by Haim Shapira
In any event, Sandra Zahler was murdered in 1974 in the same city and under very similar circumstances, and again … her neighbours heard everything but did nothing. The non-intervention phenomenon and the diffusion of responsibility are thus often named the ‘Genovese syndrome’.
Another example of the Volunteer’s Dilemma was an experiment conducted by Science, the scientific periodical. The journal asked its readers to send in letters stating how much money they wanted – $20 or $100 – and promised they would receive the sum they asked for, provided that the number of readers asking for $100 didn’t exceed 20 per cent: if it did, no reader would receive any money.
What would be my considerations if I played this game? Clearly, $100 is more than $20, but I understand that if everyone asked for $100, we’d all be left with nothing. All the rest must understand this as well as I do, and will likely write $20, not $100. Also, I think to myself that the chances of my becoming the tipping point – that is, the one who takes the number of greedy readers over the 20 per cent line – are quite low, and so I should ask for $100. Clearly, I could lose everything if enough readers thought the same. The actual result was that more than one-third of the people who sent in letters asked for $100, and the journal saved a lot of money.
In truth, that experiment was never meant to use real money, as the journal was working on a pretty safe assumption of success. Game theoreticians and, even more so, psychologists could have put the journal editors’ minds at ease, because the chance that less than 20 per cent would ask for $100 was rather slim.
Yet, as always, nothing is ever as simple as it looks. I experimented with this dilemma with my students many times, in the following way. I asked them to send me notes stating whether they wanted their grades bumped up by either 1 point or 5 points, and warned that they would each get what they asked for if less than 20 per cent of them asked for 5 points, but that nothing would happen for them if the 20 per cent limit opting for 5 points was exceeded. My students never earned that bump, except once – in a psychology class.
Chapter 10
LIES, DAMNED LIES AND STATISTICS
In this chapter l’ll provide some useful tools that will help us understand statistical data a bit better and improve our ability to detect statistical falsehoods – bad statistics, unfortunately, can be used to prove almost anything with relative ease. I’ll use funny and illuminating examples from everyday life.
When it’s time to make decisions we often turn to numbers – very many numbers. The discipline that deals with analysing and understanding numbers is known as statistics.
The novelist H G Wells (1866–1946) predicted that ‘statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write.’ Indeed, statistical data are found everywhere today. You can’t open a newspaper or watch TV news or surf the web without coming across quite a few statistical terms and figures.
DRIVEN TO STATISTICS
Several years ago, I read in a news report in a major newspaper that speeding doesn’t cause accidents. That assertion was based on statistical data, according to which only 2 per cent of all traffic accidents involve cars travelling at 100kph (some 60 mph) or faster, which could be interpreted to mean that 100kph is a remarkably safe driving speed. Of course, despite being published in the papers, this is an absolutely wrong conclusion. After all, if it’s true, why stop at 100kph? Let’s take a sad song and make it better. According to my data, no accidents at all happen at 300kph, so the state should instruct all motorists to maintain that safe speed. I’m even willing to have the law that obligates everyone to never drive slower than 300kph named after me and forever be known as the Shapira Law.
Seriously though, that report failed to provide some crucial pieces of data – for example, the percentages of driving time in which motorists maintain that speed. We need this to ascertain whether that speed is indeed safe or actually rather dangerous. For example, if motorists spend 2 per cent of their driving time at 100kph or more, and 2 per cent of all accidents take place during that time, then it’s a ‘normative’ speed: it’s neither safer nor more dangerous than other speeds. Yet if we drive at 100 kph only 0.1 per cent of the time and still have 2 per cent of the accidents, then that speed is very dangerous.
A recently published Israeli survey stated that women drive better than men. This may be right, but the survey cited a bizarre reason for its conclusion, namely that more Israeli men are involved in serious car crashes than women. This fact alone tells us very little. Suppose there were only two female drivers in the whole of Israel and they were involved in 800 serious accidents last year, while a million male drivers were involved in 1,000 accidents. That would mean that the average number of accidents for every woman was 400 per year (more than one per day). I wouldn’t have said they were good drivers on this basis. Would you?
