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.