What’s the sales tax rate in San Francisco?

There’s a restaurant in downtown San Francisco called ‘wichcraft. They make sandwiches.

This post is about their menu. Specifically, if you saw their menu could you work out the sales tax rate in effect?

It’s a silly question, at first. But consider their breakfast options. They cost, in descending order of price, $7.35, $6.90, $5.12, $4.68, $4.45, $4.01, $3.34, $2.45. (These are not the prices you’ll see on the menu at their web site, because their menu has their New York City prices.) These prices look a bit strange, but we might guess that they’re round numbers after tax. And in a US context, if you’re expecting cash transactions, “round numbers” means multiples of 25 cents.

Take the differences between these; they’re 45, 178, 44, 23, 44, 67, and 89 cents. Just looking at this sequence I can see some quantization; I guess that that 23-cent difference becomes a quarter after tax, the 44- and 45-cent differences become two quarters, and so on. So those differences are, in post-tax quarters, 2, 8, 2, 1, 2, 3, and 4, which add up to 22. In particular $2.45 becomes n quarters after tax, for some n, and $7.35 becomes 3n.

Then, $7.35 is three times $2.45; thus n is 11, and $2.45 pre-tax becomes $2.75 post-tax. The pre-tax prices, and the corresponding post-tax prices, are:

$7.35 $6.90 $5.12 $4.68 $4.45 $4.01 $3.34 $2.45
$8.25 $7.75 $5.75 $5.25 $5.00 $4.50 $3.75 $2.75

So what’s the tax rate? Each one of these prices gives us some small interval which contains the tax rate. Let the tax rate be x percent; then we must have, for example, that 7.35(1+x/100) is between 8.245 and 8.255, from which x must be between 12.177 and 12.313. We can do a similar computation for each price. The highest lower bound that we get is (4.995/4.45)-1 = 12.247 percent; the lowest upper bound is (5.255/4.68)-1 = 12.286 percent.

One last assumption – the tax rate is a round number. So it must be 12.25 percent.

But the California State Board of Equalization says the sales tax in San Francisco is 8.5 percent! And this is in conjunction with a couple places I saw on Clement Street a few weeks ago that charged $4.38 for a dim sum special, which are what inspired this post; add the tax and it’s $4.75 — I held this post back because from that single data point it’s hard to show much.

On propagation of errors

I’m reading John R. Taylor’s textbook An Introduction to Error Analysis: The study of uncertainties in physical measurements. This is meant for people taking introductory physics lab classes, but it never hurts to revisit these things. I actually double-majored in math and chemistry in college. It’s fun watching the contortions that chemists go to to avoid math.

Anyway, measurements come with uncertainities: that is, they have the form x \pm \delta_x, where x is our best estimate of the quantity and \delta_x is an estimate of the uncertainty. (We can think of this as being roughly the standard deviation of the distribution from which x is drawn.) In these intro lab classes one quickly learns some rules for manipulating these uncertainties. These can be thought of as defining arithmetic on intervals; however this isn’t the usual interval arithmetic but actually an abbreviation of arithmetic on probability distributions.

    • (x \pm \delta_x) + (y \pm \delta_y) = (x+y) \pm (\sqrt{\delta_x^2 + \delta_y^2}) – that is, the variances attached to the measurements add. Similarly, for differences, (x \pm \delta_x) - (y \pm \delta_y) = (x-y) \pm (\sqrt{\delta_x^2 + \delta_y^2}). For example, (30 \pm 4) + (20 \pm 3) = (50 \pm 5) and (30 \pm 4) - (20 \pm 3) = (10 \pm 5). Note that the error for the sum and the difference are the same – but for the difference, the error is relatively much bigger.
    • To find (x \pm \delta x) \times (y \pm \delta_y), start by finding the fractional uncertainties \delta_x/x and \delta_y/y. Then the squares of the fractional uncertainties add: the fractional uncertainty of the product is \sqrt{(\delta_x/x)^2 + (\delta_y/y)^2}. The same fractional uncertainty holds for quotients. For example, the fractional uncertainty in 30 \pm 4 is 4/30 \approx 0.133, and that in 20 \pm 3 is 3/20 = 0.15. So the fractional uncertainty in their product is $\sqrt{(0.133)^2 + (0.15)^2 = 0.201$. Thus we have for the product 600 \pm 120 and for the quotient 1.5 \pm 0.3.
    • Perhaps one learns rules for dealing with powers, logarithms, and the like. These are all easily derived from the rulef(x \pm \delta x) = f(x) \pm |f\prime(x) \delta x|.