By the way, according to an article published on Sunday 21 February 2016 on the Internet site of The Telegraph, women are, after all, better drivers than men, at least in the UK. The article states that ‘Female drivers outscored males not only in in-car tests but also when observed anonymously using one of the UK’s busiest junctions – Hyde Park Corner.’
GRAPHS AND LIES
Here’s an example of playing with data in graphic presentations. Suppose a company’s share price went up from $27 to $28 between January 2015 and January 2016. In our computerdominated day and age, people love to demonstrate such things with graphs and presentations. How can this be done well? That depends on your audience.
If the presentation is for the taxman, the first graph on the next page would be the recommended graph.
As you can see, things don’t look too good here. The graph looks like the pulse of a dead person. It could break the hearts of even the most hardened IRS agents.
If the same data were to be presented to the company’s board, I’d amend the graph slightly and make it look like this:
How about that arrow! Watch how it shows that not only did the share price skyrocket, but that it will keep rising.
The difference between these two presentations is one of scale – the particular yardstick that we choose. With a little imagination and effort, anything can be presented in a way that suits our needs.
Watching a TV commercial, I saw the following graphic presentation of client satisfaction with three service companies. Naturally, the company that sponsored that ad scored the highest – 7.5 (out of 10) – while its two competitors scored 7.3 and 7.2 respectively. The graph didn’t show the number of clients sampled, and so there’s no way of knowing whether the difference between the three companies is real. In any event, this is how the figures were presented:
The columns create the impression that the advertising company is way ahead of the competition. Well, who would have guessed it?!
Benjamin Disraeli (1804–1881) was probably right when he said that there are three types of falsehoods: lies, damned lies and statistics. In truth, however, that story is probably false too. Mark Twain (1835–1910) attributed the comment to Disraeli, but no one has ever claimed to hear the British prime minister pronounce this famous saying, and it isn’t to be found in any of his writings.
SIMPSON’S PARADOX
In 1973, investigators who followed a gender-bias complaint against the University of California, Berkeley, found that after some 8,000 men and 4,000 women applied for postgraduate studies, the percentage of men who were admitted was much higher than the intake of women. The university was sued for bias, but did it really discriminate against women? The investigators checked admission figures of individual departments, and it turned out that if there was any reason for a lawsuit, it was for the opposite bias: all of the university departments favoured women applicants and admitted more than men, percentage-wise.
If you’re not familiar with statistics (or the laws of calculating fractions), this may seem impossible. If all the departments were found biased towards women, then the university as a whole should surely ha
ve displayed the same gender bias; and yet, that was not the case.
The British statistician Edward H Simpson (b.1922) described this phenomenon in his 1951 paper ‘The Interpretation of Interaction in Contingency Tables’. Today we call it Simpson’s Paradox or the Yule–Simpson Effect (Udny Yule was a British statistician who had mentioned a similar effect already, in 1901). I’ll explain it not with Berkeley’s real-life data but with a simple hypothetical version.
Suppose we have a university with only two departments – mathematics and law – and let us assume that 100 women and 100 men apply to the school of mathematics, and that 60 women (or 60 per cent) and 58 men (or 58 per cent) are admitted. It would seem that the school of mathematics favours women. Another 100 women apply to the school of law and 40 (40 per cent of the women) are admitted, while only three men apply and just one of them is admitted. One of three is less than 40 per cent, so it would seem that both schools are biased toward women. Yet if we look at the combined figures for the entire university, we’ll find that 100 of the 200 women, or 50 per cent, who apply are admitted, while 59 of the 103 men who apply are admitted, which is more than 50 per cent.
What’s the explanation here?
