Posts Tagged ‘MATH AND MIKE’

OH WOW, I CAN’T BELIEVE I DID THAT. THANKS.

November 24, 2017

I’ve probably written about this before; if so, I apologize.  I happened to think about the incident when I awoke this morning.  I slept well, which is a blessing as I approach the last year of my sixties.  Most of us my age and older don’t sleep as well as we’d like: through the night without awakening, and awakening refreshed.  It’s a good night if I’m up once, fall back asleep, and awaken on my own, without one of the cats standing on me, yowling (that was this morning, at 4:47), or barfing up a hair ball.

Anyway, for some reason, a dream maybe, I awoke thinking of one day when I volunteered  in the calculus class at Sabino High School in Tucson.  I occasionally helped out there, if for some reason the college algebra or geometry classes where I usually worked didn’t require my presence.  I liked Dave, who taught calculus; indeed, years ago, when email was new, Dave was the only faculty member with one, which is how I contacted him and began to help.

Anyway, that particular day he began by discussing the behavior of the function y=x^2, a parabola, near where x=1 and y=1.  Specifically, he was looking at the rate of change or the slope of the function as it approached the point.

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What we learn by the slope is how fast the curve is changing at that point.  People get this concept wrong all the time.  If we hear the rate of growth of population is slowing, some people think the actual number is going down.  It isn’t.  It is still rising, not just as fast.  This is extremely important to know, and worth repeating: If the rate of increase is slowing, the actual number is still increasing, not falling.

Dave started by showing the slope where x=0.99 and y=0.99*0.99, or 0.9801.  The slope then was (1-0.9801)/(1-0.99), 1.99/0.01, or 1.99.  No problem. Then he let x=0.999 and, with a calculator, squared it, 0.999^2, which was 0.998001. The slope was now 0.001999/0.001, or 1.999.  He continued, saying that as x got closer and closer to 1, y would get closer and closer to 2, and the limit; that is, if we could take take x as 0.99999999— to an infinite number of 9s—and here he paused….

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Notice the pattern for y—first is .81 then .9801, then .998001,then .99980001.

“You’d need a big calculator to calculate y, but the slope would be 2 at the limit.”  he stated.

I sort of blurted out without thinking, “You don’t need a calculator to get y.”  The words just appeared, I swear.  Everyone in the class turned towards me.

Dave looked at me, held the marker out in one hand, and said, “come up and write it down.”   He wasn’t at all angry.  We had known each other for several years at this point. As I walked up to the board, I asked, “How many 9s are there in your number?” I have bad astigmatism.

“Eight.”  So, I am multiplying 0.99999999 by 0.99999999.

Without a calculator.

“It is one less 9, followed by an 8, followed by the same number of 0s as you have nines, followed by 1.”  That would be 0.9999999800000001.  Take that, Texas Instruments.

I then turned to the class.  “Last year, I was here when this problem was discussed. You”—-I pointed to Dave—-“said that there was probably a pattern, and you were absolutely right.  I found it in a few minutes. The pattern is one fewer nine, an 8, the same number of 0s, and a one,” repeating myself.

Dave is good: he knows what he has taught, what a student should know, makes the student think and find answers to his questions, because they have all the information available.

I could have added that Dave, as a good teacher, didn’t take my coming to the board as showing him up; indeed, he knew that I was modeling exactly the behavior he wanted in his students.

I’ve been down that road before.  A dozen years earlier, I was in grad school getting my Masters in Statistics.  I had several professors.  One was absolutely brilliant, able to teach an entire difficult upper level graduate course in linear models without consulting notes.  There may have been one time where she made an error that another student caught, but that was it.  She was brilliant.  I’m sure I mentioned that in my post-course evaluation.

What I didn’t mention, because there were only 7 of us in the class, and I knew I would be identified, is that had she more patience with those students like me (it was the only B I got in grad school, and I worked hard to get it) who were not as brilliant, she would have been a life-changing teacher, the top of the heap.  The best teachers have patience with those who don’t have their skills.  Occasionally, I approach that.  Dave was there. So was my advisor.

My advisor didn’t need notes when he taught, either.  But he had patience with me, and that mattered a lot.  He got me out of New Mexico in 2 years, which I deeply appreciated. I haven’t seen him in about 15 years, but when I emailed him asking if he could help a friend of a friend–a free favor– he replied immediately.   He knew—damn it, I was pissed so many times when he did it, but he was dead right to do so—when I had exactly enough information to find the answer of a problem I asked him about.  He either knew the answer outright, or knew how to get it, but he was not going to tell me, but rather would give me what I needed to know—and not one hint more— to solve it.  I would then struggle for hours in some instances before having an epiphany among the papers strewn on my desk, on the floor, some crumpled and near a wall where I had thrown them in anger.

By doing that, my advisor forced me to use the new tools I had learned, to make mistakes, to figure everything out, and learn that way.  It was painful, but it was learning.  It was education, and it worked.

I’ve never gotten to my advisor’s or Dave’s level: substitutes don’t have a close relationship in one meeting, and in my brief for-profit so-called university teaching experience, students wanted everything handed to them.  But when I tutor today in the advanced math room at the community college, I occasionally encounter material far beyond what I know.  Sometimes, I try to help anyway.  And as I go through the problem with the student, asking him or her at each step how they got there, there is often a pause.  The student suddenly says, “Oh, I see what I did wrong. Oh wow, I can’t believe I did that.  Thanks.”  And walks away.  I’m still wondering what the answer was.

It just dawned on me that maybe my advisor sometimes didn’t know the answer, either, at least when I asked the question.  But he knew me well enough to know that I was capable of finding it.

Oh wow.  I can’t believe I did that. Thanks.

TONS OF RAIN AND A TINY CHANGE OF pH

September 5, 2017

“Tons of rain” was the grossest underestimate I heard of the amount of water accumulating during Hurricane Harvey.  A ton of rain is not very much, roughly about 0.02 inches on a moderate sized roof of one house.

What about 11 trillion gallons of water?  The media used the number to say how much water fell on southeast Texas.  The problem is first, too many don’t know how big a trillion is and second, a trillion is not a term often used with water.

A trillion, 1,000,000,000,000, 1 x 10 ^12, is a term used to describe both the national debt and the Gross National Product.  It is roughly the number of days the Earth has existed.  It is about the number of seconds in 31,700 years.

The term we use to describe a lot of water here is an acre-foot (1 acre covered with one foot of water), and while perhaps archaic, it is useful.  An acre-foot of water is roughly 326,000 gallons, what an average family of 4 uses in a year.  Lake Shasta, the largest reservoir in California, has a capacity of 4.5 million acre feet.  Eight Shastas would have flooded Houston, or 11 trillion gallons of water would cover all of New York State a foot deep.  If the catchment area described were 10,000 sq miles, it would have covered it to a depth of 5 feet, basically what people needed to know and could see from the pictures.

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Per cent is a useful term, but often misleading.  I inadvertently misled people before the recent eclipse.  What I should have said in the talks I gave prior to the event was that there were two kinds of eclipses (there are more, but I will keep things simpler), partial and total.  They are very different experiences.  Had I said that, mentioning that where I lived was in the partial zone, I might have persuaded more people to go to see totality.

