Welcome back to An Intuitive Introduction to Probability, Decision Making in an Uncertain World. In the last lecture, we learned about the concept of an expected value. We introduced the expected value as a summary measure, because the graphical representation, or the table representations of probability distributions can quickly get overwhelming for our limited brain, our limited human mind. And so we like to aggregate all of this information, the possible variables, the possible probabilities into a single number. But there's a drawback. In the aggregation, we lose a ton of information. The number, 3.5 on the fair die, doesn't give me any indication about the variation in the actual numbers. I can't communicate with a 3.5, there's actually variation, the 3.5 doesn't actually show up. So here, as I show in this graph, in the probability distribution, I have these six bars, I see I have six different numbers. There's uncertainty, there's only a 1 in 6 chance of them coming, but the 3.5 doesn't give me any of that information. So I lost a lot of valuable information. Now look at these four graphs. I created here four very different probability distributions that all have one thing in common, the expected value is always 3.5. In the upper left-hand corner is our fair die. In the upper right-hand corner, I have the probability distribution for a die with three 1s and three 6s. But I can only roll 1s and 6s, but the long time average is 3.5. So, bottom left-hand corner, I have three 3s and three 4s. So, I have 50-50 between a 3 and a 4. Again, the average is 3.5, but all the numbers I see are just 3s and 4s. Much less variation, they're very close to the 3.5, as opposed to the 1 and 6 in the upper right-hand corner. And in the bottom right-hand corner, I have a really funny die. It has the possibilities of -5, -2, 9, and 12. All of a sudden, I may have negative numbers or double digit numbers. Again, the average of 3.5 but I have way more dispersion, way more variation, and that's a big minus. In the 3.5 or in general, the mean, the average, the expected value does not give us any information about variation, but in many business applications, that variation is important. Sam Savage of Stanford University coins the beautiful phrase, the flaw of averages. It's a game on words, not the law, the flaw of averages, that people rely too much on averages and ignore variation. And here I also have a beautiful quote by the late historian and evolutionary biologist Stephen Gould, who says, our culture encodes a strong bias either to neglect or ignore variation, and that's dangerous and can lead to trouble. So this problem, this expected value now motivates us to go a step further. We need a measure for variation. So let's think about this, how could we define the measure of variation? Here's the easiest thing you could think of. Let's take the average deviation, the probability weight sum of the possible deviations. So take the difference, x1-mu, that's the deviation, times its probability. And then take x-2 times its probability and so on. And add up all those numbers. Wouldn't that be a beautiful measure? It sure would. But it doesn't work. I have bad news. And this is in fact, I made it a theorem here. I represent it as a theorem. This concept, this idea of an average deviation does not work because it's always 0. This is actually a way to think of mu. Mu is exactly at the position that the negative deviations and the positive deviations add up, average out to 0. So the simplest idea does not work. Why doesn't it work? If you take a look at this spreadsheet with a fair die, it doesn't work because the minus and plus deviations cancel. So at this point my students always say, I have an easy solution, cut off the minus sign, use the absolute value and indeed, this is something that's possible. It's called MAD, M-A-D, the mean absolute deviation. So here we average out the absolute differences. This is possible but it's rarely done in practice. And the reason is, the absolute value function isn't as easy as it looks. It has a kink at 0. Here I have a graph of the absolute value function for you. And in the language of mathematics, it has a non-differentiability at 0 which later on creates some problems. So while this concept may be intuitive, it's not as easy as it looks like. And therefore, let's not use it. What's the next easiest concept to get rid of a minus sign? Squaring the numbers. And that's what we do. And this leads us to the concept of a variance. The variance is a probability-weighted sum of the squared deviations. So now a deviation of -2 gives us a +4 squared deviation. Just like a deviation of a +2 gives us a +4 squared deviation. So no more cancelling of negatives and positives. Here is a proper definition. If our random variable has possible values x1, x2, all the way to xk, it's a probability-weighted average of the squared deviation. We write Var for variance of X. Sometimes you will also see the notation sigma squared. In the spreadsheet for a fair die, I calculate the variance of a fair die for you, it's 2.91667. Now, however, it's necessary for us to go a step further. The variance has a big disadvantage. We squared the numbers. And then we get squared units in the applications. In our business or real life applications of probability, our random variables often have units. Remember back to the example I showed you with the euros in the insurance example. There, squared deviations, then, would be euros squared. But what the heck are euros squared? I don't know what euros squared are. So we want to go back from euros squared to euros. How do you do this? With the square root, bingo! And that's the last concept we need. We take the square root of the variance to obtain the concept of a standard deviation. That's why we have the sigma. Sigma is a standard deviation, the Greek letter s for standard deviation. The proper definition is, the standard deviation is a measure, based on the variance, for the average deviations of the values of a random variable around its mean. Here, for a fair die, you can quickly calculate the number of 1.7. To sum up this lecture, we learned about the flaw of averages. Don't just look at means and averages. There's too much information loss. We need to also measure the variation. And our preferred summary measures for variation, or in economics and finance people like to talk about volatility, are the concepts variances and standard deviation. I encourage you to play around with the spreadsheets so you get some feeling for how to calculate these numbers so we can use them in some future lectures. Thank you very much and please come back for more on an intuitive introduction of probability. Thank you.