For example,
(30 \pm 4)^2 = 30^2 \pm |(2)(30)(4)| = 900 \pm 240 – in fact, when taking nth powers, the fractional uncertainty is raised to the n power. Similarly,
\log (30 \pm 4) = \log 30 \pm |(1/30) (4)| = 1.48 \pm 0.13.
In this case, the fractional uncertainty becomes the absolute uncertainty in the logarithm. If we know a number to within ten percent, we know its log to within 0.1 unit.

But implicit in the rules for sums, differences, products, and quotients is the idea that the errors \delta x, \delta y in the measurements of x, y are independent! So these rules can’t be used if there’s correlation between the errors. More simply, they can’t be used if the quantity that you’re interested in is a function of many variables, some of which occur more than once. Consider for example the Atwood machine, as Taylor does in his problem 3.47. This consists of two objects, of masses M and m with M > m; the larger mass accelerates downward, with acceleration a = g(M-m)/(M+m). Here g is the acceleration due to gravity. We assume this is known exactly. So there may be correlation between the numerator and the denominator.

So what can we do? In this particular case it’s not hard to rewrite as
a = g {1-(m/M) \over 1+(m/M)} = g f(m/M)
where f(z) = (1-z)/(1+z), and use the rules that I’ve already discussed. (But it may be hard to see that this is worth doing!) For example (I’m taking these numbers from Taylor) say M = 100 \pm 1, m = 50 \pm 1. Then the fractional uncertainty in the quotient M/m is\sqrt{(0.01)^2 + (0.02)^2} \approx 0.022, and we get M/m = 0.5 \pm 0.011. Then f^\prime(z) = -2/(1+z)^2, so f^\prime(1/2) = -8/9, and thus we have f(M/m) = f(1/2) \pm (0.011)(8/9) = (1/3) \pm 0.01.

Alternatively, we think that m/M is likely to lie in the interval 0.5 \pm 0.011 = [0.489, 0.511]; then f(0.489) = 0.343 and f(0.511) = 0.324, so we figure that f(m/M) is likely to lie in the interval [0.324, 0.343].

But we are not guaranteed that our rewriting trick will always work. What else can we do? I’ll address that in a future post.

Weekly links for March 25

You can buy anacrylic Frabjous kit or get your very own cardboard Frabjous.

Online matchmaking is not so great, because it leads you to focus on things that are easily measured.

The hard way to solve the 8809=6 puzzle (with regression!); I solved it here. (I actually thought about using regression. But I did not.)

Statistics project ideas for students.

Cheating on homework in a graduate course is staggeringly dumb. This post and its comments are not, and attempt to answer the question: what’s the point of homework in math classes? (If I were a better blogger, or if I were not seeing an amazing lady like one of the commenters was, I’d have an answer of my own.)

The anachronism machine: the language of Downtown Abbey. (The typo is deliberate; I don’t watch the show.)

You can look at brains and see math anxiety.

A working scientific calculator, built in Minecraft. (via metafilter)

David Spiegelhalter on What does a 13% increased risk of death mean??

Cultural ontogeny recapitulates phylogeny

The MIT course 6.042: Mathematics for Computer Science has a textbook in progress by Eric Lehman (Google), F. Thomson Leighton (MIT math/CS, Akamai), Albert Meyer (MIT EECS, MIT CSAIL); various versions of it in various stages of completion are available from their course web page. (via Hacker News.)