Instead of going into technicalities, let me present the intuitive explanation. Based on the data we have, it’s the law school that’s clearly harder on its applicants. Thus, when many women (100) apply to the school of law, their 60 per cent admission rate at the school of mathematics loses quite a lot of its value. Given that the same number of women apply to each of the two departments, the combined admission percentage is the average of 60 and 40 – which is 50. Yet, aware of its strict admission policy, only three men apply to law school, and since only one is admitted (and nothing would change even if none are), it makes merely a small fall in the percentage of men’s admission to the school of mathematics.
Conclusion: although both schools favoured women, since more women than men applied to the school of law, which has a low acceptance rate, when the percentages of acceptance are combined, men do better.
To tell the truth, Simpson’s Paradox tells us just something very simple about the laws of fractions. The following lines are no more than a fractional description of the story:
60/100>58/100
and also
40/100>1/3
but
(60+40)/(100+100)(58+1)/(100+3)
A wise man once said that statistics remind him of women in bikinis: the things revealed are nice and heart-warming, but the things that really matter are concealed.
We could conjure up plenty of examples in the same spirit. For instance, we might imagine two basketball players, Steph and Michael, and see that even though Steph’s scoring statistics are higher than Michael’s two years in a row (in percentage points of their shooting attempts), the combined two-year statistics show Michael as the better scorer. See the table below.
2000 2001 Combined
Steph 60 successful throws out of 100: 60% 40 successful throws out of 100: 40% 100 successful throws out of 200: 50%
Michael 58 successful throws out of 100: 58% 2 successful throws out of 10: 20% 60 successful throws out of 110: 54.5%
I’ve made this example very similar to the previous one in order to elucidate what’s really happening here. According to the table, Steph has better scoring percentages in 2000 and in 2001, but when the data is combined it turns out that Michael is the better scorer. The main reason for this highly surprising result is that in the bad season of 2001 Michael took fewer shots.
We could also think of two investment consultants where one was better than the other (his percentage of profitable portfolios is higher) for the first half-year and again shows better results relative to the number of portfolios in the second half-year; but the other had a higher percentage of profitable portfolios when considered annually.
When I first learned about this paradox, the example presented to me concerned two hospitals with the following data. It was known that men would rather check into Hospital A and avoid Hospital B, because A’s male mortality rates were lower than B’s, and that women admitted to Hospital A also tended to live longer than in hospital B, but when the data for both genders are combined, Hospital B rated better for mortality rates than hospital A. I encourage my intelligent readers to fill in the numbers in the table below and see how it works.
Men Women Combined
Hospital A
Hospital B
PER CENT PER SE
One of the problems of interpreting numerical analysis is the fact that we tend to think of percentages as absolutes. For example, we feel that 80 per cent is more than 1 per cent. Yet, if someone offered us a choice between 80 per cent of the shares of a tiny company or 1 per cent of a giant such as Microsoft, we would soon realize that percentage figures are not the same as dollars.
What do we have in mind when we say, ‘He missed a 100 per cent sure shot’?
What’s the meaning of: ‘This medication lowers the chances of 30 per cent of smokers suffering heart attacks by 17 per cent’?
Which is the better deal: items that sell at a 25 per cent discount, or a 50 per cent discount on the second item we buy? Why?
We must exercise caution when we make decisions based on percentages.
Percentage examples can take us as far as the stock exchange. When we hear that a certain share gained 10 per cent and then lost 10 per cent, we mustn’t believe that it ends up back where it was in the beginning. If our share was worth $100 and gained 10 per cent, it’s now worth $110. When it loses 10 per cent at this stage, the share is actually worth $11 less, which means that it’s now worth $99 (curiously, you get the same result if the share first loses 10 per cent, and then gains 10 per cent). That gap would be even more dramatic if the same profile happened with a 50 per cent rise and fall (150 and 75), and we’d reach a dramatic climax if we considered a 100 per cent rise followed by a 100 per cent fall. In this last scenario, after doubling its value, the share would simply become extinct.
