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Hello, and welcome to a video on probability in python. We'll be working with the scipy.stats.norm.cdf

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function. We won't need to load in any data today. So let's start by loading in our library. So from scipy.stats,

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we'll import norm, okay? Now, we're going to be working on a very special

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kind of problem that we'll have today called finding the area under the curve. And what that means is

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we'll have a distribution. And we're looking for the area under the curve of that distribution. Here's what that

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type of problem looks like. In a normally distributed distribution with a mean of eight and a standard deviation

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of 12. What is the probability that a randomly selected score will fall below 92?

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Let's take a look at a visualization of that. You can see here

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we have a normal distribution. And this is where our 92 is.

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We have our mean of 80 and

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our standard deviation of 12. So right here would be the first standard deviation.

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where it is on the other side. So we know that 92 is one standard deviation

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And what we're looking for is all of this blue shaded area here. We

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want to know how much area this contains.

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So to do that, we're going to use norm.cdf. And it will ask us for a couple

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of arguments. We're going to tell it that we have a score

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of 92 that we're looking for. A mean of 80. And a standard deviation of 12.

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And these are called loc and scale. You don't actually

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need to include these argument labels. This line works just as well. If you just have 92 comma 80 comma

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12. All right. Let's go ahead and run this and see what our answer is.

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Okay. So this says that the probability of selecting a value below 92 is 0.8413

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This looks about right. If we go back to our image, we

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can see that this could be about 84% of our distribution. We know that we have most

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of our distribution, most of the percent of our distribution in the center here.

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So we can say that that visually checks out. So what if we want to find a value between two

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values? Here's another type of problem you might run nto. What is the probability in a normally distributed

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distribution of selecting an X value between 54 and 78 with a mean of 70 and

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a standard deviation of 8? Let's go back to our visualization this

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time. We're looking for this shaded area here.

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How much of our distribution is between these two X values. To do

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that we will subtract the smaller X value from the larger X value.

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You always want to do it in that order, because we can't have negative values in probability. You can only have

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them with a domain of zero and one or between zero and one.

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So we're going to find probability between norm.cdf. And we're taking the

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larger of our two X values of 78 with a mean of 70 and standard deviation of 8 minus

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an X value of 54 with a mean of 70 and a standard deviation of 8.

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to guess what it's going for just from looking at the picture. And if

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you get something close to 82% then you are pretty correct.

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We can take another look at this again and kind of mentally verify, because we know that not very much of our

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distribution is in these two tail ends here. Most of it falls between the center. So again, we've got about 82%

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of our distribution between two values.

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So we've got another type of problem you might run into. What if we want to find the probability of randomly selecting a value

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above X? Let's use that first problem as an example. What if we wanted to find the probability of selecting

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a value above 92 with a mean of 80 and standard deviation of 12?

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Let's go back to our visualization. This time we have that smaller part of the distribution that we're

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looking for. So we know that norm.cdf will find anything below

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So how would we tell it we want the opposite of that. And the answer is

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that we subtract from one. So, prob_above equals one minus,

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norm.cdf, 92, 80, 12.

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And this will essentially cut away the part of

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the distribution that we were looking at before. So we're saying one because probabilities only exist between zero

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and one. One minus, this value here leaves us with

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this red shaded area. Okay,

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and that comes out to .1587, roughly. Which makes sense because

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we got 0.8413 up here. And those two numbers together will equal

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about one. So what if we'd like to calculate a percentile?

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Let's say a clinic wants to identify patients who score low on a test. So the patients can be offered a new therapy.

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The scores are normally distributed with a mean of 80 at a standard deviation 12. The clinic decides

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on the lowest 40 percent of scores. What is the score that marks the 40 percentile? Now to do

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this, you just use norm.ppf, give it the percentile,

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give it the mean, give it your standard deviation. And that's it. It will find you the percentile, that information.

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And here we have that our 40th percentile, the raw score is 76.95 now.

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Okay, that's it for this lesson on finding the area under the curve, have fun coding.