Jordan Ellenberg at Slate: Six Degrees of Innovation: What Broadway musicals tell us about creativity.

From stackoverflow, How do I find Waldo with Mathematica?

A geographical existence proof

If you spend time in the Mission in San Francisco, you think of Mission Street and Van Ness Avenue as both running north-south, with Mission parallel to and slightly to the west of Van Ness.

But north of there, Mission is one of the major streets downtown, and Van Ness runs through neighborhoods to the west of downtown. That is, Mission is now east of Van Ness.

Therefore, if you assume that each street exists only in one piece, they must cross each other. A sketch of a proof, which works because the streets aren’t too curvy: any line of latitude within the part of the city in question intersects each of Van Ness and Mission, exactly once. Take the difference between the longitude (west of Greenwich, because why not?) at which that line intersects Van Ness and the longitude at which it intersects Mission. At the latitude of, say, 24th Street, this is negative (Van Ness is east of Mission, so has smaller numerical longitude) and at the latitude of, say, Geary, this is positive. By the intermediate value theorem it must be zero at some point, the latitude of the intersection.

(Inspired by being caught in a traffic jam a few days ago, near the intersection of Van Ness and Mission, which I had previously not recognized existed, despite being familiar with both streets on both sides of the intersection.)

Comments on Johnson’s predictions of Olympic medal counts

Daniel Johnson of Colorado College predicts Olympic medal counts.

The model is based only on non-athletic factors. Johnson’s semi-technical summary of the model gives the formula used. The variables used in the prediction are as follows:

  • the total number of medals available
  • per capita income
  • population
  • whether the Olympics are being held in that country this year, or in the near future or near past, or in a neighboring nation
  • a “nation-specific effect”

I’m not entirely sure what the “nation-specific effect” means, but I suspect it’s an adjustment for countries that consistently overperform or underperform the targets given by the rest of the model. I remember hearing in 2008 that it was quite strange that India, for example, did so poorly at the Olympics. (The explanation I heard a lot was that the Olympics don’t have cricket.) Australia, on the other hand, consistenly punches above its weight.

A working paper from 2002 suggests that previous iterations of the model also had climate-related variables; the press release says that it doesn’t any more, as those no longer seem to be significant. Presumably they are in the Winter Games and we’ll see them again in 2014.

For 2012, the predicted leaders in gold medals are the USA, China, Russia, Great Britain, and Germany; the predicted leaders in overall medals are the same, with Great Britain (the host country) and Germany reversed.

But is this overkill? Roger Pielke points out at freakonomics and at his own blog that the “naive” prediction that a country will do as well this year as it did four years ago has smaller errors than Johnson’s model. Johnson replies in a comment to Pielke that his model isn’t intended for predicting what each country will do, it’s intended to show which factors are important for Olympic success. In other words, he’s interested in the coefficients of his model and how they change over time, not what you get when you plug in values for any specific country.

(via Freakonomics and Forbes. more information from Johnson’s web site.)

Absence of evidence is not evidence of absence, but it helps.

A long time ago in a city far far away I was a grad student.

Now here at Berkeley it is qualifying exam season, and the following exchange took place in an elevator:

Professor X: “I was at A’s a qualifying exam today.”
Grad student Y: “How did A do?”
Professor X: “He was great!”

Of course, X wouldn’t have said that A did well if A had done poorly. But he probably also wouldn’t have said that A did poorly; instead he would have hemmed and hawed and avoiding saying anything at all. And that would have told Y what she wanted to know.

Similarly, I remember that when I was in grad school, the information that someone had passed their qualifying exam spread quite quickly among the students, whereas the information that someone failed spread less quickly. The reason for this is simple: passing is happy news, so the person who passed and their friends will tell everyone. But failing is sad news, so you can really only find out if someone failed by specifically asking them.

