Hi everyone. We are going to proceed through the next section in chapter three, which is 3.3, which deals with measures of relative position. Or relative standing is another way to sometimes think about it. So this is a pretty interesting section because what it allows us to do is to actually make some comparisons and that's really important, right? We're always concerned about where do people and things stand in relation to one another. And that's what this section is going to look at. And so the first sort of measure of relative position that we are going to deal with is the idea of a standard score. Maybe some of you may have heard of standard scores before. So that's what we're gonna take a look at. So what it allows us to do, this idea of standard scores, is actually to make comparisons between data that may not be using the same scale. And one of the classic examples is the AC T versus the SAT right. Somebody says: "I scored a 23 on the ACT or 24, and on the SAT, you know 1100 and who did better In that case?" So the z-score allows us to do this sort of a comparison. And so I'm using these words synonymously here. It states that a standard score, sometimes called a z-score, it just depends, is really about standard deviations. That's the big takeaway here and what we need you to remember is that a z-score or standard score, is the number of standard deviations the value is from the mean. And that's really the big takeaway here, okay. The value is from the mean. And of course we have formulas that we have to use. You don't have to memorize the formulas so, the formulas are provided on the formulas sheet. So the formula says Z, for z-score. What you do is you take the data value X, you subtract the mean mu, and you divide by Sigma. Okay and again that's for populations. That should make sense. So X is the data value, mu you know is the population mean, and Sigma is the population standard deviation. I probably should have pointed out that you want to make sure you have the guided notes for section 3.3. That's what I'm working from so you can get those on the Canvas website. Sorry for not mentioning that at the beginning. That's what I'm filling in the information here from. And for samples, instead of using mu, we use x-bar, and instead of Sigma you use X...I'm sorry "s". X is the data value that we. whatever it is we're studying. X-bar is the sample mean, and "s" here is the sample standard deviation. So in reality you know the formulas work in the same way. Some people will mix up the Mu's and the Sigma's, and the x-bar in the S. At the end of the day it's okay if that happens because the answers still come out to be the same. So let's take a look at how these are used. So what it's asking for here is if the mean score in a math section of the SAT test is 500 with standard deviation of 150 points, what is the standard score for a student who scored a 630? Now in this case, what you should notice is that we're told that the mean is 500. So x-bar is the 500. We're told the standard deviation is 150, and the data value in question is 630. So we've got the necessary pieces that we need. Here we go. So it says we take X, which is 630, we subtract the mean, which is 500 and we divide by 150, and that gives us about eight, I'm sorry, point eight two seven. Now as a rule guys, very important rule that we're going to need later; we always round the z-scores to two decimal places. I'm doing too many twos here...to two decimal places. Okay now one of the questions that comes up is okay, is this score of 630 good? And it's a fair question to want to wonder. And for that you know, let's kind of consider where that's at. So in the middle, we know is zero standard deviations and we've got one and two, and three right. Negative 1, negative 2 and negative 3, and so we have .87 right as a standard deviation. So that's going to be less than one, so maybe about here is where that score of .87 would be. So is that good? Well for for something like an SAT score, the higher the score the better. So in this case, since we are above the mean, that is a good sign; a very good sign that we're above the mean. Now the question is, is it far enough above the mean? Well that's relative to what it is the person is trying to do. What kind of school are they trying to get into? That requires certain scores so as long as we're above the mean, that's good. That's what we want to be. But question of it, is it far enough above the mean? This is relative to whatever it is we're trying to do. So a little bit of interpretation there. And if it was of course negative, we would be on this side. If it was negative 0.87 okay, and we don't want things to be a negative standard deviation if we're talking about scores on an SAT test. Now in the next example here, we're actually gonna do some computation in comparison. So it says Jodie scored 87 on the Calc test and was bragging to her best friend about how she had done. She said her class mean was 80 with standard deviation of 5. So that's for Jodie, and therefore she had done better than the class average. Okay seems pretty logical in her mind. Her best friend Ashley was disappointed. Ashley had scored 82 on her to calculus test and the mean for her class was 73, with standard deviation of 6. So who really did better? okay now as its stated, it's kind of hard to tell just by comparing the scores. So what we really need to consider are a little bit more information, specifically the mean and the standard deviations of the respective classes. So for Jodi. she had the 87 and her class's mean was an 80 and standard deviation was 5. And so she ended up with a 1.4 standard deviation. Now Ashley had an 82 and her class had a mean of 73 with standard deviation of 6, which actually comes out to 1.5. Okay, so who did better? This is an excellent exam question. Now since both z-scores are positive, we want the one that is farthest from the mean. So listen to what I just said, both z-scores are positive, so in this case for who did better, we want the one farthest from the mean. So who is the winner? In this case, that means that Ashley is the winner, right, her score is farther from the mean. Okay so that's how you would explain this. Now one of the questions that comes up, people wonder is, well what happens if they're negative? And this next one is going to help us to understand that. So we've got a school district which is administering a two year-end of two year- end mathematics achievement tests to its fifth graders. We've got a state test and we've got a national test. So the state test has got a mean of 22, standard deviation of 3.4. And the national test 160 is the mean and standard deviation of 22.6. So what I would encourage you to do is pause the video, give it a shot. Try to figure out the two z-scores. And Parts A and B and then maybe give letter C a shot, which one do you think did better? And I can tell you right now that the the z-scores are gonna come out to be negative, which is okay. Think about what that means and think about which then is better, the state test, or the national test. Okay so hopefully you've had an opportunity to to work those respective z-scores out, so I'll do them real quick here. So this will be 20 minus 22, divided by 3.4, so that's gonna give us a negative 0.59. And the other one here is 150 minus 160 over 22.6 and that's gonna be negative .44 okay. Now as it stood, you know, it was hard to compare the two tests with a score of 20 and 150 respectively, They're on completely different scales so we really do need the z-score here to kind of help us. And so now the question is who actually did better? Which one is better? Well let's consider this on the Normal distribution graph. So let me a graph a quick picture here. And let me kind of exaggerate things okay. We know that both of these are actually between 0 & 1 standard deviations, but the question is where do they fall? Well I think that maybe 0.4, negative 0.44; let me erase that and make this a little bit neater here; should being about in this area. Yes, 0.44, and negative 0.59 should be about there. So I hope that makes sense, at least in terms of the location. I know it may not match quite the graph. Now if the z-scores are negative, and we're talking about exam scores, we still want the higher exam score don't we? Isn't that what we're looking for? So in this case, what you want to know is which one of these scores is closer to the mean? Which one is closer to the mean? That would have to be the national score, right. That one is closer to the mean with negative 0.44. So in fact, the better score here is the national one okay, and the reason again is when we've got negative z-scores, in the context of a test, what I want is the one that's closest to the mean. Now that was different when they were both positive. Remember they were both positive, we wanted the one that was farthest from the mean. So that gives you a nice indication here of how to handle the situations where we've got z-scores that are positive. And we have to make a decision on who did better. And if they are negative. And remember that the biggest takeaway is it's always in context, and relative to what it is we're trying to study. And so if its test scores, we always want the higher one. If we were talking about cholesterol levels, then maybe we would want the one that is lower. Or the smallest or lowest level, and so that's dependent upon if they're positive, if they're negative, in thinking about that in terms of position. What we're going to look at it the next video is the next measure of relative standing or relative position, which is called percentiles. And I'm sure many of you have heard of percentiles before.