaching assistant for one of UCSD’s vector calculus courses, had put into his class notes back in 2006 a short and very cute parametric equations quiz: match the parametric equations to the pasta shape. And he (or UCSD) conveniently left his notes on the web for posterity — or should I say pastarity?

His parametric equations were pretty basic — absolutely fine for a vector calculus quiz — and I thought I might be able to touch them up a bit. Here’s what I came up with for conchiglie rigati.

This exercise is also my excuse for finally getting MathJax up and running on my blog. [Update: I’ve disabled MathJax, because it mucks up non-LaTex posts that have $ characters. At some point I’ll figure out how to configure it amicably, but for now, the pastametric equations are provided as an image file.] You might find that this page loads slowly, and I don’t yet know if I can do anything about that. If  you don’t see any equations below the picture, however, please let me know.

Conflict. Today, my writing was likened to Dan Brown’s, and I’m compelled to demonstrate at least a rudimentary grasp of grammar and its subtleties.

I write like
Dan Brown

I Write Like by Mémoires, Mac journal software. Analyze your writing!

Interlude. Let me explain how I arrived at this conflict; skip to the dénouement if the travelogue begins to bore you. [Note to self: look up or else coin the adjectival form of interlude; consider interludinous, interludinal, interludinary, interludine.]

The comparison of my writing with Dan Brown’s occurred earlier today, while I was visiting I Write Like, a momentarily amusing web¹ site at http://iwl.me. I arrived there from this CONJUGATE VISITS post (sorry, but its author yells the title). I happened onto CONJUGATE VISITS while looking up “supposably,” which I learned today is a word (note the absence of scare quotes around “word”), as opposed to a “word,” which would have been my first guess.

The next step back is a tad embarrassing. I only realized where I’d been before looking up supposably when I retraced my steps for this blog post; I’d gotten the idea to look up supposably from this article on the web site of Reader’s Digest, a generally icky place I wouldn’t have visited intentionally. A tweet from Phil Jimenez led me to the Reader’s Digest article (more specifically a bit.ly URL in the tweet, and I submit disguise-by-shortening as my excuse).

I don’t recall whether I read Phil’s particular tweet before or after I noted that he and I shared exactly one Facebook like, Dan Savage. That was no surprise, given what (or who? It’s a fictional character, so I’m not sure.) led me to Phil’s Twitter stream in the first place — Kevin Keller. Kevin, as you may know, made his appearance in Veronica #202 today; while I’ve yet to get my hands on the issue, I’d caught wind of it from Google News and consequently searched Twitter for the latest buzz, finding Phil, then Reader’s Digest, then supposably, then CONJUGATE VISITS, then I Write Like. In summary,

• I Write Like, from
• CONJUGATE VISITS, from
• supposably, from
• Reader’s Digest, from
• @philjimeneznyc, from
• Kevin Keller, from
• Google News, from
• daily routine.

Dénouement. On to my demonstration. Consider the following sentence, which I found on Amazon in a one-star review of CONJUGATE VISITS’s authoress June Casagrande’s book, It Was the Best of Sentences, It Was the Worst of Sentences, here.

Copernicus was thrilled when he discovered that the earth revolves around the sun.

Casagrande and the reviewer both prefer this to “Copernicus was thrilled when he discovered that the earth revolved around the sun.” I on the other hand, presently compelled to say something about grammar, offer an even better sentence.

Copernicus was thrilled to discover that the earth revolves around the sun.

The proposition of Casagrande’s sentence (either version) has two parts. Deconstructing the sentence rigorously, it states first that Copernicus was thrilled, and second that Copernicus’s² thrill occurred when he made his now famous discovery. However, the second part of the proposition is perplexing, if only slightly. If the writer had stopped after “Copernicus was thrilled,” I’d have felt cheated, but because she’d failed to explain why he was thrilled, not because she’d failed to explain when he was thrilled. Emotions interest readers because of their why, not their when.

For most readers, I’m sure the second part of the sentence as written sufficiently explains the why. Similarly, if the “thrilled when” sentence were part of an SAT reading comprehension question, the “correct” answer to Why was Copernicus thrilled? would be a) Because he discovered that the earth revolves around the sun., not d) It’s impossible to determine from the reading. But why explain “why?” indirectly by explaining when? The turn of phrase “thrilled to discover” isn’t the only choice — one might say “thrilled by his discovery” or “thrilled to have discovered,” but it’s the best choice, and this is my blog. Also, I might have answered d) to the SAT question, especially if I knew I’d get to argue with a teacher about it later. I don’t brag about my SAT English score, and for good reason.

