Saturday, May 23, 2015

The Criterion Criterion

Grammar is constantly in flux. In his "Twenty-Four Word Notes," which was published in 2003, David Foster Wallace insisted on the distinction between who and that:
There's a very basic rule: Who and whom are the relative pronouns for people; that and which are the rel. pronouns for everything else.
For me, too, it's the man who was driving the bus, never the man that was driving the bus. (I wouldn't call the latter usage a mistake, although like Wallace I tend to view it as a class-marker.) Wallace qualified this advice with "As of 2003," and it does seem to me that using that to refer to people has become much more common over the past dozen years. I wonder if it will ever sound archaic to say the man who was driving the bus. (Probably it will, when buses drive themselves.)

My own problem with who/that arises when talking about animals. A certain sympathy inclines me to call them who. My cat? Definitely a who. And in the children's book Click Clack Moo: Cows That Type, the word that always makes me frown. Those cows can type, for crying out loud! Give them a who! (Some pedant copyeditor probably put a stop to it.)

A train that left the station a long time ago was the use of agenda as a plural noun. Nowadays, agenda is a singular noun that means, more or less, a list. Once upon a time, an individual item on such a list would have been called an agendum, but today it would be eccentric to use that word. Now it's an agenda item.

Criteria, on the other hand, is still plural. A person looking for an apartment might have ten, twenty or a hundred criteria, but if he only has one then it's a criterion. Some people seem unaware that criterion exists, so they say Now let's look at the first criteria or This criteria is shorter than that one. People who habitually use the word criterion initially hear these statements as utter nonsense. (This giraffes is shorter than that one?). Naturally the intended meaning comes through a half-beat later, but it's apt to be accompanied by an unkind mental note about the speaker's vocabulary.

Sometimes people try to defend the use of criteria as a singular noun by analogy to the way agenda and data have become singular nouns. The analogy doesn't work, though, because agendum and datum have both disappeared, whereas criterion has not. As long as singular and plural forms are both available, there's no choice but to match them to singular and plural verbs and pronouns. Hence the usage note in the American Heritage Fourth Edition:
Like the analogous etymological plurals agenda and datacriteria is widely used as a singular form. Unlike them, however, it is not yet acceptable in that use.
This is good advice, because when a listener does perceive this error, it is perceived as catastrophic. Someday criterion will be joining agendum in the obsolete category, but right now it doesn't pay to be out in front on this one.

Sunday, May 10, 2015

An Open-Ended Problem, Part 2

Study the numbers in this photo until a question occurs to you. Then answer your question. Repeat as desired.

Reader Questions

Reader Stacy W pursued the implications of treating the word "Due" as the visible portion of "Change Due." In that case, assuming 2.59 and 0.57 were the only charges, then her question was, "How much money did the customer give the cashier, in order that the change due would be 7.00?"

Answer: 10.16, likely in the form of a ten-dollar bill and sixteen cents in change. Good question!

Reader Marni pursued the implications of treating the word "Due" as the visible portion of "Amount Due." In that case, what was the rate of tax paid?

Answer: 0.57 on a pretax amount of (7.00 - 0.57) comes to (0.57)/(6.43) = 8.864967(...)%. Good question!

Some of the questions that occurred to me, along with my answers to them:

1. How much in total were the item(s) not shown?

I took "Due" to be the visible portion of "Amount Due." But this raises a puzzle: if 2.59 is the price of a retail item (probably a drink, because of the "oz"), while 0.57 is the amount of tax levied on the sale, then as reader Marni noted, some additional item(s) must have been purchased, because 2.59 and 0.57 don't add up to 7.00. It appears the cash register display was simply too small to show all of the items purchased. So how much in total were the item(s) not shown?

Answer:  (7.00 - 0.57) - 2.59 = 3.84.

Pursuing some more questions along these lines:

2. What was the pretax subtotal?

The pretax subtotal was 7.00 - 0.57 = 6.43.

3. Were all of the purchased items subject to tax?

Certain retail items are exempt from sales tax, so it's possible that the 0.57 amount was levied on some but not all of the purchased items. Note, however, that if any of the items purchased here were exempt, then the local sales tax rate would have to be at least 14.8% (0.57/3.84), which is higher than in any U.S. city. The transaction could conceivably be in a VAT country, where rates can be high. But as discussed further below, we are going to take the transaction to be in the U.S., in which case it follows that the full pre-tax total of 6.43 was subject to sales tax.

4. What can we say about the local sales tax rate?

While 8.864967(...)% gives the rate of tax that was actually paid, this was probably not the sales tax rate prescribed by law in the location where the photo was taken. Legislators aren't going to set the tax rate at a funky, non-terminating decimal value like that. Rather, issues of precision and/or rounding are probably involved here, which makes things tricky since rounding is a non-invertible function. But let's see what we can do.

