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Normal distribution problem: z-scores (from ck12.org)

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Here's the second problem from CK12.org's AP statistics FlexBook. It's an open source textbook, essentially. I'm using it essentially to get some practice on some statistics problems. So here, number 2. The grades on a statistics midterm for a high school are normally distributed with a mean of 81 and a standard deviation of 6.3. All right. Calculate the z-scores for each of the following exam grades. Draw and label a sketch for each example. We can probably do it all on the same example. But the first thing we'd have to do is just remember what is a z-score. What is a z-score? A z-score is literally just measuring how many standard deviations away from the mean? Just like that. So we literally just have to calculate how many standard deviations each of these guys are from the mean, and that's their z-scores. So let me do part a. So we have 65. So first we can just figure out how far is 65 from the mean. Let me just draw one chart here that we can use the entire time. So it's just our distribution. Let's see. We have a mean of 81. That's our mean. And then a standard deviation of 6.3. So our distribution, they're telling us that it's normally distributed. So I can draw a nice bell curve here. They're saying it's normally distributed, so that's as good of a bell curve as I'm capable of drawing. This is the mean right there at 81. And the standard deviation is 6.3. So one standard deviation above and below is going to be 6.3 away from that mean. So if we go 6.3 in the positive direction, that value right there is going to be 87.3. If we go 6.3 in the negative direction, where does that get us? What, 74.7? Right, if we add 6, it'll get us to 80.7, and then 0.3 will get us to 81. So that's one standard deviation below and above the mean, and then you'd add another 6.3 to go 2 standard deviations, so on and so forth. So that's a drawing of the distribution itself. So let's figure out the z-scores for each of these grades. 65 is how far? 65 is maybe going to be here someplace. So we first want to say, well how far is it just from our mean? So the distance is, you just want to positive number here. Well actually, you want a negative number. Because you want your z-score to be positive or negative. Negative would mean to the left of the mean and positive would mean to the right of the mean. So we say 65 minus 81. So that's literally how far away we are. But we want that in terms of standard deviations. So we divide that by the length or the magnitude of our standard deviation. So 65 minus 81. Let's see, 81 minus 65 is what? It is 5 plus 11. It's 16. So this is going to be minus 16 over 6.3. We'll take our calculator out. And let's see, if we have minus 16 divided by 6.3, you get minus 2 point-- oh, it's like 54. Approximately equal to minus 2.54. That's the z-score for a grade of 65. Pretty straightforward. Let's do a couple more. Let's do all of them. 83. So how is it away from the mean? Well, it's 83 minus 81. It's two grades above the mean. But we want it in terms of standard deviations. How many standard deviations. So this was part A. A was right here. We were 2.5 standard deviations below the mean. So this is part A. 1, 2, and then 0.5. So this was A right there, 65. And then part B, 83, 83 is going to be right here. A little bit higher, but right here. And the z-score here, 83 minus 81 divided by 6.3 will get us-- let's see, clear the calculator. So we have 83 minus 81 is 2 divided by 6.3. It's 0.32, roughly. So here we get 0.32. So 83 is 0.32 standard deviations above the mean. And so it would be roughly 1/3 third of the standard deviation along the way, right? Because this as one whole standard deviation. So we're 0.3 of a standard deviation above the mean. Choice number C. Or not choice, part C, I guess I should call it. 93. Well, we do the same exercise. 93 is how much above the mean? Well, it's 93 minus 81 is 12. But we want it in terms of standard deviations. So 12 is how many standard deviations above the mean? Well, it's going to be almost 2. Let's take the calculator out. So we get 12 divided by 6.3. It's 1.9 standard deviations. Its z-score is 1.9. Which means it's 1.9 standard deviations above the mean. So the mean is 81, we go one whole standard deviation, and then 0.9 standard deviations, and that's where a score of 93 would lie, right there. Its z-score is 1.9. And all that means is 1.9 standard deviations above the mean. Let's do the last one. I'll do it in magenta. D, part D. A score of 100. We don't even need the problem anymore. A score of 100. Well, same thing. We figure out how far is 100 above the mean-- remember, the mean was 81-- and we divide that by the length or the size or the magnitude of our standard deviation. So 100 minus 81 is equal to 19 over 6.3. So it's going to be a little over 3 standard deviations. And in the next problem we'll see what does that imply in terms of the probability of that actually occurring. But if we just want to figure out the z-score, 19 divided by 6.3 is equal to 3.01. So it's very close. 3.02, really, if I were to round. So it's very close to 3.02. Its z-score is 3.02, or a grade of 100 is 3.02 standard deviations above the mean. So remember, this was the mean right here at 81. We go 1 standard deviation above the mean, 2 standard deviations above the mean, the third standard deviation above the mean is right there. So we're sitting right there on our chart. A little bit above that, 3.02 standard deviations above the mean, that's where a score of 100 will be. And you can see the probability, the height of this-- that's what the chart tells us-- it's actually a very low probability. Actually, not just a very low probability of getting something higher than that. Because as we learned before, in a probably density function, if this is a continuous, not a discreet, the probability of getting exactly that is 0, if this wasn't discrete. But since this is scores on a test, we know that it's actually a discrete probability function. But the probability is low of getting higher than that, because you can see where we sit on the bell curve. Well anyway, hopefully this at least clarified how to solve for z-scores, which is pretty straightforward mathematically. And in the next video, we'll interpret z-scores and probabilities a little bit more.