 Hey there, this is a video covering the standard normal distribution. The standard normal distribution is a normal probability distribution, remember that means bell shaped data, with a mean of zero mu equals zero and a standard deviation of one, sigma equals one. The total area under its density curve is equal to one because remember that area represents probability. So the x-axis is marked with z-scores, so all data values are converted to z-scores, that's what it means to standardize a distribution. That's why it's called the standard normal distribution. So the area under the curve will be one. What we're going to do is we'll be given certain z-scores and we'll have to calculate the area under the curve between two z-scores to the left of a z-score or to the right of a z-score. Not only that, but the second part of this video, we will talk about given the area under the curve, we will calculate what z-score does that area correspond to. So there's two different ways or two different things that we will approach here. First order of business, so notation. Remember p represents probability, then you have the parentheses. The probability that a z-score is between two values, a and b, that's the notation there. If you have probability, parentheses, z less than a, close parentheses, that's the probability that some data value, a z-score in this case, is less than some value, a. If you have the probability that some z-score is greater than a, that's our notation here, p open parentheses, z greater than a. So we'll be looking for between probabilities less than and greater than. So it actually doesn't matter whether you use equal to or not for standard normal distributions because when you're talking about continuous distributions, the probability of a single x value, the probability that you're equal to a single x value will always be zero and that's because remember the nature of continuous data. You can go to as many decimal places as you want. For instance, if you're looking at z-scores between negative one and one, there are so many z-scores and so many values between negative one and one or between any two data values that a single one data value has a probability of zero. That's because of the nature of continuous data. So we have two options to find the area under the curve or probabilities I should say. We can use technology, which we'll get to that, or like they did at old school style or how they did it back in the day before they had sophisticated technology, they used something called a normal probability table. So actually there's a bunch of calculus behind what is called this normal probability table. So fortunately not everyone is a big fan of calculus, so that's why they created the table that did all the hard work for you. So I'm going to introduce you to the table and then I'm going to make your life easier by introducing you to technology using Google Sheets to calculate the probabilities. Alright, so the normal probability table, which I'll show you and we'll work with in a minute, is designed and it's only good for the standard normal distribution. So that's z-score values. That's what standard means. It means z-scores. There's a mean of zero and a standard deviation of one. Positive z-scores are on one page and they're corresponding areas to the left of them, while positive z-scores are on another page with their corresponding areas to the left given. Each value in the body of the table is a cumulative area under the curve to the left of the specific z-score. So the table, our table, gives area to the left of a specific z-score value. Not area to the right, area to the left, to the left, to the left. So know the difference between z-scores and area. So z-score is the distance along the horizontal scale or the x-axis of the standard normal distribution. So below your bell curve, you'll have this axis, an x-axis, and your data values or your z-scores will be labeled below. And then when we're talking about area, we're talking about actual space underneath the curve, space or area of the region under the curve. So let's jump into an example so I can show you what in the world's going on here. So a bone mineral density test can be helpful in identifying the presence of osteoporosis. The result of the test is commonly measured as a z-score, which has a normal distribution with a mean of zero and a standard deviation of one. That's always the case when you're talking about z-scores, mean of zero, standard deviation one. A randomly selected adult undergoes a bone density test. Use the table to find the following. So we'll start with the probability that the result is a reading less than 1.27. So always have a picture in this module. You want to draw a picture for every single question. It's very visual in nature. So remember for a bell curve, the mean is zero and the standard deviation is one. So just label the bottom of your curve a little bit. The most important thing is to make sure you put that zero in there, that mean. So it wants to know what is the probability that a result is less than 1.27. Where does negative two down there? Where does 1.27 fall? Well, it's somewhere between one and two. So all that matters is that you label 1.27 and you want to find the probability that it's less than 1.27. So less than means you need to find the area under the curve to the left of 1.27. So mark that z-score and then shade the correct region. So now what is the probability that a test score, which is given as a z-score, is less than 1.27? Well we have to resort to our table here. So we need to find the area to the left of 1.27. Remember area to the left is exactly what the normal probability table tells you. So remember 1.27. So look in your table and I want to find the area to the left of 1.27. So on the very left hand side you see 0.001, 0.2, find 1.2, there it is. Then the second decimal place is seven. So along the very top find 0.07 and find out where these folks intersect. So find out where the row and the column intersect, not yet, 0.8980. The area to the left of 1.27 is 0.8980. So that's our answer, 0.8980. You can keep the zero there if you want, doesn't really matter. That is our answer. That's the area to the left, that's the probability that a result will be less than 1.27. Find the probability that the randomly selected person has a result above negative 1. So in your picture, in your bell curve, put zero in the middle, because this is a standard normal distribution, the mean mu is always zero, and then plot negative 1 somewhere to the left of zero. Your data value needs to be plotted to the left. I'm looking for the probability that a result, that a z score, is greater than or above negative 1. So is that the area to the right or to the left of negative 1? It's the area to the right. So shade it, mark your data value, shade the appropriate region, and we need to find the area. But I have a problem here, literally. My problem is, the table only tells me area to the left of a data value. Well, let's find the area to the left of negative 1, and then we'll discuss this a little more. So on your table, because we're dealing with the negatives of z score, let's look along the right-hand side, let's find negative 1. There is no second decimal place, so pick 0.00 along the top row, and look where they intersect, 0.1587, 0.1587. But how does that help me find the area to the right? Hmm, brain teaser. All right, so what has to happen here is, remember I told you that the entire area under the curve is 1? Well if I take 1, and I take away 0.1587, that's going to give me the area of my shaded region. So what's 1 minus 0.1587? Well, it's going to give you 0.8413. All right, so that's done. So we had to do a little bit of a trick there. We used the table to find area to the left, and area to the right was 1 minus that value. 