Math Confusion in the News: percent

Last week Governor Arnold Schwarzenegger proposed a temporary (3-year) sales tax increase in California to help close the budget deficit. Some newspapers, however, are mixing up the amount of the increase in an effort to get the news out.

Error #1: “Governor Proposes 1.5 Percent Sales Tax Hike” from MyFox Los Angeles

The proposed increase isn’t actually 1.5 percent (which wouldn’t be all that much). It’s 1.5 percentage points, which makes it about a 20% increase from the current 7.25% state sales tax. I suspect that most people understand what the headline intends, however, because using “percent” instead of “percentage point” is fairly common. (Kudos to the LA Times for being precise in their story!)

Error #2: “Schwarzenegger proposes 1.5-cent sales tax increase to close budget gap” from the San Jose Mercury News

This headline is just wrong. A 1.5¢ tax? And it’s not just in the title, but in the body of the story. Several other newspapers made the same mistake, either running the Mercury News story without correction or writing their own story about the 1.5-cent increase (I’m looking at you, Sacramento Bee). Indeed, these 1.5-cent increase stories were common enough that I actually double checked that it wasn’t some new terminology for “percentage points”.

Incidentally, the word “per cent” is only 440 years old, and “per centage” only 222 years old. Tidbits from the OED!

Zero isn’t nothing

When there are a lot of zeros after a number, it’s easy to add or drop some. Apparently.

Two examples: there’s an email that was going around in September about the proposed Federal Bailout Plan. The plan at the time was proposed to be around $700 billion, and the author of the email pointed out that if this was divided by the 200 million or so adults in the United States, it amounted to around $350,000 per person, and the author suggested that a better solution would be to mail a check for $350,000 to each person instead. Now before you go looking for ways to spend the money, you need to double check the math. It turns out that $700 billion divided by 200 million adults is only $3500/person. No small amount, to be sure, but not quite enough to pay off the mortgage. [Note: Variations on this suggestion appeared on several websites with different numbers, but they were all off by a factor of 10 or 100.]

The web sites that posted this quickly caught the mistake and made a correction. And all was well. Except that this pesky problem of the wandering zero just appeared again. This article in The Guardian (and repeated elsewhere) begins:

Nuclear power plants smaller than a garden shed and able to power 20,000 homes will be on sale within five years, say scientists at Los Alamos, the US government laboratory which developed the first atomic bomb….’Our goal is to generate electricity for 10 cents a watt anywhere in the world,’ said John Deal, chief executive of Hyperion. ‘They will cost approximately $25m [£13m] each. For a community with 10,000 households, that is a very affordable $250 per home.’

Feelings about neighborhood nuclear reactors aside, there’s a little problem with the math. $25 million divided by 10,000 households is $2,500 per home, not $250.* Perhaps still affordable compared to the cost of heating one’s home in the wintertime, but illustrative of the care that one needs to take when dealing with really large numbers. Especially when it comes to bailouts or nuclear reactors.

*an error originally caught by Classical Values

Math Mistake costs several hundred jobs

CNN reported on Friday (and JD2718 mentioned on Saturday) that the Dallas school system had to lay off some employees after realizing that its budget was just a tad short, where “just a tad” stands for $84,000,000 [more even than the $64 million first reported]. Whoops! According to the CNN news article:

The district laid off 375 teachers and 40 counselors and assistant principals Thursday, and transferred 460 teachers to other schools within the district.

The deficit was caused by a massive miscalculation in the budget, CNN affiliate WFAA-TV reported.

So that got me wondering what possible miscalculation could have led to that kind of error. It turns out that a large part of it was due to underestimating the average salary of the teachers (which runs on the order of $50,000 for elementary school teachers, according to this site). In this September 23 video, Dr. Michael Hinojosa (superintendent of the Dallas Independent School District) stated:

When you take, you underestimate your average teacher’s salary by $3900 and you multiply that over 11,000 teachers, then that creates a huge budget error.

