Here’s the second problem from ck12.org, AP Statistics flex book. It’s an open source. A textbook, essentially, on… I’m using it essentially to get some practice on some statistics problem.
So here, number two, the grades on a statistics midterm for our 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 could 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, how many standard deviations away from the mean, from the mean, from the mean, just like that. So we literally 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 number or 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 could use the entire time. So if it’s our distribution, let’s see, we have a mean of 81. So we have a mean 81. That’s our mean, and then a standard deviation of 6.3. So our distribution, they’re saying, they’re telling us that it’s normally distributed so I can draw a nice bell curve here. They’re saying that it’s normally distributed. So that’s as good of a bell curve as I am 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? That gets us what? 74.7. 74.7. Right? If we add six, it’ll get us to 80.7 and then 0.3, it’ll get us to 81. So that’s one standard deviation below and above the mean, and then you’d add another 6.3 that goes two standard deviations. So on and so forth.
So that’s at least a drawing of the distribution itself. So let’s figure out the Z scores for each of these, for each of these scores. So 65 or each of these grades. 65 is how far? 65 is, it’s maybe going to be here someplace. So we first want to say, well, how far is it just from our mean, from our mean? So the distance is, we just want a 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 five plus 11. It’s 16. So this is going to be minus 16, over 6.3. And take our calculator out and let’s see, if we have minus 16 divided by 6.3, you get minus 2, oh, it looks like five, four. So approximately equal to minus 2.54. That’s the Z score for a grade of 65. Pretty straightforward.
Let’s do a couple of more. Let’s do all of them. 83. So how far is it away from the mean? Well, it’s 83 minus 81. Is two grades above the mean, but we want in terms of standard deviations. How many standard deviations?
So this was part a. A was right here. This was we were 2.5 standard deviations below the means. So this was part a, one, two, 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. We’re 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, well, 83 minus 81 is two, divided by 6.3, is 0.32, roughly. So here we get 0.32. So 83 is 0.32 standard deviations above the mean. And if so, it’d be roughly one third of the standard deviation along the way, right? Because this was 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 deviation. So 12 is how many standard deviations above the mean? Well, it’s going to be almost two. Let’s take the calculator are out. So we’ve got 12 divided by 6.3. It’s 1.9 standard deviations. It’s Z score, it’s Z score is 1.9, which means it’s 1.9 standard deviations above the means. So the means 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. It’s Z score is 1.9. all that means is 1.9 standard deviations above the mean.
Let’s do the last one. I’ll do it in magenta. Part d. A score of 100. A score of a 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, well, a little over three standard deviations. And we’ll, 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. It’s 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, right here at 81. We go one standard deviation above the mean, two standard deviations of 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 would 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 and actually not just a very low probability of getting something higher than that, because we’ve learned before in a probability to density function. The probability, if this is a continuous, not a discreet, the probability of getting exactly that is zero, if this wasn’t discreet. But since this is scores on a test, we know that it’s actually a discreet probability function. But the probability is low of getting higher than that because you can see where we sit in 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.