1 minus it's opposite. The opposite or right is left. But then it gets even better when you have to do between probabilities. So they said find the probability of reading is between negative 1 and negative 2.5. So I'm going to plot negative 1, plot negative 2.5, and shade in between these two values. Oops. I want to find the area of the shaded region. So remember the table only tells us area to the left. The table only tells us area to the left. Well, let's talk about this. So the area to the left of negative 2.5, the area to the left of negative 2.5 is going to be, so you have negative 2.5, there is no second decimal place, you get 0.0062, 0.0062. What about to the area to the left of negative 1? Well, I'll find negative 1, there is no second decimal place to pick 0.00, and you get 0.1587, 0.1587. So the area of my green region of my entire area to the left of negative 1 is 0.1587. That's 0.1587. This might get a little confusing, because I'm finding the probability a data value is between negative 2.5 and negative 1, at least the greatest order. Well, I want to find the area between negative 2.5 and negative 1. Well, if you take the entire area to the left of negative 1, which is 0.1587, and you subtract the area to the left of negative 2.5, which is 0.0062, that's going to give me my area between negative 2.5 and negative 1. So I took my bigger region and subtracted my smaller region. Anyway, this is actually going to give me a probability of 0.1525. That's my area. That's my between area. So it gets more confusing depending on the type of probability you're doing. So honestly, a table is probably not always the best way to solve things. And it's not even the most accurate. It's how things used to be done. Some people still use it. Just depends on who you are. We need to introduce some technology. So I'm actually going to walk you through using Google Sheets in just a moment. The directions are on the screen, but I'm not going to read through them. I will show you. So let's do the same exact situation. And let's find the probability your result is less than 1.27. So on your bell curve, plot your mean of 0 and then label 1.27, and you're looking for less than. That's the area to the left. All right, so in Google Sheets, I'm going to require a few things that I must input. I'm going to have to type in my mean mu, 0. I'm going to have to type in my standard deviation, sigma. And then I'm going to have to type in a lower bound and an upper bound. And what this means is where is your region shaded from what to what? Well, my region is shaded all the way through 1.27. And technically, the lower bound is negative infinity, a really big negative number. So since we can't type negative infinity, we'll write negative 999999, a negative sign followed by 6 9s. That's all you need to put into Google Sheets. So let's figure this out. So I'm going to go to my Google Sheets document. You're going to go to the Compute tab. And we're going to be focused on the normal region. So your job is to type in mu, 0, sigma, 1, lower bound, negative 9999999. And my upper bound would be 1.27. And look at that nice probability, round to four decimal places, 0.898. 0.898. So 0.898. So without a table, we were able to calculate that probability. But wait, there's more. Let's now do the probability result is above negative 1. So shade your curve just like we did previously. I'm looking for the probability that some data value is above negative 1. Google Sheets. Your mean is 0. Your standard deviation is 1. Always for z scores. Always for standardized distributions. And your lower bound, your shading starts at negative 1. That's why it's really important for you to draw the picture. In your upper bound, you go all the way to basically positive infinity. So let's use 6 nines. Type these four values into your Google Sheets spreadsheet. When you do type them into your Google Sheets spreadsheet, you actually get 0.8413. Let's do that between probability now, between negative 1 and negative 2.5. So we're calculating the probability that some z score is between negative 1 and negative 2.5. We can use Google Sheets. Once again, mu is 0, sigma is 1, lower bound, upper bound. Your shading starts at negative 2.5 and stops at negative 1. Once again, the importance of having your picture. Type the four values into your Google Sheets document. Remember we're using the compute tab. The compute tab is your friend. All right, so mu and sigma are already good to go. For your lower bound, you're going to type in negative 2.5 and upper bound negative 1. And just like that, without even the assistance of the table, we get 0.1524. That's the four decimal places. 0.1524. So 0.1524. Ta-da, much easier than that nonsense we did with the table. We love technology. Well, most of the time. So let's work out one more example. Assume that thermometer readings are normally distributed with a mean of 0 degrees Celsius and a standard deviation of 1 degree Celsius. Find the probability of reading is between negative 1.26 and 1.93. So area between these two values. So I'm going to mark my 0 because that's my mean. Label 1.96, label 1.93, and you're literally shading. You're finding the area between these two values. Remember, always draw the picture. It helps. Even if you don't think it does, it helps. So I'm going to find the probability of data values between negative 1.26 and 1.93. You're already strategizing what you're going to put into your Google Sheets document, probably. You're going to put mu is 0. You're going to put sigma is 1. You're going to put a lower bound of what? Your shading starts at negative 1.26, and it stops at 1.93. Try typing these four values into Google Sheets. Remember, compute tab, normal region. 0 and 1's already in there. So my lower bound is negative 1.26, upper bound 1.93. And look at that probability right there, staring right at us, 0.8694. That's 0.8694. So 0.8694. That's the answer. That's how you do it right there.