The register in the given photo is a Micros point-of-sale system. The rounding scheme is presumably configurable in the proprietary Configurator module; however, freely available documentation doesn't seem to specify how many digits of internal precision the cash register uses for its tax computations, nor the logic by which it rounds fractions of a cent. I tried to submit a helpdesk ticket about this, but I couldn't do that because I'm not a Micros customer. I did establish that the Casio PCR-T465 cash register provides several user-configurable options: multiply and round in the usual way (based on 0.5); multiply and round up; multiply and round down; or select a state-specific tax table to compute the tax via lookup. It wouldn't surprise me if the Micros system gave managers the same options.

(And now I have to digress, because suddenly I remember ringing up customers at my parents' diner when I was a kid. The cash register at Rip's Drive-In was a clicking, clanging, chugging machine. An NCR model like the one in the photo, electric-powered but not electronic. We figured the tax by remembering the amount for common orders, or computing mentally, or consulting a laminated table. Computing sales tax was easy for eight-year-old me, but I always panicked inside while figuring the customer's change, because nobody had ever explained to me the method of counting up. So to figure the change for a ten-dollar bill on a forty-nine-cent tab, I stood there for what seemed like endless seconds while the customer waited behind me, my eyes staring into space as I crossed out zeros and borrowed ten on the chalkboard in my mind. Cashiering would have been easier if I'd been taught mental computation strategies in addition to the standard right-to-left subtraction algorithm. Things could have been worse, too: a friend tells of his sister's sidekick from teenaged years who worked a register in those days and never realized that there was any specific quantitative relationship holding between the change, the amount of the tab, and the amount of the dollar bills tendered. Her method of figuring change was to paw through the coin tray with brisk confidence until she had collected what seemed to her like a convincing enough collection of coins to present to the customer.)

For purposes of analysis, I will model the Micros register as follows:
(i) the register multiplies the amount of the sale (a decimal to hundredths) by a finite-decimal tax rate, working to full precision (exact multiplication);
(ii) the register then rounds the result to the nearest cent by rounding fractions less than 0.5 to 0 and rounding fractions 0.5 or greater to 1.
Given this model, the programmed sales tax rate must lie in the range from A to B, where
A = 0.565/6.43
B = 0.575/6.43.
To see this, let R be the programmed sales tax rate and let T be the internal result of the tax computation, noting that (by (i)) T = 6.43R. Now by (ii), T must satisfy 0.565 <= T < 0.575. Dividing through by 6.43, we have A <= T/6.43 < B. But T/6.43 = R, so the programmed rate lies in the range from A to B, as claimed,

Carrying out the division, we have
A = 8.78693(...)%
B = 8.94245(...)%.
According to the model, and assuming that the tax rate is programmed correctly, the local sales tax rate lies somewhere between these two percentages.

6. Where might the photo have been taken?

I downloaded the image file and examined its EXIF data to look for latitude and longitude. (If you don't know how to do this, you can upload the image to Unfortunately, it turns out there's no geotagging in the EXIF data, so we can't locate the transaction that way. An image search on the phrase "join us in the" plus the word childhood did not yield a match to the baby in the photo. (I'd been hoping to identify a corporate campaign.)

Dollar signs are used to denote currency amounts in the United States and many other places. The only language in the photo is English, which rules out some of these countries; the unit of measure "oz" probably rules out many others. In any case, let's assume the transaction took place in the United States.

This website shows minimum and maximum sales tax data for every state. Examining the data, one finds 14 states with minimum and maximum rates that are consistent with A and B:
Alabama, Arizona, Arkansas, California, Colorado, Illinois, Kansas, Louisiana, Missouri, New York, Oklahoma, South Carolina, Tennessee, Washington.
Thus, even though we don't learn the exact location of the transaction, we still rule out out a great many potential locations for the photo (dozens of states and foreign countries). One could repeat the analysis with other reasonable models of the cash register. I'm guessing the results won't change very much, but feel free to check that.

Sometimes a sleuth has to go with a hunch. I noticed from the sales tax data that there is only one state in the U.S. where the maximum sales tax rate falls in the range from A to B. That's New York, where the maximum sales tax rate is 8.875%. Digging a little further, one discovers that this statewide maximum is attained in only one city, New York City. So if I had to guess, I'd guess that the local sales tax rate was 8.875% and that the photo was taken in NYC.

Monday, May 4, 2015

An Open-Ended Problem

Study the numbers in this photo until a question occurs to you. Then answer your question. Repeat as desired.

Feel free to post your responses in the comments! I'll write up mine in a future post.