 Alright, we'll take a look at one of the most important results in probability and statistics. This is something known as Chebyshev's theorem. Now, just as a caution, Chebyshev has a number of different spellings. It's mainly because it's a Russian name that's been transliterated into the Latin alphabet. And so, one common spelling, if you decide to look up more about this guy, you can spell it starting with a T, S, C, H, E, B, Chebyshev. Actually, some of the older spellings have a V at the end instead of an F, and some of them even start with an S, H, Chebyshev. And the most common modern spelling seems to be to use the spelling C, H, E, B, Y, C, H, E, V. But note that there are about four or five different variant spellings, depending on who's doing the writing. And Chebyshev's theorem says, given any probability distribution whatsoever, regardless of what the mean and standard deviation is, take any value of K greater than or equal to 1. You can take K less than 1, but it doesn't give you anything useful. The probability that the random variable is within K standard deviations of the mean is at least 1 minus 1 over K squared. Now, that's quite a mouthful to process, so let's see if we can do an example here to clarify what we're talking about. Pick a value, take any K greater than or equal to 1. Well, how about K equals 3? So then, I have my random variable X with some probability distribution. I don't know, I don't care, it doesn't make a difference, because Chebyshev's applies to any probability distribution. The probability that my random variable is within K equals 3 standard deviations of the mean is at least 1 minus 1 over 3 squared. That's 8 out of 9. Well, again, the example doesn't necessarily help us too much, so let's try and apply it to an actual situation. So imagine that I have some sort of test, and the test has a mean of 70 and a standard deviation of 5. If we calculate or find those using whatever means we have, and what can we say about the probability that a given test will have a score between, say, 60 and 80? Well, the first thing we note is that if I take K equals 2, Chebyshev's will tell us the probability that a score is within 2 standard deviations of the mean, that's 70 minus that's 60, and 2 standard deviations above the mean, that's 80. And Chebyshev's will tell me that the probability that I land in this interval is at least 1 minus 1 over 2 squared. That's 1 minus 1 quarter, 3 quarters, or 75%. And one of the things that this does is it allows us to make very, very general comparison with very little in the way of required foreknowledge. So, again, let's take our test of the mean of 70 standard deviation of 5, and suppose I find somebody who scores above a 90, or maybe somebody who scores below a 55. What can we say about such people? Beyond the obvious statements like, this person did really well and this person didn't do so well. And so what I might begin with is I might note that this score 90 with the mean 70 standard deviation of 5, this 90 is 4 standard deviations above the mean. Now, Chebyshev's tells us if we have K equals 4, then the probability that a score is within 4 standard deviations of the mean, 1 minus 1 over 4 squared works out to be 15 out of 16. So that tells us that at least 15 out of 16, the probability of a score being within 4 standard deviations of the mean is pretty high. So the probability of being outside that range is at most 116. Now, these are probabilities, so it's helpful to think about them from a frequentist viewpoint. Now, what this means is that a score of 90 or higher is very unusual. 15 out of 16, about 94% of the scores are within 4 standard deviations of the mean, which means they're going to be less than this score of 90. So just from the mean, standard deviation, an actual score, plus Chebyshev's, we have a remarkable conclusion. Most of the people who took the test, at least 94% of the people who take that test are going to score below this 90 value. So this person who scores a 90 is in a group that constitutes at most 6% of the test takers. Well, I can do the same sort of analysis for the person who scored 55. So in this case, we'll note that 55 is 3 standard deviations below the mean. So Chebyshev's tells me that the probability that I'm within 3 standard deviations of the mean, 1 minus 1 over 3 squared, that works out to be 8 9ths. And so the probability of being more than 3 standard deviations below the mean is at most 1 9th. If 8 9ths of the scores at least are within 3 standard deviations, then at most 1 9th is going to be outside of that 3 standard deviation interval. And the bad news for the person who scored the 55, most of those scores are actually above theirs. So again, if we look at the problem from a frequentist's viewpoint, this means that most of those who took the test 8 9ths, because they're within 3 standard deviations of the mean, they're going to be higher than the score of 55. So about 89% at least of those who took the test are going to have done better than this person who scored a 55. And again, we can consider this as another case again, keeping our test of the same mean, same standard deviation. What if I get somebody who scores higher than 80? So I'll note that 80 is 2 standard deviations above the mean, and by Chebyshev's probability of being within 2 standard deviations of the mean is going to be 1 minus 1 over 2 squared, about 75%. So again, taking our frequentist's viewpoint, at least 75% of the test scores will be within 2 standard deviations of the mean. So the probability that somebody is going to have a score higher than 80 is going to be at most 25%. Now, the thing to remember about Chebyshev's is that it applies to any probability distribution whatsoever. The only thing we know, the only thing we need to know is the mean and the standard deviation. If we know nothing else, we can still apply Chebyshev's. Now, a little bit later on, we'll start to look at more specific distributions, but Chebyshev's is a good fallback position. If you know nothing else, you can apply Chebyshev's. If you do know something else, you probably want to apply that because it'll give you more precise results.