Instead, I and many others said the eclipse would be 99.4% (or 99%) and most people figured, reasonably enough, that they would see almost the full event from their house.  I have since had several people tell me that they wish they had gone to totality.  One poignant comment was that 0.6% made all the difference in the world, since it was still sunlight and not totality.  If the eclipse isn’t total, it is partial.  It may be a little darker, a little more of the Sun will be covered if it is a deeper partial, but it is not total.  Next time, if there is a next time for me, I won’t make that mistake.

Per cent shouldn’t be used when counts are a better measure.  I have said in 2001 that the per cent of domestic flights not hijacked was 99.999996%.  Counts matter, especially when the counts should be zero.  When I was medical director of a hospital, we had a surgeon operate on the wrong side of the head.  Actually, we had three wrong side cases that I knew about—one was the wrong knee, and the other was the wrong side of the colon.  With the craniotomy, the OR head said that 99.9% of the time they did it right.  No, I retorted, we did 99.99% of them right, and that wasn’t the issue. There should be zero wrong side cases;  99.9% of landings done right means a plane crash every other day at O’Hare.

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Not every measurement is interval or ratio, meaning that the difference between 10 and 20 is not the same as the distance between 20 and 30.  Most of us realize that with temperature, that 110 is not twice as hot as 55.  That is because the Fahrenheit scale has an artificial zero that is 459 Fahrenheit degrees above absolute zero.  Therefore 55 degrees is 514 above absolute zero and 110 is 569 above absolute zero, a difference of 10.7%.  With Celsius, these numbers would be 273 above absolute zero, 13 and 43, respectively, making the temperatures 287 and 316, or 10.1%, really the same, given the rounding in the conversion from one to the other.  It’s important to recognize what is ratio data, meaning that multiples make sense, and what isn’t.  Money is ratio data, as are height and weight.  Others are ratio data, but they are used in ways where one has to be careful.  Height is ratio data, but the Body Mass index is a function (depends upon) the square of the height, or the height multiplied by itself.

This concept of squaring something is important in many areas, such as the energy of a moving body, which is proportional to or depends upon the square of the velocity.  With hurricanes, velocity of winds increasing from 100 mph to 120 mph, 20%, is a 44% increase in energy, (120/100)(120/100).   A car moving at 60 mph has four times the kinetic energy it had at 30 mph.

Cubing something is to the third power or multiplying it by itself 3 times.  While 1 yard is 3 feet, 1 cubic yard is 3*3*3 or 27 cubic feet.  A meter is almost 10% longer than a yard (9.4%), and a cubic meter is 31% more than a cubic yard.  Gravitational attraction is inversely proportional to the square of the distance between objects; tides are inversely proportional to the cube of the distance between objects, which is why the Moon, so much less massive than the Sun, is responsible for 45% of the tidal pull on the Earth.

Fourth power?  Yes.  The radiation from a star can be considered to be equal to the fourth power of the temperature, useful for determining the temperature of distant stars.  And closer to home, the damage large vehicles cause to roads is roughly equivalent to the fourth power of the load equivalent factor, having to do with axle number and weight.

Other relationships?  Yes, too.  the acidity of a liquid is the negative log of its hydrogen ion concentration (pH), which is a nice way to call 0.0000001 moles/liter of hydrogen ion a pH of 7.  Therefore, what seems like a minor fall in the ocean pH from 8.2 to 8.1 represents a 26% increase in acidity.

It’s not always the magnitude of a number that matters—99.99% is not always good, and a pH’s falling from 8.2 to 8.1 will see the end of most coral reefs on Earth.

QUITTING BRIDGE…AGAIN

March 5, 2017

I like bridge, but I have found many who play it often less than charitable to those of us not skilled.  I started reading the bridge column fifteen years ago, read a few books about the game, liked it, and on a cruise ship to the 2005 eclipse, played a little. During the last few months of my father’s life, I played with him and his group.

I played “party bridge,” often disparaged by those who play duplicate, members of the American Contract Bridge League (ACBL) who get Master’s Points from tournaments, hoping some day to become a Life Master. I would be careful to disparage party bridge.  Good players can be anywhere, and dissing one style of play is like my saying that somebody who just learned basic algebra doesn’t know real math, because I know much more.  Good for the learner.  They learned something.

I wasn’t good at bridge.  Occasionally, I would do the right thing, because bridge is a game of probabilities, and sometimes the stars align.  With time, I did a few more good things, meaning I was learning, but too many people with whom I played were neither helpful nor nice.  “Four points?  I’ve never seen a response with four points.” I had four trumps, a side ace, and distribution, for those who know the game.  Or “Why didn’t you bid xxxx?”  There are a lot of mistakes one can make in bridge, and there is no shortage of critics, many of whom are dead wrong.  They have to be, because I was criticized on the same hand and being told contradictory things.  That happened a lot when I played basketball in city league, too, and I found it annoying.

At the beginning of the session, the head of the club said that comments about play were not to be offered unless asked for.  It was a nice thought, but it failed in practice.  One man was particularly nasty.  I didn’t understand his bidding, and he always had to have his style followed.  One day, he finessed me correctly for the king of hearts, knowing that eventually he would capture it.  I had 4 cards in hearts, including the king, and rather than not playing the king last, played it on the third round.  He took it and continued his play, rather surprised when I turned up with the missing heart at the end, sinking his contract.  I had learned through reading the technique of playing a “dead” honor sooner than expected.  I remained silent.  He hadn’t counted trump.

The last day I played at the club, my partner made a bid that I misinterpreted.  Had she passed, which she should have (she preempted over a preempt, for those who know the game, and one doesn’t do that), I would have defeated the contract four tricks.  Instead, my misinterpretation cost us being set two.  I was loudly criticized by the other three people at the table and never returned.

I read the bridge column every day; my wife and I occasionally deal out hands and play them.  These allow me time to safely think and process.

After a seven year hiatus, on a cruise to the 2016 eclipse in Indonesia, I decided to play on board, not surprisingly finding myself the worst player in the room.  Bridge is a sedentary game, and while I try not to be too judgmental, many there needed to do more physical activity.  I played duplicate bridge three afternoons, calling it quits after the third.  I was paired with a different person each day, and with a partner one doesn’t know, bridge is even more difficult. I wasn’t the only one who made mistakes, and the tone of voice may not have sounded critical to the owner, but it did to me, whether I was being criticized or somebody else.  There is a way to correct people that works, and good teachers know it.  Unfortunately, there are not many good teachers.

I ran into my last partner later in the cruise.  He had played for years and explained bridge players clearly, so clearly I wondered why I never figured it out.  You see, the irony is that I am good at numbers, probability, and have a decent memory, which should make me a great bridge player.  But I have a big deficit: I process slowly, bridge is a timed game, and most play it even faster.  I can’t keep track of cards when they are played quickly.  My partner simplified matters: “The best bridge players are options traders: they have to be quick with numbers and risk averse.”  That doesn’t describe me at all.

There are those who teach bridge, but I am reluctant to seek them out, because frankly, not many are good at teaching.  I am. I understand different styles of learning, I understand that not everybody knows something as well as I, and I try to be patient.  I do this when I tutor math, show people the night sky, or explain medical conditions.  I’m enthusiastic, not critical.