So assume that rumors spread according to a logistic model, and furthermore that the information of a success spreads twice as quickly (for small populations) as the information of a failure. That is, the proportion of the population that knows that person X succeeded at time $t$ after their exam is

P_1(t) = {1 \over 1+C_1 e^{-t}}

and the proportion that knows that person Y failed at time $t$ after their exam is

P_2(t) = {1 \over 1+C_2 e^{-t/2}}

for some different constant C_2. Furthermore P_1(0) = P_2(0) = 1/(n+1), where n+1 is the number of students; solving gives C_1 = C_2 = n.

So say it’s time t, and you haven’t heard if the person who had their exam at time 0 passed or not. Then by Bayes’ theorem, the odds in favor of their passing are that they’ve passed is

O(pass|N) = O(pass) {P(N|pass) \over P(N|fail)}

where O(\cdot) denotes the odds of \cdot N denotes the event of not having heard yet. Typical pass rates at the time I was there were perhaps 3 out of 4, so O(pass) \approx 3. The conditional odds are therefore

O(pass|N) = 3 {P(N|pass) \over P(N|fail)}.

But P(N|pass) = 1-P_1(t) and P(N|fail) = 1-P_2(t). So, after some algebra,

{P(N|pass) \over P(N|fail)} = {{1 \over n} e^{t/2} + 1 \over {1 \over n} e^t + 1}

and indeed this decays (exponentially fast!) as t \to \infty. So the longer you go without hearing the news of someone’s exam outcome, the more likely it is that it’s bad news.

Of course reality is more complicated. This doesn’t take into account the structure of the social network. For example, for any given person there’s probably a ring of people at a certain social distance who would hear very quickly if they passed but not if they failed. If you know that you are in that relationship to a person, you can probably guess with pretty near certainty that they failed if you don’t hear right away.

A puzzle from James Tanton

James Tanton asks in a series of tweets (which I’ve modified slightly): Write 20 numbers. Erase any two, a and b, and replace with f(a,b), where some possible choices of f(a,b) are:

Repeat 19 times, until you get a single number. The final result will be independent of the choices of pairs made. Why?

For example, consider f(a,b) = a+b+ab. And say we start with the numbers 2, 3, 6, 7. Then we could choose to do the replacement as follows. In each case the two bolded numbers are replaced by one.

  • 2, 3, 6, 7 becomes 3, 7, 20
  • 3, 7, 20 becomes 7, 83
  • 7, 83 becomes 671

and so we’re left with 671. Or we could have

  • 2, 3, 6, 7 becomes 2, 7, 27
  • 2, 7, 27 becomes 23, 27
  • 23, 27 becomes 671

and again we’re left with 671. What’s going on here? It’s not immediately apparent, but if you try this starting with lots of small integers, you often get results which are one less than some number with many small factors — in this case, 672. And in fact 672 = (3)(4)(7)(8), which we can rewrite as

671 + 1 = (2+1)(3+1)(6+1)(7+1).

We can think of the whole process, starting with x_1, x_2, \ldots, x_n, as computing the product (x_1+1) (x_2+1) \cdots (x_n+1) by combining two factors at a time; of course the order doesn’t matter.

Similarly with the function f_2, if we start with x_1, x_2, \ldots, x_n we end up with \sqrt{x_1^2 + \cdots + x_n^2}. With f_4, we get \log \left( e^{x_1} + e^{x_2} + \cdots + e^{x_n} \right). The result with f_3 is a little harder to see but if we start with x_1, \ldots, x_n we eventually get 1/(x_1^{-1} + \cdots + x_n^{-1}). This is a bit easier to see if we realize that f_3(a,b) = (1/a + 1/b)^{-1}; therefore applying f_3 conserves the sum of the reciprocals.

What others are there? Of course there are trivial examples like f(a,b) = a+b and f(a,b) = ab; iterating these functions will just give the sum or the product of the original numbers. But what other nontrivial examples are there? Can you say what all of them are?