Yes. Yes it does. About 43 million dollars. The rest of the error was apparently due to not following a formula for how many administrative positions to have at each of the 225 DISD campuses, resulting in 1-2 “extra” people at each campus, for a total of 338 extra positions.

Moral: Being close is not always good enough.

Per mille measurements

Surely you’ve heard of percent, and probably have a sense of its etymology: per centum = per 100, for describing fractional quantities in their equivalent form as parts out of 100. For example, since , we say that 3/4 = 75%.

But are you familiar with per mille ? Denoted by the symbol ‰, per mille measures parts per 1000, or equivalently tenths of a percent. Thus 3/4 = 750‰.

Per mille measurements occur in a variety of contexts, typically ones in which small-scale changes in value are relatively significant, and so are worth the extra scrutiny that per mille units allow. Typical examples include roadway grades (or gradients), water salinity, and (perhaps most familiarly for our US readers) baseball batting averages.

Chipper Jones (of the Atlanta Braves) currently leads the Major Leagues in batting average: he has a total of 156 hits, out of 427 at-bats, for a batting average of

Given that he’s only had 427 at-bats, not all that many digits are significant digits in the decimal expansion; baseball statisticians universally round off such expansions at the third place: his batting average is .365, which one could call 365 per mille.

A perfect batter, one who got a hit every single time at bat, would have a batting average of 1. But saying that the batting average is “one” might be misleading, in a context in which baseball fans typically describe a .365 hitter as batting “three hundred sixty five” — it seems “one” might be thought of as .001 as a batting average.

So the culturally accepted practice is to say that a perfect batter is “batting 1000″. And indeed, their batting average of 1 is also 1000 per mille.

Sadly, as the following photo attests, the phrase “batting 1000″ is still sometimes subject to misinterpretation:

Perhaps we should incorporate the phrase “per mille” into our daily lexicon, for the common good. (Or maybe this just means that it is really hard to hit a baseball when you’re wearing a big furry mouse costume.)

Google Calculator has a little trouble

Seriously, we don’t mean to write only about math mistakes and promise to have something non-mistakey tomorrow and even Thursday. But for today, it turns out that Google Calculator is having a spot of trouble. If you type in a big calculation like 800000000000010-800000000000007 you get the expected answer of 3, but if you become all sneaky and change that to 800000000000010-800000000000008 then you get 0. The article “Google’s calculator muffs some math problems” on cnet news yesterday has other examples to play around with. Google released the statement:

We are aware that the calculator tool in Google Web search is not working properly for certain calculations, and we are looking into this problem further. We apologize for any problems that this causes our users.

which suggests that this might turn out to be more than floating point error. Maybe.

As a bonus mistake, the article above refers to Google wanting to raise $2,718,281,828 in its IP0 several years ago [those digits look familiar to anyone?] but calls it $2,718,281,828 billion instead of $2.7 billion. As one commenter points out, $2,718,281,828 billion is a lot of money.

Regretting the Error

We often post Math Mistakes on this site: times when a little checking could have gone a long way. The same thing applies to journalism, and on the site Regret the Error Craig Silverman posts “media corrections, retractions, apologies, clarifications and trends regarding accuracy and honesty in the press”. They’re often amusing, like the recent correction:

A picture purporting to show Apple’s corporate headquarters in Cupertino (Google pipped – Apple the new king of Silicon Valley as market value overtakes hi-tech rival, page 3, August 15) in fact showed Symantec’s headquarters nearby

Not surprisingly, there are a lot of mistakes that have to do with numbers and math. A LOT of mistakes. These are a little different than the math mistakes we usually post, because the problem is typically in the reporting rather than the original story, but they still include such stories as:

The Star Ledger, August 20, 2008
Due to an editing error, the For Collectors column in Saturday’s Abode section reported incorrectly that the 2009 Double Eagle gold coin would sell for $20. While the coin will have a $20 denomination, it will contain an ounce of 24-karat gold and will sell for approximately $900, depending on the value of gold at… (Story)