What I need is a bridge hand where everything is played slowly.  I need a chance to figure out who has what and decide what to play next, being gently guided with tips how to keep track.  The bridge I have played isn’t this way.  I know it exists somewhere, but not where I’ve been.  In a sense, bridge reminds me of learning German.  I was always in a group of better speakers, but I couldn’t find one who would work with me to make me better.  It is why after 3 years I eventually gave up trying to be fluent, yet can understand it well enough to teach beginners how to go about learning it, because the teachers and online methods I know are insufficient.

I will return to reading books on bridge and watching German videos alone. I enjoy both.  I will continue to devote efforts to volunteering as a math tutor both at the community college and online, where the comments about my teaching are “You are awesome,” or “Thank you for explaining everything so clearly <3.”

I understand math.  I understand that people have different learning styles, so I teach to the person.  Perhaps most importantly, I realize that many don’t “get” math the way I do and never will.  I am neither a language person nor a bridge person.  I can improve, but I am no longer going to hit my head against a wall trying to be something I cannot be.

Better I break down math walls and save some heads.  I’ll avoid options trading, too.

STATISTICS AT THE PINNACLE-PART 2

February 15, 2017

I’d look at the audience and find two rows where I saw 22 people.  “What do you think the probability in these 22 there is at least one pair with the same birthday?”   I’d ask that because most people would think it is quite unlikely.  Our brains tell us that, but our brains can deceive us, not only the brains of others.  Turns out, the probability is about 50%.  I might even start with 7 rows containing 70 people, where the probability is 99.9%.  “This does not make intuitive sense,” I would add, “but is easily proven by taking the approach of the probability that two people don’t have the same birthday.”  My statistics advisor did this in a class where I happened to be attending as a graduate student, and the first student’s birthday matched mine.  He said the look on the face of the students was priceless. “Why does this matter?  It matters because sometimes the way you think is wrong, flat out wrong!  Your brain lied to you.  Your brain said the likelihood of two people in the same room’s having the same birthday was small, very unlikely with 22 people, which is not true.  Our brains lie to us about speed, direction, and up and down. They worry more about improbable losses than probable gains, and how certain major events in our lives shape our thinking, even if they are very unlikely to ever happen agan.   The solution to the birthday problem is also a good life lesson:  figure out what you don’t want and whatever is left over is what you want.”

I’d talk about the lottery and expected values. People play the lottery, because eventually somebody wins.  We can predict quite accurately the probability that somebody will win.  “You see,” I’d say, “low probability events happen; they just happen with low probability.  Take the lottery with a 1 in 110 million chance of winning.  If 330 million tickets have been bought, the expected value of jackpot winners is 3.  That doesn’t mean that 3 will win, but it is expected.  We can easily, and I mean easily calculate the probability of 0,1,2,3, and 4 with a calculator and a few key strokes.  Three people nationwide might win.  Three people, in the entire country.  Yes, it has to be somebody.  But do you think it is going to be you?”

If you have a disease, you have certain symptoms.  Medicine is the study of people who have certain symptoms and tries to figure out the probability of their having a disease.  Physicians and others would do well to understand the idea that not all who test positive for a disease have the disease.  “Suppose a disease has a 0.1% prevalence in the population, or 1 in 1000 people has it.  We would do well to teach percentages early in math and often, too.  Suppose if you have the disease, you test positive for it 98% of the time.  If you don’t have the disease, you test negative for it 99% of the time.  You test positive.  What is the likelihood you have the disease?

What is important here is the background frequency of the disease.  The fact the disease occurs in only 1 of 1000 means that it is unlikely somebody who tests positive will have the disease: only 9%.  

Anybody remember W. Edwards Deming?  He was ignored here but found the Japanese receptive to his ideas about data analysis and optimizing systems.  The Japanese cleaned our clocks in the automotive industry before the Big Three caught on, not because Japanese cars were fancy, but because they worked.  There is an apocryphal story about how a Japanese company was told by an American buyer that no more than 4% of ball bearings should be faulty.  In the next shipment, 4 at the top of every box were faulty.  When asked why they were there, the company spokesman said, ‘you didn’t want more than 4% faulty.  Here they are, on top.  The rest are perfect.’

“Deming taught that variability could be classified as “common cause” (noise) and “special cause” (signal, important).  It was he who said that considering every variation as significant was not only wasteful, such “tinkering” made the process worse.  How often do we hear comparisons of say a murder number in a city being more than last year’s and hearing somebody pontificate an explanation?  Have you ever heard that this is common cause variability, and that if you want to lower the murder number, you need to address the entire system?

Samples have to be random, which is a way of saying everybody in the population, the group of people one is studying, has a definable non-zero chance of being chosen.  That doesn’t mean, ‘They didn’t ask me, so the sample is no good.’  It’s no good if the sample is done in the Deep South and the sampler wants to extrapolate it to the whole country.  One living in New York or Ohio never had a chance of being sampled.  Most people think large samples mean more useful results, but bias in a sample of 200 continues to be bias in a sample of 200,000 if the methodology doesn’t change.  The mathematics of sampling are not difficult to understand, and if one wishes to be a little less confident, 90% rather than 95%, and the margin of error for a dichotomous (yes-no) question allowed to rise to 8 or 9%, rather than 1-2%, the sample size needed decreases dramatically.

“I’ve worn out my welcome, but let me finish by mentioning the concept of 2-3 standard deviations from the mean, which most people take as being a significant outlier.  That all depends whether the curve is Bell-shaped.  If it is, then the probability of something more than 2 standard deviations from the mean is 5%.  But it is possible, depending upon the distribution of the data, to have up to 25% of items more than 2 standard deviations from the mean, hardly a significant outlier.  For 3 standard deviations, it is 3 in 1000 chance with a bell-shaped curve, but with some distributions, up to 11% of the observations.  I wonder how many who have been 2-3 standard deviations on the wrong side of the curve have been punished unjustly.  Five standard deviations?  4%.  It is 1/25, which is 5 squared.  This is known as Chebyshev’s Inequality.

“Finally, I would like to see students learn how to make good graphs instead of the ones I see today.  I would make Edward Tufte’s books required reading.  I would like to see more line graphs, dot plots, and box-and-whisker graphs with fewer multi-color pie charts.  I said I could go on for three more pages about statistics.  I have. Statistics has many day-to-day encounters, it is often used poorly, both by those who don’t know it and worse, by those who want to fool you.  It’s not lies, damned lies, and statistics but rather lies and damned people who lie using statistics.” 

SAY MORE, ARTHUR BENJAMIN!–STATISTICS AT THE PINNACLE–PART I

February 7, 2017

One of my good hiking friends posted a TED talk by Arthur Benjamin on why we should teach statistics at the pinnacle of math education, rather than calculus.  I had only two complaints with his talk: first, it was too short, fewer than 3 minutes.  He should have gone on for an hour with that audience.  They would have learned a lot from him. Second, I’d add that statistics can teach us a lot about life lessons.

I commented briefly, saying that I could easily write for 3 pages.  Then I thought, “Why not?” Few will read it, because it’s math, and well….

Anyway, I’d start off with a deprecatory statement about my field:  “We statisticians are almost never right.  That’s remarkable. Never right.  BUT, we know how wrong we are likely to be, because our estimates have a margin of error.  Any estimate that does not have a margin of error is, to us, worthless.  If that fact went to the Halls of Congress, if somebody said that “Social Security will be bankrupt by 2028,” I’d like someone to ask, “What is the margin of error?”  Why?  Because somebody made a prediction about the future with data.  If somebody made a different prediction with slightly different assumptions, they would have gotten a different answer.  How different?  That is what margins of error are all about.  We’re talking about the future, and we can’t predict the future with utter confidence.

