 Chapter 2 deals with measures of relative standing and density curves. We're going to start by talking about measures of relative standing. Now, relative standing, we're not talking about your relative's standing. Instead, we're talking about a way to compare apples and oranges. Relative standing. In other words, how does an individual score relative to the distribution of scores compared with others? Let me explain with an example. Let's say you just scored 85 on a statistics test and your friend scored 91 on an English test. Let's say both scores are out of 100 points. So the natural question is, which one of you did better? Clearly, your friend did better, but that's not the point. A more interesting question from a statistics standpoint is, which of you scored better relative to the other students in your respective classes? What does an 85 mean? What does a 91 mean? In order to answer that question, we need to know more information. The mean of all scores helps. The mean score of the statistics test was 77. The mean of the English test was 82, which means both of you scored above the mean. Your score was 85, which is 8 points above the mean. Your friend scored 9 points above the mean. Well, that doesn't quite answer the question either, because we need to know about the spread of all the scores. In general, how consistent were scores on the statistics test and the English test? The standard deviation allows us to answer that question. For the statistics scores, standard deviation was 3.5, which means on average that's how much scores deviated from the mean of 77. The standard deviation for the English test is larger, which means there's more spread, there's more variation in the scores. And so it's perhaps more likely that your friend would score as far away from the mean as you did. Another way to answer this is with how many standard deviation units away from the mean is your score versus your friends. Well, 3.5. Your score of 85 is more than two standard deviations above the mean. Your friend's score of 91 is exactly two standard deviations. And so that kind of gives us an answer to the question. You scored better, you scored more than two standard deviations above the mean. Now, rather than saying that over and over, there's a faster, a more concise way of saying number of standard deviations above or below the mean. And that's a standardized score, or a Z-score. To calculate a Z-score, you'll take the individual raw score, subtract the mean, and divide by standard deviation. Now, just a notation piece. Sorry, I forgot to mention this. If you have a positive Z-score, of course, that means you scored greater than the mean. Negative Z-score means you would score less than the mean. So a notation piece. When we're dealing with sample statistics, you will remember that the mean we denoted as X bar, X with the horizontal line over top. When we're dealing with samples, that's what we'd call our mean. When we're dealing with population parameters, in other words, the truth about the entire population, then we'd use Greek letters. We'd say mu, or mu of X. For standard deviation, we said it was S, or S of X. The population parameter version of standard deviation is the Greek letter sigma. So we would say sigma, or sigma of X. It's a small notation piece, but to make this a more compact calculation, we say the Z-score is calculated with X, the original score, minus mu over sigma, which means X minus the mean over standard deviation. And this is a formula that's going to come up often. In fact, throughout the entire remainder of this course. So now we can answer this question in a more compact way. You scored approximately 2.3 standard deviations above the mean, whereas your friend scored exactly 2 standard deviations above the mean. Which means that you did better relative to the other students in your class than your friend.