New Scientist, August 4, 2008
We said that Australian companies “forecast spending $800 between 2002 and 2013 on geothermal exploration” (19 July, p 24). That should have been $800 million. (Story)

The New York Times, July 15, 2008
An article last Monday about the United States Olympic swimming trials, including the accomplishments of Dara Torres at age 41, misstated the age of a Canadian swimmer from the 1972 Games. Brenda Holmes was 14, not 44, when she competed for Canada. (Story)

For a list of many of the mistakes (often titled “Fuzzy Numbers”) see here.

An old math mistake: crab boat buyout

I was just looking some stuff up on Google (always eager to find math mistakes in the news), and I ran across this story from June 2004 of a math mistake that caused all sorts of trouble in the crab boat industry. Apparently there was a program in which federal fishing authorities would buy back crabbing boats [to compensate for their being too many boats for the amount of crab available], with the buyouts related to the fishing history of the individual boat. The problem occurred with boats that had multiple owners: if a boat had three owners, for example, then the catch was counted three times (once for each owner) and the boat’s total was incorrectly listed as being three times as large as it actually was. The article continued:

This led to another mistake. The NOAA officials then used the flawed history totals to calculate how much crab would be divided among the remaining boats — and how the fleet would repay the buyback loan.

According to a later article the mistake was apparently rectified and 25 boats bought out (compared to the original 28 that were planned).

Speaking of math mistakes, on God Plays Dice last week there was a quote from The New York Times which referred to 300,000 million Chinese playing basketball. That’s a lot of people.

How to lose inches without even trying.

I was just looking over this morning’s paper, and reading the story “Russian champion disses Jenn” about how pole valuter Yelena Isinbayeva was pretty sure that she was going to win the gold [which she did later today], and that Jenn Stuczynski was unlikely to surpass her. The third paragraph in the story read:

Asked if she was annoyed by media suggestions that Stuczynski was a challenger after her U.S. record vault of 16 feet, 3/4 inch (4.90 meters) earlier this season, Isinbayeva was utterly dismissive.

This was followed shortly by a quote from Isinbayeva:

“They said, ‘Wooooo’ when she jumped 4.90 (16 feet, 1 inch), but I jumped this height four years ago. It is nothing special.”

Personally I think that vaulting over 16 feet is pretty special indeed: I believe these are the only two women who have ever done it. But what caught my eye was that 4.9 meters was stated as the equivalent to 16 feet, 3/4 inch in the first case, but was translated to 16 feet, 1 inch in the second.

So I checked. It turns out that 16 feet, 3/4 inch is 489.585 cm, which does round to 4.90 meters. Furthermore, 4.90 meters is 16 feet, 0.91 inches, which rounds to 16 feet, 1 inch. So my initial thought was that everyone was just rounding.

Then I checked the USA Track and Field conversion site which had the same hedging, but in the opposite direction — everything is rounded down instead of up. It says 16′ 1″ should be converted to 4.90 meters, but 4.90 meters should be converted to only 16′ 3/4″ . And what should 16′ 3/4″ be converted to? To 4.89 meters. Which converts to 16′ 1/2″. Which converts to 4.88 meters. Which converts all the way down to an even 16′. And of course 16′ converts to 4.87m, which converts to 15′ 11 3/4″, which — hold on to your hats here — also converts to 4.87m. Finally, a fixed point!

And by transitivity of conversion, we have that 16 feet, 1 inch is equivalent to 15 feet, 11 3/4 inch.

Photo (cropped) of Yelena Isinbayeva by Eckhard Pecher, published under Creative Commons Attribution 2.5.

Math Fails

Two photos from The Fail Blog:

see more pwn and owned pictures

see more pwn and owned pictures

The Fail Blog is a fantastic place to visit, though not all the photos are safe for work [and certainly many of the comments are safe for neither work nor home.]