“What is confidence?” I would ask. “Let me first define probability: Probability is the likelihood an event will occur in the future, not the past.  It can be 0, no possibility at all; 1, certain it will occur, or any number in between those two.  Those who gamble know something about probability.  Good bridge players know probabilities of various distributions of cards; six missing cards in a suit are more likely to divide 4-2 than 3-3.  What we need to learn in this society is the idea that probability is not always equal if there are only two options.  Heads-tails is 50-50; boy-girl is close enough, although not exactly 50%. Millions of illegals voted in the last election or did not, or vaccines cause autism vs. they don’t, and you still have two possibilities, but now they aren’t equal.  I wish the media would learn that and not assume all sides deserve equal billing.  As a corollary, I wish the media would remember that strong statements require strong evidence.

“Roll a die, and there is 1/6 chance a 3 will come up; all 6 possibilities have equal probability.  But when you roll two dice, there are 11 possible sums, from 2-12 inclusive, and their probabilities are not all 1/11.  If you disagree, please see me with your wallet in hand and we will play, because the expected value of my winnings, which is the likelihood of my profit or loss over a period of time, will be in my favor.  If I can bet on the fewest sums that will in the long run pay me money, I will choose 7, which has a 1/6 probability, 6 and 8, which each have (5/36) probability, and either 5 or 9, each of which has 1/9 probability.  In the long run, the probability will be 20/36 in my favor.  We need to teach that competing ideas do not necessarily have the same probability.  That means we shouldn’t give equal time to people who think alien abduction occurs, because it either does or doesn’t, and they feel they should have equal say.  When we get to more significant probabilistic questions, such as smoking significantly increases the likelihood of lung cancer or heart disease, or that polio vaccination dramatically decreases the likelihood of contracting polio, we can and should make appropriate public policy.  Liberal theories?  Nope, just laws of mathematics that can be proven and which may be applied to everyday life.

“Furthermore, probability can be independent or dependent, and failure to remember that was in part was behind the Challenger shuttle disaster. Independence means that the results of one trial don’t affect the next.  Dice don’t have a memory.  Dependence means that they do.  When one O-ring fails, the likelihood of another’s failing increased.  Pull three aces out of a deck of cards, and the probability I will draw an ace from the remaining cards is now 1/49.  That is a conditional probability.

“When we make an estimate of something, we need a margin of error, a wonderful concept which teaches us to be humble and say, “I could be wrong,” four words every man ought to learn before getting married, and a breath of fresh air again in the Hallowed Halls of Power.  A caution, however, in that a margin of error doesn’t mean anything goes, that “anything is possible.”  Anything is possible if one’s idea of possibility is a one in a trillion event matters.  Statistics discusses things like million, billion, and trillion, so let me describe likelihoods for various scenarios:

  • 1 in 1000: about the likelihood of getting a straight flush in poker or correctly picking a second at random that I have chosen which occurred in the last 17 minutes.
  • 1 in 10000: about the likelihood of guessing right a kilometer I am thinking of between Chicago and Tokyo, or picking a minute correctly that I am thinking of that occurred in the past week.  
  • 1 in 100,000: correctly picking a millimeter at random that I am thinking about on a football field from the back of end zone to the back of the opposite end zone.  Correctly pick an hour chosen at random in the past 12 years.
  • 1 in a million: Correctly pick a person chosen at random in a large city; a second chosen at random in the last 12 days; an acre I am thinking of in a large wilderness area 50 x 30 miles size.
  • 1 in two billion:  Correctly pick a second, chosen at random, from the 1 January 1955 to now.  A single second. Correctly pick a randomly chosen acre in the US.
  • 1 in a trillion: Pick a day at random since the Earth was formed.  

I think that every legislator be compelled to know the differences among million, billion and trillion before they are allowed to run for office, so we don’t get silly statements of “billions and billions, and billions of acres are locked up by the federal government.”  The whole country has fewer than 2 billion acres.  If you don’t have the sense of what a billionaire is, you can’t appreciate how much money that is.  A billionaire could spend two thousand dollars a minute for a full year, day and night, before they would run out of money.  Ten million dollar house bought Monday morning?  Paid off Thursday evening.

“We use something called a confidence interval.  That is a range around an estimate where we state how confident we are that the true value lies in the interval.  It isn’t probability, it’s confidence.  You see, there exists a true value, but it is unknown and unknowable.  The range we have will either contain that true value or it won’t.  That is a 100%-0% question and not helpful.  We have 95% confidence intervals to explain that if we were to take 100 different samples, obtain 100 different estimates and confidence intervals, 95% of them would contain the true value, but we wouldn’t know which 95.  See?  We don’t know the answer.  But we are highly confident we can construct an interval wherein it lies.

Knowing confidence intervals would have been useful for journalists who reported on the once famous 44,000-98,000 deaths annually due to medical errors.  They rounded the latter figure up to 100,000 and used it, but the point estimate of 71,000 was the single best number.  Zero was not possible, nor 10,000, nor a million, not possible if we are going to remain sensible about the world.

“Global climate change likelihood is prediction, which lends itself to statistics and to confidence intervals, and the IPCC was more than 95% confident years ago, a strong statement of science.  It means that the interval they calculated was highly likely not to contain 0, no temperature rise.  It is incumbent upon those who disagree to come up with a confidence interval so that we can look at their data and see what assumptions and calculations their models have.  This would prevent a lot of unnecessary arguing, and the arguments we have would be more appropriate.

“Means and medians are basic concepts people should understand, because a mean, the average, is affected greatly by outliers, whereas the median is not nearly as sensitive.  Housing prices and salaries are much better described by the median.  

People talk about a non-existent term called the Law of Averages.  I’d not teach it, and maybe it would go away. There is The Law of Large Numbers, which says frequencies of events with the same likelihood of occurrence even out, given enough trials or instances.”

“I can see that a lot of you are yawning and looking fried.  I’m giving you a year’s curriculum in a few minutes.  Imagine, however, how useful all this stuff might be if I had a year to teach it to students.  I actually tried to do that in Tucson in 2011, for free, as a trial course, my swan song before leaving town 3 years later.  But I didn’t have an education degree, and the school had other priorities.  Such a shame, really.   OK, let’s take a break, and come back and I’ll finish the summary.”

A TIDE IN MY AFFAIRS

January 6, 2017

Warning: This post will contain some mathematical formulae and terms, which may scare or otherwise turn off some.  I hope such formulae do not detract from the beauty of what will be seen, because indeed, mathematics is beautiful.  It answers questions.  Is that not beauty?  In a week, pictures of the result will be shown.

I’m going over to Newport, Oregon next week to see the King Tides, something I had once never heard of.  I am almost a true Oregonian, but when I led a trip to the coast the last week, I forgot to look up the tides. That’s inexcusable.  Always know the tides when you are at the ocean.

Tides matter.  A lot.  In nature, many species thrive at border zones between one ecosystem and another.  They allow for organisms to live in varying degrees of wetness, rather than always wet or always dry.  They allow for tidal pools to become cut off from the ocean, where periodically they get refilled or organisms shuffled.  Without tides, the Earth would be a very different, far less diverse place.

