EASY????

Some calculations are easier than you think! I’m drawn to math that can be done on a napkin at a restaurant. Quick mental calculations and methods of approximation that give you the ball park quick answer…”It’s about …blah.” and then I try to look as if I don’t think I’m really clever. I had a couple of these come up recently, so I thought I would share…

One was at a backyard barbecue…an electrical engineer, a graduate physics student, and a math teacher, beers in hand as the hamburgers burned and the hot dogs turned to charcoal… One had just come back from New York City and pointed out that the Empire State bldg. was 1224 feet high to the upper observatory… then someone brought up the old conjecture that a penny dropped from the Empire State building would kill you. The engineer suggests that in truth the penny would reach some terminal velocity that would keep it from actually killing anyone. The physics student suggests that it would not be too hard to figure out using differential equations if we just ignored the air resistance, but he didn’t have his calculator (which apparently is his main resource for doing integration). The math teacher offers, “Well, with only high school math and physics, and a little mental calculation, it should be about 280 feet per second.”

A long silence, followed by “How did you get that number?” So I explained. In physics the big ideas often provide great simplifications…and in this case the big idea was the conservation of energy laws…. The total energy remains constant. The Penny on the 102nd floor has a potential energy equal to its weight times its height. When it hits the ground the potential energy has all been converted to kinetic energy (and a very small amount of heat due to air friction which I ignore) . Epot= m g h and Ekin = ½ m v2. Cancel the mass on each side and we have gh = ½ v2. If we use the common value of 32 ft/sec2 for the gravitational force, then we can simplify this all down to v2= 64 h.. or taking square roots, we get v= 8 times the square root of h…. Now the square root of 1224 is about the same as the square root of 1225 which is 35 (one of those multiplication tricks I teach my students each year) and 8 x 35 = 280, so the answer is about 280 ft/second.

They both look incredulous… so I offer a second approach… OK, an object dropped from rest will fall 16t2 feet in t seconds. To fall 1224 seconds it would take t=sqrt(1224/16) and taking the square root of top and bottom we get 35/4 seconds… OK, so velocity is 32 times t, so 32 times 35/4 = (wait for it…. ) 8 x 35 = 280 again… “Easier than you thought… right”

The second event corrected a misconception I have had for years. I have mentioned earlier that one of my preoccupations on the road is looking at mile markers and license plate numbers and trying to factor them. I know all the easy tricks for divisibility by small numbers, except seven…. The rules for divisibility by seven always seemed harder than just dividing by seven, so I was surprised to read a note from a guy who seems to have invented (I never saw it before) an easy way to test divisibility by seven

Without going into the derivation, here is the rule. Take the number, call it N, and think of it as 10A + B. For example if the number is 2345 the A = 234 and the B= 5. Now the method is just to calculate A-2B and if that answer is divisible by seven, so is the original. And if you don’t know, just reapply the same rule again. For example 2345 is divisible by 7 only if 234-10 = 224 is also….which can be applied again with 22-8 = 14… ohh….. I KNOW that is a multiple of seven, so 2345 is also a multiple of seven, in fact it is 7 x 335.

This is getting a little long, so on a later day I’ll explain how he found it, and even show you a method I found for testing 13 which is almost as easy, and another for testing divisibility by 17… as the tv folks say….stay tuned

Two Problems

OK, take off your political blinders, whether you are conservative or liberal, and read this comparison like a mathematician/statistician. It is a comment about large multi-national companies and their relation with smaller nation-states. Be warned, there is a major flaw (and many minor ones) in this argument… find it :
”In Nigeria, a relatively economically strong country, the GDP is $99 billion. The net worth of Exxon is $119 billion. ‘When multi-nationals have a net worth higher than the GDP of the country in which they operate, what kind of power relationship are we talking about?’ asks Laura Morosini. [Source: ‘Impunity for Multinationals’, ATTAC, 11 Sept 2002”

Ok, little things that niggle (yes, that IS a word) at me; Exxon’s net worth includes the value of off-shore oil rigs in the Gulf of Mexico and office buildings in downtown Houston (or wherever). It would not be practical to think they could use all of that value to bring to bear in a competitive situations…. And GDP is the value of goods and services sold in the country…. You sell me a watermelon for $3.00 and it goes into the GDP, you spend that $3.00 for (part of ) the payment on a car, the $3.00 counts again…. so Nigeria couldn’t use all that in a conflict with Exxon (footnote, when the Exxon Valdez went aground and spread crude oil over large parts of Alaska’s waterfront, the GDP of Alaska went up… all those people cleaning up sludge, even if they were volunteers were eating food, renting motel rooms… driving up the GDP… see, disasters are GOOD for you)….. but neither of those are the BIG one…. Think… what is being compared??…

OK, here is a similar comparison, I have $1000 in my bank account (not really, I’m practically broke, but hypothetically) and you make $8.50 an hour at McDonalds, so I have an economic advantage in trade with you…… see it, one is a rate and one is a fixed value. In the Exxon/Nigeria situation, the GDP is measured each year…just by custom, and the value of Exxon is just Dollars, a fixed value. What if it was the custom to measure GDP on a ten year basis. Now we would have a GDP of $990 billion for Nigeria, and poor old Exxon looks like a chump with only $119 Billion. The idea in Physics and math class was called units analysis. If you needed newtons, which is Kg meters/sec², then you needed to multiply mass times distance and divide by the square of time.. and with comparisons, they don’t make sense if you are not comparing the same thing…. My car gets 26 miles to the gallon and the Eiffel Tower is about 324 meters high??? Yeah… and so what? The trouble is, when people like Ms. Morosini (a wonderful person I’m sure) make comparisons like this, you have to do more than just nod and smile…. THINK, ask questions…..

