 The Z-score is fundamental to our ability to compare items from different sets of data. So when I was younger I had the choice to make whether to go into comedy or to go into mathematics. Then one day I was walking down the street and I saw a man playing checkers with a dog. And I ran over there and the man made a move and surprisingly the dog made a perfectly good counter move. And after they were playing for a few minutes I said to the man, wow that's amazing your dog can play checkers. And the man said, nah he's not so good he only wins one game out of five. And now you know why I became a mathematician. Still the lead up to not become a comedian does leave me with a number of stories like this which illustrate an important mathematical point. To properly evaluate a quantity it's important to consider how extreme it is relative to the population it belongs to. And so a score of 80 on an exam looks better than a score of 60. Unless the mean on the first exam is a 90 and the mean on the second exam is a 40. But the mean is not enough. Even if the scores and means are the same one might be more unusual than the other. So for example consider two classes. Class A has scores of 69, 69, 69, 70 and 73. And if you calculate the mean score in this class you find that the mean is 70. Or we might take a different class, class B with scores of 45, 73, 74, 76, 82. And if you calculate the mean for this class you'll find that the mean is 70. Now the classes have the same mean and in both classes there is a student who got a 73. But a 73 in the first class is the highest score in the class. But a 73 in the second class is the second lowest score in the class. Now Chebyshev's theorem guarantees that most scores fall within a few standard deviations of the mean. And so this allowed us to compare different scores in different classes. So back in this problem we were able to determine that the first student did better than at least 75% of the students in their class while the second student did better than at least 96% of the students in their class. What this suggests is using the standard deviation as a measure of how far a score is from the mean. And this leads to the notion of a z score. Let x be a data value from a set with a mean of mu and a standard deviation of sigma. Where if you haven't seen these letters before this is the Greek letter mu which is their equivalent of an m and this is the Greek letter sigma which is their equivalent of an s. I'm not entirely sure why you'd want to use m to represent the mean and s to represent the standard deviation but there it is. In any case once we have these values we can calculate the z score x minus the mean over the standard deviation. And one way to look at this is the numerator x minus the mean tells you how far away from the mean the value is when we divide by the standard deviation we get that distance in terms of the standard deviation. For example suppose a population has a mean of 15 and a standard deviation of 3 find the z score for a value of 11 and for a value of 20 and interpret the significance. Again definitions are the whole of mathematics all else is commentary so let's bring in that definition. The z score for a value of 11 is 11 minus the mean over the standard deviation and we'll calculate that value as minus 4 thirds. The z score for a value of 20 is going to be 20 minus the mean over the standard deviation which is 5 thirds. And now we can interpret these values. The first z score minus 4 thirds tells us that the first value is 4 thirds of a standard deviation below the mean. The second z score 5 thirds tells us that the second value is 5 thirds standard deviations above the mean.