What are tides, anyway?  They are common throughout the universe.  If one object tugs on another, it can deform the latter due to gravitational attraction, which may cause buckling or movement of the surface of the attracted object.  Jupiter’s moon Io gets tugged by massive Jupiter, causing volcanic eruptions on its surface.  The first was spotted by a woman, Linda Morabito, who saw a plume on Io, which had been once thought once to be dead, then had volcanism predicted.  Io is the most volcanically active place known in the solar system.

Both the Sun and Moon tug on the Earth.  While the Moon is much smaller, a mass 1/27,000,000 that of the Sun (mass is the amount of “stuff” something has; weight is the effect of gravity.  Diet removes mass; being in zero gravity does not, but it makes you weightless), the Moon exerts a majority (55%) of the tidal activity on the Earth.

For a long time, that 55% bothered me, because gravitation is proportional to the product of the masses but inversely proportional to the square of the distance, the distance between the two centers, or d, and the numbers didn’t work.

F=G m1 m2/d^2.

where G is the gravitational constant, m1 the mass of one body, m2 the mass to the second, and  d^2=d*d, the distance between them multiplied by itself.  The Moon is smaller, less massive, but it is much closer than the Sun.  Still, if one compares the large mass of the Sun with its admittedly larger distance from us (400 times further from the Moon, and the distance varies, which is important), the Sun ought have an effect 170 times greater than the Moon upon us.  It doesn’t, and that bothered me.  I show this below.  Gravity is the reason we circle the Sun and not the Moon; the Moon circles both of us.  I did not consider tidal forces, those which work differentially on a body, more on the near side than the far side.  These Ah-hah moments are one of the joys of life, when one understands a concept that has been murky for years.

The Moon tugs on the Earth, the oceans are pulled towards the Moon. Tides are maximal in general when the Moon is either overhead or at the opposite side, although that can vary considerably due to other factors and local conditions, which give rise to enormous tides at the Bay of Fundy or tidal bores on Turnagain Arm in Alaska.  The tide is greater (spring tides, nothing to do with the season) when the Moon is lined up with the Sun and the Earth, occurring about every 15 days, and lesser (neap tides) when the Moon is not aligned.  The square of the distance means that anything decreasing distance increases the tide, so when the Moon is close to us, which happens every 27.5 days, even not well aligned with the Earth and Sun, the tides are significantly affected. The Earth is 3 million miles closer to the Sun in early January compared to early July, and this increases tides as well, because while the Sun’s force is slightly less than the Moon’s, its distance from us is the least for the year. That’s why we’re going to Newport.

In Newport, king tides occur at full Moon in January, near perihelion.  The full Moon is opposite the Sun, meaning that it is in the northern part of the celestial sphere, over the northern hemisphere, and therefore is closer to the coastal cities there.

I also didn’t know why the Moon had a greater pull, given the gravity equation.  The numbers didn’t work. I thought—incorrectly— it was all gravity.

The tidal force looks at slight changes in the distance between the two bodies; the force is proportional to the cube of the distance between the bodies, d^3, or d*d*d, and a simplified proof is shown below.  Cubes are volumes, and the three factors are length, width, and depth.  When we compare the gravitational equation using the cube of the distance and twice the mass product, the Sun is responsible for about 45% of the tidal force; the Moon the rest.

Additionally, the lowest tide is not in January, as one would think, but is in the late spring early summer and at New Moon.  Why?  In May, the Earth is further from the Sun, so the Sun’s pull is less.  But at New Moon, which aligns with the Sun, the Moon is over the northern hemisphere. There are issues with the lunar nodes and the tilt of the Earth’s axis at different times of the year.  Tides are more complex than I thought, not due to simple gravitational pull but to a differential force that must be accounted for. When I go to Newport, I will be watching a 3 meter high tide and the -0.5 meter low tide, both a full meter higher than normal.

 

 

F(S-E)=Gm (S)*m(E)/d(S-E)^2. The Sun-Earth gravitational force is proportional to the product of the masses and inversely proportional to the distance between their centers. The same holds for the Moon-Earth.  It also holds between you and your computer, too.

F(M-E)=Gm (M)*m(E)/d(M-E)^2

Let’s take the ratio of the Sun-Moon forces which is dividing the top by the bottom.  Stay with me, because G and m(E) will disappear when we divide, because they are part of both.

Ratio=m(S)/d(S-E)^2 divided by m(M)/d(M-E)^2

When we divide, we invert the divisor, which is the value that is “going into” something.

If we divide 1 by 1/3, we invert the 1/3 and have 1 *3/1 or 3.  One-third goes into 1 three times.

If we do this math, we invert the denominators and have

Ratio=m(S)*d(M-E)^2 divided by m(E)*d(S-E)^2

We know these ratios.  The mass of the Sun is 27,000,000 that of the Moon.  The distance to the Moon is about 1/389 the distance to the Sun.  Let’s call it 1/400.  By the way, in the sky, the Moon is about the same angular size as the Sun, which is why we can just have total solar eclipses. The Sun is about 400 times the diameter of the Moon and is about 400 times further away, so they have about the same size when viewed from the Earth, one of the greatest cosmic coincidences there is.

The ratio of forces is about 27000000/400^2, or 169.  But the Sun is actually less powerful as the Moon in producing tides.  Tidal forces are differential and work differently on one side of the body versus the other.  Tidal forces are not the same as gravitational forces. They work as the inverse cube, not as the inverse square.  A cube here is d*d*d or d^3.  We measure volume when we know three factors—length, height and depth.

The ratio can be done by subtracting the force of the two objects from the front by the force from  the back.  Or, and this is why calculus was invented, we can take the derivative of the gravitational force with respect to the distance, because only the distance is changing, not the masses, and derivatives of constants are zero, making life a lot easier.  Here, we deal with the change of distance.

The derivative of Gm1m2/d^2 with respect to d is -2Gm1m2/d^3.  The bottom line, literally, is a cube, and the differential force for tides is a function of the cube of the distance, not the square.  If we look at the above ratio, we get 27,000,000/400^3 and it is 0.42.  If we use the average figure of 389 times further away, we get 0.46.  Tides are much more complex, but the idea of the inverse cube ratio is why the Moon exerts a greater tidal force on us than the Sun.

A second proof for tidal forces being proportional to the inverse cube of the distance is abbreviated, but goes something like this:

Force of Sun (Fs)= G(SE)/d^2, where G is the gravitational constant and SE is the Sun Earth distance.  We could make it the lunar distance if we wanted to.

The distance is slightly different on the other side of the Earth, so we will call that p.

F(SE-near or s1)-F(SE far or s2)=G (SE)/d^2-G(SE)/(d+p)^2

=G(SE){(1/(d+p)^2)-(1/(d^2)}, d is much greater than p or d>>p.  We have factored out G(SE), which is common to both.

Look at the parentheses, and using common denominator subtraction,

(d^2+2dp+p^2-d^2)/(d+p)^2d^2

=2dp+p^2/(d^4+2d^3p+d^2p^2)

=2dp/d^4,  skipping some steps, since as d gets very large, the denominator approaches d^4,

=2p/d^3

From earthsky.org, which is nowhere near scale but shows where tides come from.

