We have a handy rule, called the empirical rule, that allows us to calculate the probability of various events, given that the underlying data or distribution is approximately normally distributed. And so, I'm going to first of all state the rule, and then I'll show you a picture that captures graphically what the rule is showing you. So, here's what the empirical rule says. It says that if I draw an observation from a normal distribution, then there's a 68% chance that that observation is going to fall within one standard deviation of the mean. That's about a point two-thirds probability. So two-thirds of the data would lie within one standard deviation of the mean. If I go out two standard deviations from the center of the distribution, then there's a 95% chance of the data would fall within that range. So that's about 19 out of 20 of data points would lie within plus or minus two standard deviations of the mean. And finally going out to three standard deviations, well three standard deviations will capture essentially all of the data, to be more precise, about 99.7% of the data. And so you can see what the empirical rule is doing. It's taking the mean and the standard deviation of a normal distribution and telling you the probability that an observation lies within that particular range. So it's useful for doing what one might call back of the envelope calculations. So here is an illustration of the empirical rule. So what you're looking at now is a graph of a normal distribution. It is centered around its mean. Remember it's symmetric, mu is sitting there in the middle, and the graph has been labeled in units of standard deviation going out from -3 to +3 standard deviations from the mean. If you were to find the area under this graph between plus or minus one standard deviations, plus or minus sigma from the mean mu, you would find that area was approximately 68%. So that's what you can see in the middle. Then if you go add another standard deviation you'll capture 95% of the area under that graph. Another way of saying that is the probability that an observation falls in that range is 95% and if you go all the way out to three standard deviations, then you're going to capture 99.7% of the distribution. It's very likely an observation is going to lie within three standard deviations of the mean. So that's the illustration of the empirical rule. Now once you have that empirical rule, you're able to use it to calculate probabilities of various events or at least approximate those probabilities. So I want to finish off this class with an example of such a calculation, a place where a normal distribution can come in useful. And what I'm going to consider is finding the probability that a particular stock, and I'm going to consider the company Apple stock moves by certain amount. So let's assume, so here's a probability model that the return on Apple's stock is approximately normally distributed and that it has a mean mu of 0.13% and a standard deviation sigma of 2.34%. So those are sometimes termed the parameters of the normal distribution. Now that I've got the normal distribution defined, I can ask questions so here is an example question. What is the probability that tomorrow Apple's stock price increases by more than 2.47%. So I'm saying what's the probability that it has a return of more than 2.47% tomorrow. Of course, this is a future event. I don't know the answer with absolute certainty but I can put a probability around that with the aid of a probability model and my probability model here is the normal distribution with the parameters as specified. Now in practice, you can be sitting there saying well where on earth did those numbers come from, the 0.13% and the 2.34%? Well those could potentially come from looking at historical data. So I've got the question stated. What's the probability that Apple stock goes up by more than 2.47% tomorrow? Well the way that I approach that Is to look to see how many standard deviations 2.47 is away from the mean. So, that's the calculation that's happening under the term technique. What we do is count, we count how many standard deviations 2.47 is away from the mean. The mean being 0.13 and that counter is sometimes called a Z score. So I calculate the Z score for this particular instance, I get Z equal to 2.47 minus 0.13 divided by 2.34. So 2.47 is the value that I'm interested in. 0.13 is the mean. And 2.34 is the standard deviation. So, there's the Z score, what that tells me is that 2.47 is exactly one standard deviation above the mean, so I didn't choose this number to make the calculation convenient, to be honest. So 2.47 is exactly one standard deviation above the mean, and I can now use the empirical rule to say that the probability of an observation is one or more standard deviations above the mean is approximately 16%, or 0.16. And if you say, well where did that 16% come from I'll show you on the next slide. So, here's what I'm doing with my normal distribution. I have now centered it around 0.13 which is the mean. I have units of standard deviation on the horizontal axis, and the value of interest which is 2.47 is exactly one standard deviation of the mean. And I've just partitioned the probability based on the empirical rule into the various components. There's a 50% chance that you're less than 0.13. Remember there was a 68% chance you're within one standard deviation so therefore there's a 34% you're between zero and one standard deviation above the mean. That's the second area. And that means that the area to the right, because the areas have to add up to one must be 16%. So, that's an example of a calculation using the empirical rule. So, the empirical rule It's a very handy back of the envelope or back pocket calculation based on knowing a mean and a standard deviation and believing some approximate normality you can quickly get at some probabilities. Now if the event you're interested in doesn't fall exactly one standard deviation or two standard deviations or three standard deviations away from the mean, then what you would do in practice is use a calculator or a spreadsheet to calculate the exact probability. It corresponds to an area under a curve, so you can certainly do the calculations for other values. But the empirical rule, as I say, is typically used as a back of the envelope type calculation technique and we state it in terms of the whole number moves in standard deviation. With in one standard deviation 68% with in two 95 percent and with in three essentially your on 99.7%.