Edited 8/28 to add: Apparently that second photo is actually a promotional poster for The Simpsons — the KWIK-E-MART in the corner should have tipped me off.  But the fact that one can buy a twin pack for more than twice the price of one (as mentioned in the comments below) meant that I never question the reality of this supposed promo!  And the first picture is real, I believe.

Order of operations: does it really matter?

George Orwell

While surfing the webpages of a variety of newspapers this morning, I stumbled on the following….

On the staged-reality-tv show Big Brother (UK version), they gave the housemates a mathematical task: they had to compute three different sums, then use the three resulting answers as the combination to a safe.

Implicit in the problem was that the three calculations should each result in a two-digit integer.

Video posted to the Channel 4 website shows the contestants muttering, struggling, and having an extremely difficult time of it. And with good reason!

Here are the three calculations they were given, as posted on the Channel 4 website:

Sum#1: 3 x 17 – 24 + 78 x 9 ÷ 5 – (13²) + (65 – 29) ÷ 4 + (4²) – (7 x 3) + (3²) + 99 – (7²) – 49

Sum#2: 1396 x 2 ÷ 4 — (12²) + 46 x 2 ÷ 40 x (5²) – (7 x 99) x 3 – (11²) x 5 – 219

Sum#3: 100 – 33 x 5 + 665 ÷ (5²) x 17 – 248 x 3 ÷ (4²) + 52 ÷ 7 + (273 – 217)

From the look of things on the on-line video, I’m guessing that the contestants had no writing implements, and had to do all of this in their head. That makes this challenging enough, I suppose.

Making matters worse is that none of the three sums is integer valued; they work out to 62/5, -4583/2, and 26189/70. Rather, these are their values if one computes using the usual order of operations, where exponents have precedence over other operations, where multiplication and division take precedence over addition and subtraction, where calculations are performed left-to-right, and parentheses can be used to override this sequencing. (“PEMDAS” is a popular acronym with my students, standing for “Parentheses, Exponents, Multiplication and Division, Addition and Subtraction”, and sometimes recalled using the mnemonic “Please Excuse My Dear Aunt Sally”)

Apparently the folk who created this puzzle expected their contestants to work left-to-right, ignoring operator precedence, in the way that a $1 calculator might do. (Calculators that do pay heed to order of operation conventions are often marketed as “scientific” calculators.)

For example, the first sum should go as follows:

3 x 17 – 24 + 78 x 9 ÷ 5 – (13²) + (65 – 29) ÷ 4 + (4²) – (7 x 3) + (3²) + 99 – (7²) – 49

= 3 x 17 – 24 + 78 x 9 ÷ 5 – 169 + 36÷ 4 + 16 – 21 + 9 + 99 – 49 – 49

= 51 – 24 + 702/5 – 169 + 9 + 16 – 21 + 9 + 99 – 49 – 49

= 62/5

But I suspect the intended calculation was instead:

3×17 = 51, 51-24 = 27, 27+78 = 105, 105×9 = 945

945÷5=189, 189-(13²)=20, 20+(65-29)=56. 56÷4=14,

14+(4²)=30, 30-(7 x 3)=9, 9+(3²) + 99 – (7²) – 49 = 19

Similar (incorrect!) computations for sum #2 and sum #3 yield 31 and 75, respectively.

Clearly it is important that we agree on our order of operations.  But why do we prefer one over the other? Is this merely a cultural convention?  One stock answer to this is to point to the algebra of polynomials: our conventions regarding operator precedence play a central role in how we interpret linear equations (e.g. what is the slope of the line ?), how we interpret polynomials (e.g. is a quadratic or a cubic polynomial?), and how we compute sums and products of polynomials.