Many people don’t understand that if their share gained 90 per cent and then lost 50 per cent, they’d actually lose money. Hard to believe? Let’s work it through. Suppose your share is worth $100 and then gains 90 per cent: it’s now worth $190, right? Next, it loses 50%, which means that it’s now worth only $95. When a financial manager brags about a share he recommended to you, saying that it gained 90 per cent and lost only 50 per cent, many people would think they made a 40 per cent profit that year. No one believes they could ever lose.
But if we lose our way around percentage points, just imagine what might happen when we move to the realm of probabilities (mainly of future events).
PROBABILITY, THE BIBLE, 9/11 AND FINGERPRINTS
A scholar once showed me this slick trick. The 50th letter of the Book of Genesis in Hebrew is T. Count another 50 letters and you arrive at an O. Another 50 letters lands us on R, and the 200th letter (another 50) is A. What does that spell? TORA, the Hebrew word for Pentateuch or The Teaching. Is that an accident or was it premeditated? Combing the Bible for all kinds of meaningfully coded intervals used to be a popular pastime, and several articles and books have been written on the topic. So does Scripture really contain secret messages like that? Ignoring the theological aspects of this theme, it’s primarily a statistical question that we could ask about other voluminous books, such as War and Peace. Do they contain such interesting combinations? Well, they probably do. Plenty of interesting arrangements can be found in Moby-Dick, Anna Karenina and many other big, big books. (Just think what we could find in In Search of Lost Time, the seven-volume novel by Marcel Proust.)
After the 9/11 terror attacks, many New Yorkers were startled by the coincidental ‘facts’ that emerged around the atrocity. For example, the flight number of the first plane that hit the World Trade Center was 11. The name New York City comprises 11 letters, as do Afghanistan and George W Bush. Also, 11 September is the 254th day of the year. What about it, you may ask? Well, 2+5+4 = 11! E
ven the shape of the Twin Towers reminds us of the number 11. Now that’s really scary!
Another interesting issue, tangentially related to this, is the art of solving crimes using fingerprints. I maintain that when courts prepare to convict people because fingerprints identical to theirs have been found at a crime scene, they should first consider the size of the population of the neighbourhood. To the best of my knowledge, fingerprint matches are never absolutely certain, but refer to a certain number of identical shapes. (As may be recalled, Benjamin Franklin said that only two things are certain: taxes and death. He didn’t mention fingerprints.) The odds of finding a wrong match are 1:100,000 or 1:200,000, depending on which book you read. Thus, when fingerprints are found at a crime scene in a township with a population of 200 and we have a suspect whose fingerprints match them, the chance that we have the perpetrator is quite high, because it would be very unlikely to find another resident in that town with similar fingerprints. Yet when this method is applied to a crime perpetrated in a city such as New York or London or Tokyo, it’s reasonable to assume that we could find a greater number of people there with similar fingerprint patterns.
ON AVERAGES AND MEDIANS
Although averages are mentioned very often in various everyday contexts, I feel that the ‘average’ is one of the most confusing issues in the world of statistics. For example, suppose we’re told that the average monthly salary in Happystan is $100,000. What does that mean? I asked several intelligent people and it turned out that many understand this to mean that some 50 per cent of Happystanians earn more than $100,000 per month while the other half earn less than that. This, of course, is a mistake. The datum that splits the population in two is known as the median, not average. As for the average figure presented above, it’s very likely that a select few people earn a whole lot more and everyone else – the majority – make less. For example, seven people work in a hypothetical bank branch. There are six who collect normal salaries, but the manager makes $7 million. That would make the average salary in that bank more than $1 million. After all, if we divide only the manager’s salary by seven, we’d have a million each, so the real average must be higher. In this example, only one person earns more than the average, while everyone else earns less – thus, many more than half the employees make less than the average salary. It’s a known fact that in many countries only some 30–40 per cent of workers earn more than the average salary.