OK, second problem… What is the least likely group to be hurt by the increased gasoline pump prices? Did you say “People who don’t own cars”? You would think that, but I just read in an AARP article (OK, I’m old, put your big girl panties on and deal with it) that gas prices are causing a big drop in volunteers to deliver meals-on-wheels programs to shut-ins. Without sufficient manpower, many have resorted to delivering one hot meal and several prepared frozen meals so that they don’t come as often. In addition, as the economy goes south, the people who donated money and food for the program and reducing their contribution. Ok, children; Ignore the signs…DO feed the old people. One of those fundamental rules of Economics is called the Law of Unexpected Consequences. It’s a good one to keep in mind, especially if you think you have the solution to any economic problem.

BI-MODAL???

Several years ago at the AP Stats reading I remember Ann Watkins (Cal State, Northridge) talking about how rare bimodal distributions are, at least that lots of things we think might be bimodal, really aren’t. The classic example we always talked about in class (before enlightenment) was distribution of heights when both men and women were included.

Turns out it just isn’t so… according to Ann’s speech . So I took some statistics from the National Health Service in the US that said the average height of white males over 20 was 70.2 inches, and for females it was 64.6 inches. Then I pulled out my trusty TI-84. It seems incredible that almost none of the pages which give average height pay any attention to the standard deviation; but since the distributions shown led me to believe it was around 2.5 inches for both groups, I used a generous 3.0 inches for the std. dev. of both groups. In the first image shown the two distributions are shown separately with a window ranging from 60 to 75 inches, with a cursor on the male curve at 67 inches

. If we combine the two, (I simply added the functions and divided by two) as if we measured the nations adults without regard to gender, the distribution looks like this,

So how much would we have to separate them to get a double mode? Well, a lot it seems. Remember that the two means, 70.2 and 64.6, are already almost two standard deviations apart. I decided to see what would happen if we made the males even taller, so I moved them up to THREE standard deviations above the female height, 73.6 inches (ever girls dream, all the guys are over six feet). The resulting combined distribution looked like this.

OK, so why bring this up now? Well, I recently read a report about the entry level salaries of newly minted lawyers (and minted is right for most of them). Most lawyers start out in private practice and make a pretty good salary, (way more than teachers), but some start out in public law and often (almost always) make considerably less. The image posted on the distribution is shown below, and is one of the most striking bimodal distributions I have ever seen for real data.

Yeah, you want to know.. the lower mode is a bridge between the $40,000 and $50,000 (the data seems to have been reported in $5,000 increments). This was 11% of the total data set. And the fat-cats???, there modal hump was at $135,000 to $145,000 (fresh out of the box... wow!) with about17% of the data set.

OK….. TRUTH in Statistics time… data like this is biased by selective reporting… Some of the folks who got low salaries just didn’t send it in…..”Yes, I have the lowest starting salary of anyone in my class.” Others may have lied ( Yes, people do that) and made them a little higher than the truth; but even with very accurate reporting, this looks like one of those true bimodal distributions. OK, keep your eye out for others like this, and when you find them, send me a link… (wow only a month till school starts… )

Picturing statistics

Statisitcs is about data in context, but great statistics, convincing statistics, requires more. In a world where nearly half the people are mathematically illiterate, and way more than half believe the old saw about, "Liars, damned liars, and statistics" you have to be able to present information in a dramatic way. A way that shakes people out of the cold dull stares we reserve for passive entertainment. I think much of what Chris Jordan does falls into that "great" statistics category.

The Ted Talk description says, "Artist Chris Jordan shows us an arresting view of what Western culture looks like. His supersized images picture some almost unimaginable statistics -- like the astonishing number of paper cups we use every single day." How about this for a shocker...we use 4,000,000 plastic cups a day on airlines alone... A Day! Other things that may surprise you...one out of every four people in prison in the world, are in the USA...2,300,000... "Land of the free?"

Remember 9-11? Of course you do... but on that day 3000 (or so) people died from a terrorist attack. Chris points out that on that day, and every day since, 1100 people die from cigarettes, over 400,000 people every year, (rule of thumb, ALWAYS check statistics.. people lie, so I checked with the Center for Disease Control,
.."Cigarette smoking is the single most preventable cause of premature death in the United States. Each year, more than 400,000 Americans die from cigarette smoking. In fact, one in every five deaths in the United States is smoking related. Every year, smoking kills more than 276,000 men and 142,000 women."... …
and 157,000 are JUST from lung cancer). When people think of cancers caused by smoking, the first one that comes to mind is always lung cancer. Most cases of lung cancer death, close to 90% in men, and 80% in women are caused by cigarette smoking. There are several other forms of cancer attributed to smoking as well, and they include cancer of the oral cavity, pharynx, larynx, esophagus, bladder, stomach, cervix, kidney and pancreas, and acute myeloid leukemia. The list of additives allowed in the manufacture of cigarettes consists of 599 possible ingredients. When burned, cigarette smoke contains over 4000 chemicals, with over 40 of them being known carcinogens. Kind of like the lottery of death, except there is a “winner” every time…I am reminded that I teach my students that in statistics we calculate binomial probability using the term p for the probability of success, even if the “success” in question is death.

I'm a teacher. I work with teenagers every day, and it scares me that 65,000 teenagers will begin smoking this month. Maybe a visual impact on their senses will stop one... or two?

A good conclusion, "How do we change?" WATCH !!