Video transcript
Here's the second problemfrom CK12.org's AP statistics FlexBook. It's an open sourcetextbook, essentially. I'm using it essentially toget some practice on some statistics problems. So here, number 2. The grades on a statisticsmidterm for a high school are normally distributed with amean of 81 and a standard deviation of 6.3. All right. Calculate the z-scores for eachof the following exam grades. Draw and label a sketchfor each example. We can probably do it allon the same example. But the first thing we'dhave to do is just remember what is a z-score. What is a z-score? A z-score is literally justmeasuring how many standard deviations away from the mean? Just like that. So we literally just have tocalculate how many standard deviations each of these guysare 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 outhow far is 65 from the mean. Let me just draw one charthere 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 standarddeviation of 6.3. So our distribution, they'retelling us that it's normally distributed. So I can draw a nicebell curve here. They're saying it's normallydistributed, so that's as good of a bell curve asI'm capable of drawing. This is the meanright there at 81. And the standarddeviation is 6.3. So one standard deviation aboveand below is going to be 6.3 away from that mean. So if we go 6.3 in the positivedirection, that value right there is going to be 87.3. If we go 6.3 in thenegative direction, where does that get us? What, 74.7? Right, if we add 6, it'llget us to 80.7, and then 0.3 will get us to 81. So that's one standarddeviation below and above the mean, and then you'd addanother 6.3 to go 2 standard deviations, so on and so forth. So that's a drawing ofthe distribution itself. So let's figure outthe z-scores for each of these grades. 65 is how far? 65 is maybe going tobe here someplace. So we first want to say,well how far is it just from our mean? So the distance is, you justwant to positive number here. Well actually, you wanta negative number. Because you want your z-scoreto be positive or negative. Negative would mean to the leftof the mean and positive would mean to the right of the mean. So we say 65 minus 81. So that's literallyhow far away we are. But we want that in termsof standard deviations. So we divide that by the lengthor 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 beminus 16 over 6.3. We'll take our calculator out. And let's see, if we haveminus 16 divided by 6.3, you get minus 2 point--oh, it's like 54. Approximately equalto minus 2.54. That's the z-scorefor a grade of 65. Pretty straightforward. Let's do a couple more. Let's do all of them. 83. So how is it awayfrom the mean? Well, it's 83 minus 81. It's two grades above the mean. But we want it in termsof standard deviations. How many standard deviations. So this was part A. A was right here. We were 2.5 standarddeviations 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 isgoing to be right here. A little bit higher,but right here. And the z-score here, 83 minus81 divided by 6.3 will get us-- let's see, clearthe calculator. So we have 83 minus 81is 2 divided by 6.3. It's 0.32, roughly. So here we get 0.32. So 83 is 0.32 standarddeviations above the mean. And so it would be roughly 1/3third of the standard deviation along the way, right? Because this as one wholestandard deviation. So we're 0.3 of a standarddeviation above the mean. Choice number C. Or not choice, part C, Iguess 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 termsof standard deviations. So 12 is how many standarddeviations above the mean? Well, it's goingto 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 standarddeviations above the mean. So the mean is 81, we go onewhole 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 is1.9 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 needthe problem anymore. A score of 100. Well, same thing. We figure out how far is 100above the mean-- remember, the mean was 81-- and we dividethat by the length or the size or the magnitude of ourstandard deviation. So 100 minus 81 isequal to 19 over 6.3. So it's going to be a littleover 3 standard deviations. And in the next problem we'llsee what does that imply in terms of the probability ofthat actually occurring. But if we just want to figureout the z-score, 19 divided by 6.3 is equal to 3.01. So it's very close. 3.02, really, ifI were to round. So it's very close to 3.02. Its z-score is 3.02, or a gradeof 100 is 3.02 standard deviations above the mean. So remember, this was themean right here at 81. We go 1 standard deviationabove the mean, 2 standard deviations above the mean, thethird standard deviation above the mean is right there. So we're sitting rightthere on our chart. A little bit above that, 3.02standard deviations above the mean, that's wherea score of 100 will be. And you can see theprobability, the height of this-- that's what the charttells us-- it's actually a very low probability. Actually, not just a verylow probability of getting something higher than that. Because as we learned before,in a probably density function, if this is a continuous, not adiscreet, the probability of getting exactly that is 0,if this wasn't discrete. But since this is scoreson a test, we know that it's actually a discreteprobability function. But the probability is low ofgetting higher than that, because you can see wherewe sit on the bell curve. Well anyway, hopefully this atleast clarified how to solve for z-scores, which is prettystraightforward mathematically. And in the next video,we'll interpret z-scores and probabilitiesa little bit more.