Screenshot 2017-01-06 09.52.28.png

GREAT EXPECTATIONS

March 23, 2016

It all started with a packaging error on some soy burgers I bought.  At home, I discovered a slit in the bottom of two that I didn’t see when I bought them.  Who knows how long the contents had been exposed to air?  How often do you check the packaging when you buy something?  For me, not often enough.

I returned to the store and told the manager, who was shocked.  I said I was going to look at the other packages they had to see if they, too, were defective.  When I checked the box that had the identical soy, sure enough, the two remaining packages in it had rips.  I took them back to the manager’s counter.

I had to wait, however, as I watched a woman with bleached blonde hair, maybe mid 40s, and a young man, who might well have been her son, discuss lottery tickets.

“We can’t afford too many,” she said.  “We need $20 for xxxx.”  I couldn’t hear what “xxxx” was.

I was stunned, as I watched them purchase six lottery tickets, some sort of scratch type, for $60, which the woman counted out using twenties, fives and ones.  It might have been a Raffle. You can’t win if you don’t play, right?  That was the catch phrase in Arizona.  You can’t lose if you don’t play would have been more accurate, and I can prove it.

Let us look at the expected value, which anybody playing the lottery should understand.  We ought to teach this in schools. If the expected value of something done is positive, over the long run it will be successful; if negative, over the long run, it will not be.

Here’s an example.  At a roulette table, you plunk down $1 on a roll of two dice.  If they come up double 3s, you get a 30-fold return—$30.  Well, it isn’t 30 fold, because you paid $1 to begin with.  It is 29-fold.  Don’t laugh, casinos get rich on these “minor points.”  You have a 1 in 36 probability that the dice will both be 3: 1 in 6 for each, and we multiply the probabilities when one doesn’t affect the other, a term called  independence. The probability of winning $30 is (1/36), and the probability of losing $1 is (35/36).  If you multiply these and then add, you get +$(30/36)-$(35/36)=$(-5/36).  That fraction comes out to about MINUS 14 cents per dollar bet.  Are there winners?  Yes.  The consistent winner is the casino.  Not only is it consistent, they can predict very accurately how much they will make, because in the long run (not a player’s time horizon), the casino will make 14 cents on each dollar wagered for that bet.

What I saw for a $10 Raffle ticket:

1 in 250 chance of winning $100

1 in 25,000 chance of winning $20,000

1 in 250,000 chance of winning $1,000,000

The exact calculations are more complicated, but the simple way works just fine.

You spend $10, and you have a 1/250 chance to win $100, an expected value of $100/250 or $0.40, 40 cents.  You also have a 1/25,000 probability of winning $20,000, and that is $0.80, so the expected return of both is $1.20. Add to that the 1 in 250,000 chance of winning $1 million, $1,000,000/250,000=$4.  The total expected return on $10 spent is $5.20, give or take.  That’s the plus.  The minus is $10(249/250)=$9.96.  The expected value is $5.20-$9.96=  MINUS $4.76.  The pair spent $60. If we looked at a lot of people buying, all of them, they would on average lose about $28.50.  One of those $20s that one comes in to the store with, and nearly a ten spot as well, won’t be seen again.  Do that weekly for a year, one will lose nearly $1500 on average.

The lottery is very coy about posting long lists of numbers that won $100.  Sounds like a great deal, except that first, they really won $90, because they paid $10 for the ticket.  Secondly, the list of those who lost is about 120 times longer.  We don’t see that one.

Here in Oregon, 93% of the money spent on the lottery goes to payouts.  Sounds good, except that the big payouts go to very few people.  BUT A LOT OF PEOPLE WIN A LITTLE, LIKE $100, AND THAT’S A HUGE PROBLEM.  This is called variable ratio reinforcement.  Winning occasionally keeps one playing.  Never winning at all causes one to quit sooner.

There is a very strong correlation between low income and high use of the lottery. Why do people play?  Answer: they see this as the only way out of poverty.  The probability is exceedingly high, however, that they will only go deeper into poverty. The lottery is a regressive tax levied on those who can least afford it.  The lottery steals from those who don’t understand math, probability, or how our brain can lie to us.  When the money goes to education, a noble cause, it is being paid for by those who have the least money: the very poor spend 9% of their income on the lottery.  If one makes $13,000 a year, that is nearly $1100 they spend on the lottery.  If one has difficulty making ends meet, this is going to push them over the edge—with high probability—and I can define that probability exactly.

I never forgot what my statistics advisor said about expected values:  “If it is positive, I will beg, borrow, steal every dollar I have to play.”

It becomes absolutely certain at some point that somebody will win Powerball.  We can predict that as well.  If there is a Powerball with a probability of winning equal to 1 in 110 million (roughly equivalent to your guessing correctly a minute I choose between the signing of the Declaration of Independence in 1776 and now), and 330 million play the Powerball, the expected value of winners is 3.  Three people are expected to win; the probability of exactly 3 is about 22%; the probability that between 1 and 5 will win is nearly 7 in 8, and the probability nobody will win is about 5%, quite small.  In other words, one can make remarkably accurate probabilistic statements what is going to happen.  Did you guess my minute? (It was 3:32-3:33 a.m. 15 August 1846).

I’m lucky; I live comfortably.  Still, I bend down to pick up a dime or a penny if I see one.  If PetsMart gives me a $3 coupon for doing a survey, I will do it.  If REI gives me a dividend, and I am planning to buy something, I will buy it with the dividend and furthermore try to get it when the item is on sale.  I use coupons when I shop, I comparison shop, I don’t drive 30 miles for cheaper gas, because it’s more expensive to do so, I pay my credit cards off every month, I try not to get a tax refund, because it means I loaned the government money, and I DON’T PLAY THE LOTTERY.

But occasionally, I do silly things with money.  The store agreed that the ripped packages I returned, and the two others in that I found that were also ripped, needed to be removed.  I found two good packages, but they were not on sale, so I actually had to pay to replace them.  I paid $4.40 for doing the store and any customer who bought those a favor.  Small price for what I learned about the lottery.

The expected value for doing a good deed was negative.  In the long run…

ONLINE, ON COURSE

November 29, 2015

I received the following letters the past few weeks.  They made my day.

Thank you so much, I was struggling and your answer made it simple and understandable. UR GR8.

You are Amazing!

You have helped me in the past and always have accurate answers, I am so grateful you took your time out to help me today, thank you, I appreciate it so much!

Thank you, from the bottom of my heart, you have written this in a way that I totally understand. I will differently (sic) praise you to all my friends and family. God Bless you,

What did I do to deserve these?

Reason for the first:

The area of a triangle is 30 sq in. The base of the triangle measures 2 in more than twice the height of the triangle. Find the measures of the base and height. 

The area of a triangle is (1/2)*base*height.  Remember that?  Therefore, the base*height must equal 60 for the area to be 30.  Let the height= x inches, then the base is 2x+2, two more than twice the height.  Then, 

x*(2x+2)=60, 2x^2+2x=60, and dividing by 2, x^2+x=30.  We can write that as x^2+x-30=0 and factor it as (x+6)(x-5)=0.  That is 0 if x=-6 (not possible for a length) or if x=5.  So, the base is 5 inches and the height is 2(5)+2, or 12 inches.

For the last comment,  I answered the following, taking about a minute in my head, writing it down as I thought.

Write the slope-intercept equation for the line that passes through (-12, 10) and is perpendicular to 4x + 6y = 3. 

