0:05

Welcome back to An Intuitive Introduction to Probability,

Â Decision Making in an Uncertain World.

Â In the last lecture, we learned about the concept of an expected value.

Â We introduced the expected value as a summary measure,

Â because the graphical representation, or the table representations of

Â probability distributions can quickly get overwhelming for

Â our limited brain, our limited human mind.

Â And so we like to aggregate all of this information, the possible variables,

Â the possible probabilities into a single number.

Â 0:39

But there's a drawback.

Â In the aggregation, we lose a ton of information.

Â The number, 3.5 on the fair die, doesn't give me any

Â indication about the variation in the actual numbers.

Â I can't communicate with a 3.5, there's actually variation,

Â the 3.5 doesn't actually show up.

Â 1:04

So here, as I show in this graph, in the probability distribution,

Â I have these six bars, I see I have six different numbers.

Â There's uncertainty, there's only a 1 in 6 chance of them coming, but

Â the 3.5 doesn't give me any of that information.

Â 1:20

So I lost a lot of valuable information.

Â Now look at these four graphs.

Â I created here four very different probability distributions that all

Â have one thing in common, the expected value is always 3.5.

Â In the upper left-hand corner is our fair die.

Â In the upper right-hand corner, I have the probability distribution for

Â a die with three 1s and three 6s.

Â But I can only roll 1s and 6s, but the long time average is 3.5.

Â So, bottom left-hand corner, I have three 3s and three 4s.

Â So, I have 50-50 between a 3 and a 4.

Â Again, the average is 3.5, but all the numbers I see are just 3s and 4s.

Â Much less variation, they're very close to the 3.5, as opposed to the 1 and

Â 6 in the upper right-hand corner.

Â And in the bottom right-hand corner, I have a really funny die.

Â It has the possibilities of -5, -2, 9, and 12.

Â All of a sudden, I may have negative numbers or double digit numbers.

Â Again, the average of 3.5 but I have way more dispersion,

Â way more variation, and that's a big minus.

Â In the 3.5 or in general, the mean, the average,

Â the expected value does not give us any information about variation,

Â but in many business applications, that variation is important.

Â Sam Savage of Stanford University coins the beautiful phrase,

Â the flaw of averages.

Â It's a game on words, not the law, the flaw of averages,

Â that people rely too much on averages and ignore variation.

Â And here I also have a beautiful quote by the late historian and

Â evolutionary biologist Stephen Gould, who says, our culture encodes a strong bias

Â either to neglect or ignore variation, and that's dangerous and can lead to trouble.

Â 3:35

So let's think about this, how could we define the measure of variation?

Â Here's the easiest thing you could think of.

Â Let's take the average deviation,

Â the probability weight sum of the possible deviations.

Â So take the difference, x1-mu, that's the deviation, times its probability.

Â And then take x-2 times its probability and so on.

Â And add up all those numbers.

Â Wouldn't that be a beautiful measure?

Â It sure would.

Â But it doesn't work.

Â I have bad news.

Â And this is in fact, I made it a theorem here.

Â I represent it as a theorem.

Â This concept,

Â this idea of an average deviation does not work because it's always 0.

Â This is actually a way to think of mu.

Â Mu is exactly at the position that the negative deviations and

Â the positive deviations add up, average out to 0.

Â So the simplest idea does not work.

Â Why doesn't it work?

Â If you take a look at this spreadsheet with a fair die,

Â it doesn't work because the minus and plus deviations cancel.

Â 4:44

So at this point my students always say, I have an easy solution, cut off the minus

Â sign, use the absolute value and indeed, this is something that's possible.

Â It's called MAD, M-A-D, the mean absolute deviation.

Â So here we average out the absolute differences.

Â This is possible but it's rarely done in practice.

Â And the reason is, the absolute value function isn't as easy as it looks.

Â It has a kink at 0.

Â Here I have a graph of the absolute value function for you.

Â 5:17

And in the language of mathematics,

Â it has a non-differentiability at 0 which later on creates some problems.

Â So while this concept may be intuitive, it's not as easy as it looks like.

Â And therefore, let's not use it.

Â 6:06

If our random variable has possible values x1, x2, all the way to xk,

Â it's a probability-weighted average of the squared deviation.

Â We write Var for variance of X.

Â Sometimes you will also see the notation sigma squared.

Â In the spreadsheet for a fair die,

Â I calculate the variance of a fair die for you, it's 2.91667.

Â Now, however, it's necessary for us to go a step further.

Â 6:40

The variance has a big disadvantage.

Â We squared the numbers.

Â And then we get squared units in the applications.

Â In our business or real life applications of probability,

Â our random variables often have units.

Â Remember back to the example I showed you with the euros in the insurance example.

Â There, squared deviations, then, would be euros squared.

Â But what the heck are euros squared?

Â I don't know what euros squared are.

Â So we want to go back from euros squared to euros.

Â How do you do this?

Â With the square root, bingo!

Â And that's the last concept we need.

Â We take the square root of the variance to obtain the concept of a standard

Â deviation.

Â That's why we have the sigma.

Â Sigma is a standard deviation, the Greek letter s for standard deviation.

Â 7:31

The proper definition is, the standard deviation is a measure,

Â based on the variance, for

Â the average deviations of the values of a random variable around its mean.

Â Here, for a fair die, you can quickly calculate the number of 1.7.

Â To sum up this lecture, we learned about the flaw of averages.

Â Don't just look at means and averages.

Â There's too much information loss.

Â We need to also measure the variation.

Â 8:06

And our preferred summary measures for variation, or

Â in economics and finance people like to talk about volatility,

Â are the concepts variances and standard deviation.

Â I encourage you to play around with the spreadsheets so you get some feeling for

Â how to calculate these numbers so we can use them in some future lectures.

Â Thank you very much and please come back for

Â more on an intuitive introduction of probability.

Â Thank you.

Â