0:05

All right, well let's take another potential example,

Â how would we go about calculating this?

Â All right, we've got our same distribution, mean of 500 for demand,

Â standard deviation of 100.

Â We currently have enough seats to meet demand of 450.

Â If we think about adding another flight to this route,

Â now we're going to have a capacity for 600 people, all right?

Â Well how likely is it, that demand's going to fall between 450 and 600?

Â So how do we go about calculating this?

Â 0:51

So we've got our z-scores for the two,

Â we look up the probabilities associated with both of these values, and

Â we can get, what's the probability that demand falls in this range.

Â 1:02

Another way of doing this,

Â want to just jump into Excel, is to use this particular set of commands.

Â So here's, what's the probability that demand is less than 600.

Â So that's saying that demand is anywhere below 600, and

Â what I want to do is subtract from that, the chance that demand is below 450.

Â So, if I kind of take this piece out, this is the range that I want to focus on.

Â So, the probability of demand being below 600,

Â minus the chance of demand being below 450, that's going to tell me how

Â likely is it, that demand falls between those two values.

Â And it turns out, that it's about a 53% chance of happening.

Â Now, that's not telling us exactly what the level of demand is going to be.

Â 1:50

Might be 451, might be 559, 599, we don't know.

Â And this is not enough information for us to make the decision of,

Â should we add another flight to this route.

Â You know, think about what's going to go into that decision.

Â Well, what's it cost us to operate that flight?

Â How much revenue are we going to generate?

Â What's the cost associated with not offering enough seats,

Â versus offering too many seats?

Â You know, these are some of the issues that airlines, they have to grapple with,

Â when they're allocating their resources.

Â And that's what our next exercise is going to be based on,

Â is an over booking problem, another situation that airlines have to deal with.

Â 2:28

Another example that airlines might have to contend with,

Â would be a case where they say, I want to make sure I have enough seats for

Â passengers, a particular percentage of the time.

Â So, for example, in this case, what we're looking at is saying,

Â I'm only willing to run a 10% chance of not having enough seats.

Â Now that's something that the airlines might look at,

Â what about in the context of a retailer?

Â You know, a new product coming on the shelves, or it's the holiday season,

Â the hot product.

Â I want to make sure I have enough of my product available to meet demand 90%,

Â 95% of the time.

Â You know, there's a lot of costs involved, if I don't have that product in stock,

Â I potentially put a customer's business at risk.

Â So, how much do I have to plan to have available?

Â 3:33

Well, if I know that, if I know that I want to make sure that I've

Â got seats available at least to cover 90%.

Â That's where this norm inverse function's going to come into play.

Â So we know the probability, 0.9, we have our mean of 500,

Â our standard deviation of 100.

Â Tells me, I need to have seats available of at least 628.15.

Â Can't have 0.15 seats available, let's bounce it up, call it 629 seats available.

Â If I only want to have a 10% chance of not having enough seats.

Â 4:09

Right, so using characteristics of the normal distribution,

Â as well as using these distribution and inverse commands within Excel, we can

Â perform these calculations, to tell us how many seats we need to have available, how

Â likely is it that we can expect business to fall within these particular ranges?

Â 4:28

Again, we could look this up on our z tables,

Â to find out what's the appropriate value that will correspond to this.

Â In these case, we'd say, well let's find 90% in our z table, and

Â then reverse engineer what the appropriate number of seats would be.

Â 4:51

So, we've talked about being able to characterize the extent of uncertainty,

Â being able to characterize the variance, if you will, that we're observing.

Â The amount of dispersions, the range and frequency of possible outcomes.

Â The other piece that's remaining for

Â us is, well, how do we go about making predictions.

Â And if we're talking about sales forecasting,

Â there are a number of factors that we might take into account.

Â We might consider seasonality, we might consider competitive actions,

Â we might consider our marketing mix.

Â And all of those factors, they're going to be assumed to influence that best guess,

Â that prediction for what the level of demand is going to be.

Â Everything that we're talking about, in this case, the normal distribution,

Â is the variation, or the fluctuations, around that best guess.

