It's Labor Day. Cadence is closed in the US. Unfortunately, I'm in India and it's not a holiday here. It's CDNLive India later in the week. As is traditional, I will post about...whatever I feel like. Everyone seems to like these off-topic posts. About the only thing they have in common is that they are not about anything to do with semiconductors: visual illusions, the Kansas walkway collapse, how to tell fake Englishmen, surprising results about medical tests, and more. Today, let's look at some mathematical paradoxes and oddities. The Will Rogers Phenomenon I mentioned somewhere in a recent post about jobs that the median income in the US is $30K/year (and, by the way, the cutoff for being in the 1% is $250K, which I find surprisingly low). The median income in Mexico is $10K/year. But here's something that seems a little paradoxical. If a relatively high-earning Mexican making $15K/year (50% above the median income) moves to the US, then the median income in both countries falls. If our hypothetical Mexican makes $25K now he or she is in the US, then the median income in both countries still falls. The headlines seem bad: median income falls. But the only change is that the individual is making a lot more money (but is now counted in the US statistics and not the Mexican ones). Moral: be careful what you wish for. Why is this called the Will Rogers Phenomenon? Because of a remark attributed to him. He was a comedian from Oklahoma, who died in 1935, so this remark must have been pretty late in his life since the dustbowl is usually dated from 1930, the "dirty thirties": When the Okies left Oklahoma and moved to California, they raised the average intelligence level in both states. Interesting that in 1930 Californians were considered to be stupid. Legs I used the median rather than the mean above, since those are values that are easy to find. Medians are also less affected by outliers (if Bill Gates walks into a party, the mean net worth might go up to millions of dollars, but the median just ticks up a notch). Another fun paradox about means: Almost everyone has more than the average number of legs. I told this recently to a woman who has a degree in applied math and even she didn't get it. So I had to explain it. So just in case, here's the quick explanation. Nobody has 3 legs. Not many, but some, have 1 leg. So the average number of legs is something like 1.999999 and so almost everyone, at 2 legs, has more than the average. Moral: the mean does not have to be "somewhere near the middle of the population." Simpson's Paradox Here is another paradox, known either as Simpson's Paradox or as the Yule-Simpson Effect. It is easiest to just show an example. Here are the baseball batting averages of Derek Jeter and David Justice during the 1995 and 1996 seasons. 1995 1996 Combined Jeter 12/48 .25 183/582 .314 196/630 .310 Justice 104/411 .253 45/140 .321 149/551 .270 Here is the paradox: if you look at the numbers, Justice has a better average in both 1995 and 1996. But add the two years together, and Jeter has the better two-year batting average. A similar effect was noticed with a study of gender bias at UC Berkeley in 1973. The numbers showed that men were more likely to be admitted than women. But for all the big departments in the university, women were more likely to be admitted than men. The reason for the anomaly was that it seemed that most of the women applied to very competitive big departments, such as English and Psychology, where they lost out (to other women, mostly), whereas men applied to less competitive departments (like engineering, what today we call STEM but didn't in 1973) where they mostly got admitted, even though women who applied to do engineering were even more likely to get admitted than men who did (even in 1973, engineering departments were desperate to find the few women who were interested in engineering, and any woman with good enough math skills would be accepted). Here's another one (all these facts are true): smoking during pregnancy can cause low birth weight babies, and low birth weight babies have higher mortality than normal birth weight babies. But low birth weight babies born to smokers have lower mortality than overall low birth weight babies. The reason is that smoking is a relatively benign reason for a low birth weight baby, compared to really serious diseases. The main reason that low birth weight babies have a higher mortality rate is that some really serious problems cause both low birth weight and premature death. A smoking mother may cause a low birth weight, but it won't kill you. Moral: the moral is not that you should smoke during pregnancy! Non-transitive Dice It is possible to design 3 dice such that the orange one will, on average, roll a higher number than the yellow one. The yellow one will, on average, roll a higher number than the green one. And here's the weirdness. The green one will, on average, roll a higher number than the orange one. This means you can play against someone, let them choose the first dice, and no matter which one they pick, you can pick one that will (on average) beat it. We are just not used to non-transitive things like that. It seems that if the Giants normally beat the Dodgers, and the Dodgers normally beat the Rockies, then you wouldn't expect the Rockies to beat the Giants. But it's actually not that hard to set up 3 attributes of each team in such a way that each team beasts the other on two attributes, and not unreasonable that if a team beats another on two attributes and loses on just one, that it would win the game. At least in mathematics land. The dice in the image to the right has the property that I described above. Demonstrating this in a blog post doesn't really work, so here is a video with the marvelously named David Spiegelhalter, (which means "mirror holder") who has the even more marvelous title as the Professor of the Public Understanding of Risk at Cambridge University (the real one, not the upstart Cambridge where there are a couple of other minor universities). By the way, Dr Spiegelhalter is a regular guest on the BBC program/podcast More or Less which I highly recommend. https://youtu.be/zWUrwhaqq_c Tomorrow I hope you had a great Labor Day, and we'll be back to semiconductors tomorrow. In fact, tomorrow's post is about something that happened 20 years ago today, but since today is a holiday, I'm covering it a day late. It is both a personal story, and a Cadence story. Sign up for Sunday Brunch, the weekly Breakfast Bytes email.![]()
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