One gets the slope first by rewriting the equation as 6y=-4x+3 and dividing by 6 to get y=-(2/3)x+1/2.  The slope is -2/3. The perpendicular line has a negative reciprocal slope.  Turn the fraction over (-3/2) and change the sign (3/2).  That is the slope of the perpendicular line.   Using the point slope formula where we know the slope and a point, x=-12, y=10, y-y1= (3/2) (x-x1).  That is y-10=(3/2)(x+12).  This becomes y-10=(3/2)x+18, and finally y=(3/2)x+28.  Also, y=mx+b, so 10=(3/2)((-12)+b.  That is 10=-18+b, so b=28.  Both methods work; the more ways one knows, the more ways to explain it to students.  One of the ways is likely to stick.

For the one who called me amazing?

Find the accumulated value on an investment of $15,000.00 for 9 years at an interest rate of 11% if the money is compounded 

a) Semi- annually b) Quarterly c) Monthly d) Continuously 

 Here, one uses the formula 

Principal=Starting Principal{1+ rate/compounding per year} raised to (the number of years*compounding per year). P=Po{1+r/t}^nt.  Semi-annual is P=Po{1+(0.11/2)}^18, because it compounds twice a year and there are nine years.  This is $39,322.  For continuously compounding, it is easier, P=Po*e^rt.  e^rt= e^(0.99), because 9*11%=0.99.  Po*e^0.99=$40,368.52.  Continuously compounding gives you more money, although the difference between it and monthly is only $200 less than continuously.  The last formula allows one to prove that the doubling time of money in years is 70 (or 72, which is easier to work with) divided by the interest rate in per cent.  I grew up in the age before calculators, and we had to do this by logs.  On a calculator, it takes about 15 seconds. Dividing 72 by 11 gives a doubling time of about 6 1/2 years, so $15,000 should double once and be well on its way to doing it again.  The answer makes sense.

This is an online math help site.  More than 2000 tutors take part, some of whom have solved one problem, one nearly 70,000.  I’ve solved 2000.  About one in four thanks me.  That’s nice.

Several tutors offer their services for pay, $1 per answer, $2-$5 to show the work.  I do it to relax.  Yes, relax.  This stuff is fun for me, and I have learned the easier the problem for me, the more grateful people tend to be.  I don’t need to hear anything, unless my answer is wrong or not understandable.  I’m there to help.  I don’t know names; I do know I have helped parents help their children.

I’ve learned much.  It has been a great review of my statistics, I now deal with ellipses better, and I understand geometric series better than I ever have before.

I usually want a challenge, so I choose what I want to solve.  I have a big advantage:  I grew up in the era of no Internet, Chemical Rubber Company tables of integrals, no calculators, only log tables to do complex calculations.  In other words, I learned math from first principles, from the ground up.  Yes, it helps to have a genetic ability to do this stuff.  I can’t play the violin, but I can find the vertex of a parabola mentally and write it in three different forms.  Kids need someone to help them understand how to do it, not in their head, but to allow them to understand these and similar problems.

The current list has perhaps 50 problems, and I often work down it until I find a problem I feel like doing.  If interested, I go to the list of unsolved problems.  Last I checked, statistics had about 40,000.  A lot of those are tough, and if I don’t have pen or paper around, I don’t do them.

When I tutor at the community college, I answer algebra questions online while waiting for non-virtual students to ask for help.  I guess I am volunteering, but I am having a lot of fun.  It’s nice to lay out quickly an answer in simple form for a person who is struggling.

The other day at the CC, I was asked to go into the higher level math room to help out.  That was a compliment, because I was felt to be good enough to help out there. I’m the go-to guy for statistics.  The other tutors are really smart, yet all of us at one time or another have trouble with something.  I may struggle at the high levels, but I often find myself pulling stuff out of the air from the past and making sense out of it.  Or better yet, I ask a student where he got a specific term in an equation.  The student looks puzzled then suddenly says, “Oh, wow, I didn’t see that before.  OK, I understand.  Thanks a lot.”  And he leaves.

I hadn’t a clue how to solve the problem, but I think I helped him.

Math is mentally taxing.  After doing about a dozen problems, I take a break.  It helps me later solve troublesome problems.  In the math lab, I have concentrated so deeply that one day when I walked out of the room, I forgot whether it was Tuesday or Friday.

I think the absent-minded professor was probably working overtime on a difficult problem.

CONCATENTATION AND EXPECTED VALUES

July 18, 2015

A bloody picture of a cyclist adorned my Facebook page.  The writer was succinct:

How I joined the walking dead:

1. Rented a bike with defective brakes.

2. Started riding through a long dark RR tunnel.

3. Encountered a multi-family group with very small children in tow coming the other way.

4. Wiped out trying to avoid scattering kids like bowling pins.

This is a classic description of a fortunately not tragic accident.  Each one of those incidents alone might not have been sufficient, but together they caused a bloody rider. There was a concatenation of events.  Sometimes, we have a concatenation of errors.

I had my own sixteen years ago this month:

  1. Took part in a long distance bicycle tour only a few months after starting to ride a road bike.
  2. Ended up on a rainy day wet, tired after crossing 3 Colorado passes, and eager to get to the school where we were going to be camping.
  3. Saw a car in the turn lane headed towards me.  I had limited experience riding a bicycle in traffic.
  4. Assumed the driver saw me.
  5. The car suddenly turned in front of me.
  6. Too late, with wet brakes, I skidded and landed on my right hip, trying to avoid him.  I wasn’t the walking dead, but I didn’t walk normally for several months, and I’m lucky I can walk today.

It’s worth discussing the concept of the expected value of an event, like the lottery.  People see 2 winners in the last lottery and buy tickets, because after all, they could win.  It has to be somebody.  This is usually true.  If not, eventually the probability becomes so high that when the lottery has an unusually large payoff somebody (or several people) almost certainly will win.

If the probability of an occurrence is extremely small, invariable, and not zero, and the number of times the occurrence may happen is very large, the expected value is their product.  A probability of 1 in 110 million of winning x 440 million lottery tickets sold has an expected value of 4 winners.  It’s that easy.  Low probability events, like automobile fatalities, occur every day, because so many people drive. Expected values are just that.  They are expected, but they are not necessarily going to occur.

Aviation, perhaps more than any other endeavor, has taken safety to heart, because aviation is so unforgiving of errors.  Additionally, aviation has a large number of events, called flights, where there is a low but non-zero probability of a crash.  Aviation has tried to improve the probabilities and in commercial aviation, there have been multiple years, often consecutive, without a fatality.

Non-commercial aviation isn’t as safe.  Nearly two decades ago, a 7 year-old was trying to be the youngest person to ever fly across the country.  Being the youngest, oldest, first, most disabled, fastest, —st is often the first cause in a cascade of events that leads to tragedy.

A 7 year-old had no business being at the controls of an aircraft.  Period.  One of the last things to mature is judgment.

  • They took off to try to beat a thunderstorm, poor judgment, because wind shear is unpredictable in thunderstorms.  One must wait.
  • They were overloaded.
  • The runway was at a higher altitude where there is less lift for aircraft.
  • Rainwater on the wings diminished lift.  Airfoils are delicate; distortions of shape diminish performance.
  • They turned to avoid part of the thunderstorm.  Turning decreases lift.  The overloaded, slow moving, distorted airfoil plane stalled and crashed, killing all aboard.