But this morning, having not yet had my first cup of coffee, I wonder: is it possible to change the rules of arithmetic, so that all operations have the same precedence (unless exceptions are forced by parentheses), and to develop a meaningful algebra based on similar principles? It seems to me that the answer is yes, and I wonder exactly what is lost by doing so, other than familiarity.

The Trouble with Units

There have already been a couple examples of what can go wrong when you mix up Metric and Imperial units on a Boeing 767 or a spacecraft to Mars. But it turns out that even nonstandard units can cause a little bit of trouble. By “nonstandard units”, I mean units such as spork and by “little bit of trouble” I mean an accounting error of $66,500,000. That’s a lot of cutlery.

The error is related to the aftermath of Hurricanes Katrina and Rita. Back in 2005, a large number of house supplies (pots, pans, toilet paper, etc.) were donated or purchased for people who had lost their homes. The items were stored in a warehouse in Louisiana, and then in Texas, and then…nothing. They sat there in Forth Worth, and FEMA paid about a million dollars a year to keep them stored. And not long ago FEMA decided to give them away rather than continue to pay that storage (plus the warehouse was apparently going to be torn down). At this point the story became quite public, and Louisiana said that HEY they still needed those, because three years later there are still people who are recovering, and why have all these things been collecting dust anyway? And FEMA said, Well we offered them to you and you didn’t want them, so don’t blame us. Etc.  Etc. And during all this arguing, the reported value of the supplies was listed at $85 million.

But now it turns out that it wasn’t $85 million after all — it was more like $18.5 million. Because when they were counting things, they counted single items such as  one spork the same as entire cases, as in one case of sporks. As the General Services Administration explained last week, “The final adjustments reveal there was a significant overstatement in the total asset valuation.”

So on the one hand, this is good because that’s $66.5 million dollars that wasn’t wasted after all (indeed, it never existed). And on the other hand, the fact that there was such a large accounting error on top of the revelation that all this stuff was just sitting there doing nothing isn’t really making anyone feel very good.

(See CNN for more details.)

Math Mistakes in the News: Calculator Time

I found this story on Eric Berlin via God Plays Dice.

The Herald reported last week that a Traffic Warden was incorrectly ticketing cars in a Devon, England parking lot because of how he was using a calculator. In this parking lot, drivers would pay for a certain amount of time and then post a slip in the windshield with the time they’d entered and how long they’d paid for. One driver, for example, entered at 2:49pm and paid for 75 minutes.

Now 75 minutes is 1 hour, 15 minutes so the driver was covered until 4:04pm. But the Traffic Warden figured out the expiration time by entering in 14.49 into his calculator (for 1449 military time, which corresponds to 2:49pm) and adding on 0.75 (for the 75 minutes). He got 15.24, which he interpreted as meaning that the driver was only covered until 3:24pm. Since it was already 3:41pm, he issued the car a ticket. The car owner saw all this and tried to explain the error — that hours have 60 minutes, not 100, so standard decimal addition doesn’t apply — but the Traffic Warden didn’t see any problem and continued to ticket cars.

In good news, after appeal the incorrect tickets were repealed and a letter of apology sent.

Two recent Math Mistakes in the News

Neither mistake was horrendous, but one was public nationwide and the other cost a company a $134,000,000 job (and provided the reporter with the best Freudian slip I’ve seen in a long time!)

Back in April, Cynthia McFadden talked about gas prices on Nightline.

In this video she explains:

Tonight, $3.51: that’s the average price nationwide of a single gallon of regular unleaded gasoline. That means a 15-gallon tank now costs more than $50 to fill. As a little reference point, the week George W. Bush was sworn in as president, the price of a gallon of gas was $1.47. Now even accounting for inflation, that’s almost a 200% increase. And since we’re guessing your paycheck hasn’t grown 200% in the past eight years, we asked Vicky Mayberry to head to Arlington, Texas, to see the creative ways people are paying their [bills?].