Â 5:44

All right, so we've talked a fair amount already about the normal distribution.

Â And it's great if your data actually looks like a normal distribution,

Â it's an appropriate distribution to use.

Â We've got the commands built into Excel,

Â we can visualize it easily with a bell curve.

Â We have a good sense for one standard deviation,

Â two standard deviation, but, does your data actually look like a bell curve?

Â 6:11

That's not always going to be the case, so here are just some

Â examples of places where other distributions might be necessary.

Â If we think about the number of employment offers made,

Â versus the number of employment offers that are accepted.

Â If we think about people making brand choices.

Â These are choices, these are not following the normal distribution,

Â they're not continuous outcomes.

Â If we look at how technology or new products diffuse over time,

Â that diffusion curve does not look like a normal distribution.

Â It might start off slow and then speed up, until when it hits the mass market.

Â If we look at product failure times, or

Â if we look at customer lifetimes, these tend to be skewed.

Â You have a lot of customers who have short lifetimes, and very few customers,

Â potentially, who have long relationships with companies.

Â Well, if that's what the data looks like,

Â the normal distribution isn't going to be appropriate.

Â And, we can actually make very serious mistakes from a financial standpoint,

Â if we don't choose the appropriate distribution.

Â The normal distribution,

Â 99.7% of that data falls within three standard deviations.

Â Well, what if the tails are a lot fatter?

Â What if there's more of the mass out in those extreme observations?

Â Well, those extreme observations could be what we would consider

Â catastrophic losses and blockbuster successes.

Â Well, if we use a normal distribution,

Â we might under weigh the likelihood of those outcomes occurring.

Â 7:40

So just to give you some examples using different distributions,

Â suppose we're dealing with a company that's looking to hire new employees.

Â And they want to get between 5 and 7 new employees in a given year.

Â Well, how many offers do they have to make in order to do that?

Â So, let's say in the past, 80% of the people they made offers to,

Â accept those offers.

Â If I'm looking to net between 5 and 7 employees, all right,

Â well I've got to make at least five offers, that's the easy part.

Â But if I want to maximize my chances of falling in the range of 5 to 7,

Â what do I consider a success?

Â I would consider it successful if I got five, six, or seven new hires.

Â So I need to figure out, how many offers do I make,

Â in order to fall within that range?

Â 8:29

Well this is a case where the binomial distribution is going to be

Â appropriate for us.

Â The binomial distribution is used when we have, for a fixed number of trials or,

Â in this case, employment offers, how many successes do I observe, how many yeses?

Â How many people said yes to that employment offer?

Â Again, we have formulas that we can use if we want to calculate the mean,

Â the variants, and the standard deviation.

Â You'll notice these differ considerably,

Â 8:57

compared to what the equations look like for the normal distribution.

Â Again, it's a different distribution, it's a different

Â distribution that we could use to fit our data, so it really depends on the context,

Â in terms of, which distribution is going to be appropriate.

Â Now, if i'm dealing with very large numbers of trials,

Â the binomial distribution will actually end up looking like a normal distribution.

Â So for large values of n,

Â we can approximate the binomial distribution with a normal distribution.

Â But for small values of n, we want to be careful, and let's take a look to see

Â what this distribution's actually going to look like.

Â 9:34

All right, so within Excel, these commands are going to look pretty similar

Â to what we saw before, in terms of their syntax.

Â Instead of NORM.DIST, now we have BINOM.DIST.

Â So, if I want to know the probability of my outcome being below

Â a particular value k, that's where BINOM.DIST is going to be used.

Â I input n, the number of trials, I input p, the probability, and

Â i input that true statement, to find out how likely is it,

Â that the number of successes falls below the value k.

Â If I want to know, how likely is it that I get exactly k successes, that's where

Â changing that true statement to false is going to come into play for us.

Â And just like we had the inverse statement for the normal distribution,

Â we have the critical value statement for the binomial distribution.

Â So, what is that cut-off probability, if you will.

Â So in this case, if I know that I'm looking to fall above a particular

Â probability, a cut-off probability, I'm going to look for,

Â what's the smallest possible value to make that happen.

Â