Epilog. Dare I paste this blog post into I Write Like? And if I do, then post the result here, then paste it in again, will the result be the same, and if not, and I repeat the process… [Update: The result is … H. P. Lovecraft. I’ll leave it at that. Tear from the fabric the threads that are old!]

I write like
H. P. Lovecraft

I Write Like by Mémoires, Mac journal software. Analyze your writing!

Postscript. You, dear reader, are a mensch for getting to this point. Let me know how I can return the favor. You are almost as much of a mensch as Itzik, who hired me as an editor … twice, the second time after knowing how I go on about things like this.

¹ By writing web and not Web, I comport with one of the “Significant Rule Changes” in the latest edition of The Chicago Manual of Style. The interested reader (which is to say You, because you’ve read this far into my footnote) can find the full list here. This footnote is not an endorsement of The Chicago Manual of Style.

² Ibid. Among the Significant Rule Changes are rules on the possessive forms of two kinds of names: those ending with an unpronounced “s” and those ending with an “eez” sound (in the latter case presumably when the name also ends in “s,” because there can’t be any debate on possessives like Lise’s). Copernicus falls into neither category, and I don’t know the latest rule on his possessive. My rule is to always add ’s to form a possessive (as in This is Steve Kass’s blog.) except maybe for Jesus, Moses, and princess. Even for them I’m not certain what I’d do, but they don’t come up in my writing much.

Web sites about mathematics should help people understand and appreciate mathematics, not confuse the crap out of them with misinformation. Unfortunately, Wolfram Mathworld does the latter.

Example 1. MathWorld explains here that “The numbers of palindromic numbers less than a given number are illustrated in the plot [below].”

So the left plot tells us that there are about 100 palindromes less than or equal to 20. But there are only 21 nonnegative integers less than or equal to 20, so there can’t be 100 palindromes among them. In fact, there are 11 palindromes less than or equal to 20: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 11. My guess is that the left plot illustrates the n-th palindromic number as a function of n. In any case, it’s not what MathWorld describes.

MathWorld begins its list of the “first few palindromic numbers” with 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (these 10 numbers are palindromes and are all less than 10), but in the next paragraph, MathWorld states that the number of palindromic numbers less than 10 is 9. There are 9 if you don’t count zero for some strange reason, but if you don’t intend to, give a definition that excludes it (MathWorld’s definition is less than clear), and then don’t list it.

Still confused? Read the Wikipedia article.

Example 2. Pascal’s Triangle shouldn’t be hard to screw up, right? Wrong. Here’s MathWorld’s Pascal’s Triangle:

This triangle needs to go to the shop for an alignment. The numbers are neither lined up in columns nor staggered (the latter being the usual presentation). What are the numbers in the column containing the rightmost 4? What numbers are along the diagonal through the top? (1, 1, 1, 1, 1, 5, 6?) As shown, MathWorld’s anyway-ill-worded “each subsequent row is obtained by adding the two entries diagonally above” is meaningless.

Example 3. In its article on Mersenne numbers (numbers that are one less than a power of two), MathWorld attempts to explain why “[i]n order for the Mersenne number [2ⁿ-1] to be prime, n must be prime.” MathWorld’s justification: “This is true since for composite n with factors r and s, n = rs. Therefore, 2ⁿ–1 can be written as 2^rs–1, which is a binomial number and can be factored.” That’s sloppy to say the least. First, if a composite number n has factors r and s, it’s not necessarily the case that n = rs. Furthermore, the fact that a number can be factored doesn’t prove it’s composite. Every Mersenne number 2ⁿ–1 can be factored. It’s just that when n is composite, there’s definitely a factorization into positive integers neither of which equals 1. Explaining it isn’t hard: In order for 2ⁿ-1 to be prime, n must be prime. For if not, n = rs where r and s are integers greater than 1 and less than n; then 2ⁿ–1 = 2^rs–1 has a factor between 1 and 2ⁿ–1, namely 2^r–1.

Example 4. MathWorld describes prime numbers as “numbers that cannot be factored.” Prime numbers, like all integers, however, can be factored, and elsewhere, MathWorld gives the factorization of several prime numbers, such as 7: 7 = 7×1.

Example 5. Any of MathWorld’s articles on statistics.

In the article on the Central Limit Theorem, what is lowercase n? What is f? The “limiting cumulative distribution function” of X_norm is limiting in the sense of what approaching what? (It’s not clear to me that MathWorld’s statement of the theorem is even correct, but it’s clearly unclear.)