No, my paycheck hasn’t grown that much, but neither has the price of gas. The difference between $3.51 and $1.47 is $2.04, which is just over 138% of $1.47. In other words, the average price of gas today is more than 238% of what it was eight years ago, but that only amounts to a 138% increase.

And that’s not even taking inflation into account, as the quote indicates we should. According to this site, to convert 2001 dollars into 2008 dollars I need to divide $1.47 by 0.831, giving about $1.77. And now $3.51-$1.77=$1.74, which is approximately 98% of 1.77. So with inflation taken into account, that’s almost a 100% increase. Which is still more than my paycheck has done, but then again according to this article “the price of lettuce, broccoli and apples increased much more than the price of gas” (at least in 2006-07), so it’s hard to know what to compare to what.

A less public but perhaps more distressing (to one company, anyway) mistake was reported on two days ago in the Miami SunPost. The company Skanska USA submitted a bid for a job at the South Dade Wastewater Treatment Plant. They had the lowest bid, but lost to Poole & Kent Company because Skanska USA’s bid had arithmetic mistakes in it. In particular, a decimal point was put in the wrong place. (Yet another example of how real life doesn’t always award partial credit!) They were given a chance to resubmit, but didn’t fix the mistake so they were out of luck. Remember folk: check your work.

As a final note, the SunPost article appears to have misreported the name of the company. Instead of calling it Skanska USA, want to know what they called the company that bid on this Wastewater Treatment job? Skanka USA.

“Green guilt”, a study in statistical significance

An article dated May 7 2008 in the on-line edition of USA Today describes the results of the second annual “green guilt” survey, conducted at the behest of the Rechargeable Battery Recycling Commission. (“Green guilt” is a reaction to the belief that one ought to live more ecologically than one does or attempts to do.) As the USA Today article puts it, people’s guilt levels are rising! And we’re recycling more! Unless you’re male, that is.

The Monty Hall paradox and the design of experiments

M. Keith Chen, an economist at Yale, questions whether half a century of research on cognitive dissonance is fundamentally flawed. (The New York Times has a nice article on his work, and the TierneyLab blog has continued the discussion at the Times.)

What the party line on Cognitive Dissonance has been: Well yadda, and then yadda yadda. Or so it seems.

(I originally wrote that as visual filler, but in hindsight it kinda works.)

One of the early studies of cognitive dissonance showed that when faced with three equally enticing options, if a subject was forced to choose between two of them and was then given a choice between the rejected item and a third item (as equally enticing as the first two), the subject would reject the same item again about 2/3 of the time. This was interpreted as the subject justifying their first choice by downgrading the estimation of the initially rejected item.

This phenomenon has been repeatedly tested in an array of experiments since the late 1950s, has been observed to occur in a variety of contexts, and even in several species.

Professor Chen points out, though, that the same phenomenon would be expected to occur if we assume that the subject regards the three objects as not all being of precisely equal interest, but rather if the subject has slight preferences for one over the other.

Suppose (the New York Times explains) a monkey is presented with three M&Ms, colored green, red, and blue, and has an internal ranking of these three colors.

At the first selection, the monkey is asked to choose between the red and the blue M&Ms. Suppose they prefer red to blue.

Of the six possible color rankings the monkey might have, three of them involve a preference for red over blue:

Notice that in two of those three cases, the monkey also prefers green over blue.  In other words, if we now ask the monkey to choose between blue (the rejected color) and green (the untested color), we see we should expect them to reject blue again about  2/3 of the time. This is a result of simple combinatorics rather than the monkey deciding to reject blue again to justify its earlier rejection.

Professor Chen argues that the accepted experimental designs for measuring cognitive dissonance are unable to distinguish between rational behavior on the part of the subject, and the cognitive dissonance effect they set out to measure, and that much of the empirical evidence gathered to date for CD could be accounted for by this Monty-Hall-esque phenomenon.

Professor Chen’s paper is still a working paper; it will be interesting to track its progression through the peer review process.