Boreal B[l]og

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.” 

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.”