 I'd like to walk you through a couple of sample problems dealing with the first part of chapter 2, measures of relative standing and density curves. In this first example, let's say that there are a thousand students taking a standardized test and your score, you got a 63, and that sets you in the 73rd percentile. Well what does that mean? Remember percentile is at or below a certain score. So if your score was 63, that means 73% of students scored 63 or below. This next example problem deals with a scenario, pause the video and read this please. So we're looking at these summary scores and a histogram. Here's the histogram of those scores. You can see we have a left skewed distribution, in other words there are a couple of scores that are low. Most of the scores are in kind of this middling range of between let's say 55 and 75 and 70. Most of the scores are in there. We've got I guess like the Charlie Browns of the class down here, and then a couple of high flyers. But most of the scores are up in here with a kind of a long tail to the left. We've got a bunch of different summary statistics here, and this is kind of typical of what a computer would give you if you asked for summary statistics. So the variable we're looking at is self-cons, which is an abbreviation of that self-concept score. And always refers to the number of individuals. Here we have 78 students, we have the mean, median, this is called the trimmed mean, and we can talk about that more later in class, but suffice it to say we can safely ignore that. Likewise the standard error of the mean is not something that we'll really deal with until later on in the course. So we've got our mean, median, standard deviation, and then we've got the minimum, maximum, Q1 and Q3. Together with the median, these would make up the five number summary. So if we were asked to kind of sketch in a density curve, first off we would need for the axis to be instead of frequency, relative frequency. In other words, these would have to be percentile percentage bars instead of frequency bars. But otherwise you just kind of smooth a curve over the distribution. So that would be considered a density curve provided that the area underneath was equal to one. Next, we're looking at this one student who had a self-concept score of 62, and we see that 34 students had scores higher than that. Well there were 78 total students, so 34 students were greater than that, which means 44 students had scores at or below. So in terms of percentiles, we know that percentile means at or below, and there were 44 students who were at or below 62 out of 78. And so that puts us in the, we can just kind of round the 56th percentile. Now in asking for standardized scores, that's a Z score. And remember a Z score is calculated as raw score minus the mean over standard deviation. So in this case, our raw score was 62 minus 56.96 all over 12.41. And that gives us a Z score of about .406. In other words, this student, I can't get the 6. So in other words, this student is about .4 standard deviations greater than the mean score. This last question is asking for a typical score. Typical in this case sort of means, where is the center of this distribution? Now, because we have this skew in the distribution, because we have a left skew, that means that the mean is sort of being dragged down by these low outliers. In other words, the mean is not resistant. The median is. And so the median would be perhaps a better choice for describing a typical score. And as a result, we would say that the typical score is somewhere around 59.5.