The article “explaining” the p-value has perhaps the worst definition of p-value I’ve ever seen when not grading exams. MathWorld says it’s “[t]he probability that a variate would assume a value greater than or equal to the observed value strictly by chance: P(z > z_observed)” (wrong). Wikipedia says “In statistical hypothesis testing, the p-value is the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true” (right).

In today’s number news (State-by-state cremation rates in U.S.), we learn that “slightly more than a third of all persons who died in 2006 were cremated, according to the Cremation Association of North America.” Happily, the article contained the raw data, but only as an alphabetical-by-state table of numbers.

Here’s an illumination, as MapPoint is my amanuensis. Click to embiggen.

Explanation: Pie areas are proportional to the number of deaths; the yellow slice is cremations, the red non-cremations.

Pies for our nation’s two newest states are not shown. Alaska’s looks like a two-thirds size Vermont pie; Hawai’i’s looks like a one-third size Oregon pie.

Today’s eLetter from the folks at Fine Cooking began “Baked Pasta 259,200 Ways. We did the math.” As you can imagine, I did the Baked Pasta Recipe Maker math, too. I figure it’s 16,128,000 ways, or about 60 times the number Fine Cooking found when they did the math. Here’s the calculation. The recipe maker walked me through the following steps:

• Choose one or two of four Flavor Bases
• Chose one of three Sauces
• Choose two or three of nine Sauce Enhancers
• Choose one of eight Pastas
• Choose zero, one, or two of five Vegetables
• Choose two or three of six Cheeses

Assuming no choice combinations are forbidden (the recipe maker doesn’t appear to prevent you from adding olives and sherry vinegar to sausage and chicken in pink sauce, for example), you find total number of different ways to make a choice at every step by multiplying together the numbers of choices at each step.

It’s easy to count the number of ways to “choose this many of those Things.” If this many is k, and those Things are n in number, the number of ways to choose k of the n things is “n choose k,” sometimes written as C(n,k). These numbers can all be found in Pascal’s triangle. As it’s shown here, C(n,k) is in the row labeled with the n value, under the column labeled with the k value. Here’s how to use the triangle to find the value of C(9,3):

• To choose one or two of the four Flavor Bases, there are C(4,1) = 4 ways to choose one plus C(4,2) = 6 ways to choose two, for a total of 10 ways to choose this item.
• To choose one of the three Sauces, there are C(3,1) = 3 ways.
• To choose two or three of the nine Sauce Enhancers, there are C(9,2) = 36 ways to choose two plus C(9,3) = 84 ways to choose three, for a total of 120 ways.
• There are C(8,1) = 8 ways to choose a pasta.
• There are C(5,0) + C(5,1) + C(5,2) ways to choose up to two vegetables, or 1 + 5 + 10 = 16 ways.
• There are C(6,2) + C(6,3) to choose the Cheeses, or 15 + 20 = 35 ways

Multiplying these numbers of choices for each step yields 10·3·120·8·16·35 = 16,128,000 ways, about 60 times as many as Fine Cooking found when they did the math. Counting ways isn’t standard recipe math, and I’d like to note that Fine Cooking’s math is generally fine when it comes to ounces, grams, cups, servings, and calories.

Scientific American, you ruined my day, but thanks, I needed it.

Silly me for thinking the Math Wars ended when Mathland bit the dust a couple of years ago. Last May, according to this month’s Scientific American, the Seattle School Board adopted the “Discovering Mathematics series, a reform-math high school text that uses student investigations as a means of discovering math principles—such as using toothpick models to derive recursive sequences.”

I looked at it for as long as my stomach could bear — at least at the one chapter that’s available online as a .pdf file here. It’s wretched. Wrong. Not only wrong like in I-don’t-like-it wrong (which it also is), but falselike wrong. And bad, stupid, dumb, and foolish, among other things. It would take me too long to point out all the things wrong in just the first few pages. (I won’t lie. There were some good things, but not many.)

I don’t think the students who wouldn’t have gotten much out of mathematics curricula in the ‘60s will do any better with this. For the students who want to learn mathematics, unfortunately, school will be even more of a waste than it used to be. They should do their best (especially if they go to public school in Seattle) to learn mathematics from the Internet, which is not nearly so wrong as Discovering Mathematics. With luck, any poor grades they get in stupid reform math courses won’t count against them, and if College Board caves and reforms the SAT to correlate with grades in stupid reform math courses, there will hopefully still be pressure for them to keep the AP and SAT II tests. If everything falls apart, kids that like math can drop out of school, learn from the Internet, then make a living tutoring the hapless victims of the new reform math.

Oh, and if you ever see an elevator whose “control panel displays ‘0’ for the floor number,” when it’s at the basement, please take a photo and send it to me.