Remarkable finding of evidence and piecing it together led to understanding why Air France 447 crashed in the mid-Atlantic in 2009.  Here’s a root cause analysis:

  • Why did the plane crash?  It stalled.
  • Why did the plane stall?  It was in the nose up position for the last part of the flight, reducing lift.
  • Why was the plane in the nose up position?  Because the co-pilots had taken control and saw that the altitude was low.
  • Why did the co-pilots take control? Because the autopilot had shut off.
  • Why did the autopilot shut off?  Because it wasn’t getting useful information from the pitot tubes, like altitude and speed; the altitude reading was faulty.
  • Why didn’t the co-pilots keep on the same course as the autopilot? Because they trusted the instruments.
  • Why weren’t the pitot tubes sending useful information?  Because they were faulty and needed to be replaced, but the airline was phasing them in.
  • Why was the airline allowed to phase them in?  That ends the questions.  That’s where action needed to occur.  Additional causes included the pilot’s napping (not wrong) so he was not in the cockpit when called.  There were other crew miscommunications.
  • What could have been done?  As soon as the “stall” alarm came on, the crew needed only to push the nose of the aircraft down.  Planes stall when they climb too rapidly.

**********

This root cause approach to errors is what medicine needs.  When a surgeon operated on the wrong side of the head, he got a letter telling him not to do it again.  Nothing changed.  Here’s what happened.

  • Patient in ED had a subdural hematoma and needed emergency surgery.  There are emergencies where one must act in a matter of seconds, and there are emergencies where one needs to act quickly, but can take a few minutes to think about the necessary approach.  A lot of people in and out of the medical field don’t understand that there is a huge difference between the two.  Unnecessary hurry is one of three bad things in medicine (others are lack of sleep and interruption).  A subdural hematoma needs to be evacuated, but unlike its cousin an epidural, it doesn’t need to be done in the emergency department, and there is time to plan the procedure.
  • CT Scans were relatively new and had changed the left-right orientation opposite to traditional X-Rays.  I practiced when CT scans showed this orientation, and it was extremely confusing.
  • Many people have trouble distinguishing left from right.  It isn’t a personality flaw, it is a biological issue, akin to being shy.  Approximately 15% of women and 2% of men have this problem.
  • Nobody spoke up to tell the surgeon they were concerned upon which side he was operating.

Without going into more detail, I reiterate the comments I made to the head of the operating room, who assured me that 99.9% of the time they did it right.

“No,” I replied.  “You get it right 99.99% of the time, and that isn’t good enough.  Counts matter, and wrong side surgery cases must be zero.”

We need better system design to decrease the probability of the wrong thing’s happening.  The stronger our systems, the more events will have to occur for something to go wrong, and that means people will be safer.

We will never know if a better system saved a life.  But probabilistically, it will increase the expected value of success, and I trust expected values.

PUSH BUTTON EDUCATION

June 2, 2015

“You guys have all given me different answers, and I don’t know what to do.”

The math tutoring room at the local community college has two parts, one for advanced math—trigonometry, pre-cal, and calculus; the other is for basic math, from carrying and borrowing up to college algebra.  I work in the latter, but as somebody with a Master’s in statistics, I am often the “go to” person for statistics questions.  The fact I have seldom used statistics in the last decade has made me rusty, but the material comes back, so long as one learns it well the first time.

When the individual came to me stating the conflicting opinions she had received, I should have either turned her down or told her she was going to have to decide up front whom to believe.  If I were not that individual, she should leave, and not waste her time.  The issue itself was a 1-sample proportion test, one of the M &M problems, where a certain proportion of different colors are put into the bag, people count out the number of each type to see if the proportion corresponds with the claimed proportion, within a reasonable margin of error.

The student had used the instructions given to her what to input into the calculator and found a probability that made no sense to me.  I looked at the question and came up with the correct probability.  The example she copied looked at the probability’s being greater than a specific number; the problem she asked me looked at the probability’s being less than a specific number.  She didn’t understand that the example given to input and the problem were asking opposite things.

I tried every way I could think to explain the issue to her.  I have become more adept at calculators, finding them fast and helpful.  This woman, as are so many students today, was faster with the calculator than I.  Her problem, however, was something that it took me some time to figure out.  I had drawn a diagram of the probability curve, the Bell-shaped normal or Gaussian distribution, and she had looked confused.  That led me to finally ask a simple question:

“Have you ever computed these probabilities using a normal probability table?”

“No.”

I now understood her problem.  She was being asked to input data and push a lot of buttons.  Unfortunately, she had no idea what was being done to the data and why.  A lot of statistics is finding the difference between the sample and a postulated or known mean/average, then dividing by the standard error, a measure of variability.  The concept of variability is critical to understanding not only statistics, but everything statistics is used for, be it political campaigns or climate science.  Natural processes, like heart rate, body weight, stock market prices, or temperature, are not the same when measured over a period of time.  They fluctuate, and statistics helps us understand the fluctuation.

Dividing the mean by the standard error normalizes the data, allowing it to be compared to one standard, this instance to a table to find a probability.  By doing many problems where I had drawn a bell-shaped curve and looked at probabilities, I understood the concept well enough to teach it to undergraduates in Las Cruces for two years and in Tucson for another four.

This woman was from another generation of students, however, and in the decade where I have not been heavily involved with statistics, drawing a picture of how the data were distributed and having a sense of what the data were trying to say has atrophied, at least where I am tutoring.  The argument I was having with the student had a lot to do with the arguments I needed on the calculator;  she did not understand them, only that she was obtaining different answers.  Put simply, she did not have the background to be using a calculator.  I could say that about many students.  When I taught for a private for profit college, when a student saw a probability “6 E-4,” they wrote “6” as a probability, both impossible and showing no sense of what E-4 means, which is a power of 10 to a minus number:  6 E-4 =0.0006.  I don’t expect the average person to know that; I do expect somebody taking statistics and using a calculator to understand it.

That is only my opinion, from one who learned the material from first principles and is still slow to pick up a calculator, because I am often more comfortable performing my own calculations.  It remains to be seen whether we will continue to teach by calculator or teach by understanding the material, using the calculator as a tool to speed up the process.  I fear that in our rush to educate people, we are giving them instructions as to what buttons to push in a lot of subjects, without any idea of what is going on inside a calculator or more importantly, inside the system we are analyzing.

This is not idle philosophical musing.  When I taught, more than half the class did not understand what “the rate of increase in health care costs is declining” meant.  To them, the statement meant that the costs were decreasing, rather than the number was still increasing, but less rapidly than it was before.  This term is commonly used. The concept of statistical error to many people means that statistics is wrong, so it doesn’t matter.  Statistics is unable to tell us the exact results in a population, because from a poll, we do not know what the exact result is for the people in whom we are interested.  Where we differ from other fields is that we quantify the error in terms of confidence and probability, and we know the difference between the two terms.  We reject the concept that “anything can happen,” because we define a priori what “can happen” means.

We need to learn what calculators can do; equally importantly, what they cannot do.  Data that are not collected randomly have limitations what we can say about them.  Calculators do not have the ability to discern that. Calculators answer only what we ask them; they neither ask questions, nor do they tell us what we might want to know.

Calculators and computers are wonderful tools to get information that one needs, but education, critical thinking, and understanding remain timeless.