Rod: When introducing the class in the use of Z-scores (read "Zed Scores" in the Great White North, eh?) I think that it's important to stress the idea that we are talking about how an individual does _relative to the rest of the distribution_.
An example I use involves a bet between two roommates, Pat and Chris, who are in two different statistics courses. They have midterms coming up, and they want to make a bet as to who will "do better" on the exam. Pat suggests that whoever gets the higher score on their exam wins the bet (no dishwashing for a week). Chris points out that the two exams may have different numbers of questions, and/or a different maximum score. So, looking at the raw scores may not be a way to compare their two performances [e.g., 45 out of 55 versus 49 out of 60]. Instead, they should each convert their grades to a percentage, and compare percentages. [This introduces the concept of converting or transforming scores] That's an imporvement, says Pat. But what if your prof gives you an easy exam with a high class average, while my prof gives a toughie with a lower class average? That wouldn't be fair to me. [Must take distribution average into account] Let's make it so that who ever does better relative to their class average wins the bet. In other words, subtract the class average from your score [X - Xbar], and whoever has the higher positive difference wins. Chris replies with, I like the idea of comparing scores to the respective class average rather than raw scores, but if your exam is out of 100 and mine is out of 20, you're much more likely to get a high positive difference score. [The scores come from distributions with different properties] And even if the two exams are out of the same total and have the same average, maybe your class will have much more variability around that average, giving you a better chance of scoring further above the mean than I would have in my class of less variable scores. [Must take variability into account] So how do we handle this, they ask. Let's figure the standard deviation (a measure of variability) for each class distribution, and see who scores more standard deviations [standard scores or Z-scores] above the mean. That will give us a standardized measure of our scores relative to our respective class distributions. [This is exactly what a Z score is: a standardized score indicating relative position in a distribution, i.e., where a score stands relative to all of the other scores in a distribution, taking into account each distribution's mean and standard devaition]. I then provide the two roommates' scores and their respective class averages and standard deviations. Next, we calculate the percentage, difference score (X - Xbar), and Z score for each raw score. I try to work it out such that the "Better" score goes back and forth depending on which measure is used. For example, raw score: Pat wins; percentage score: Chris wins; difference score: Pat wins; Z-score, which I explain is the fairest way to compare scores from two distributions: Chris wins the bet (Chris did better relative to the rest of the scores in her class's distribution) and doesn't have to wash dishes for a week! For a pictorial representation, I demonstrate by drawing two normal distributions (with differing variability) centred one under the other, indicating the two means and where one standard deviation lies in each distribution, i.e., under the point of inflection. Then I ask students to estimate how far along the respective X axis each of the two raw scores would be found. If we've talked about percentile scores, I'll expalin how Z scores can be transformed into percentile scores. The same example can be used when comparing one person's scores from two different tests (e.g., first and second midterm exams), to check on improvement realtive to the rest of the class. Also takes into account the difficulty (mean) and variability (standard deviation) of each of the two midterms (distributions with different properties). As with many of the examples and demos I've collected over the years, I can't recall where I found this demonstration (textbook, instructor's manual, TIPS, or if I made it up). With a student such as Rod's who doesn't see the need to include standard deviations in a calculation, I'd draw out two distributions one over the other with the same mean but with different spreads, and show how it is easier to get a higher raw score on the distribution with the greater spread. Perhaps Rod's student is having difficulty understanding why it's not "cheating" to convert to Z scores because the examples of running and swimming speeds are too dissimilar (minutes versus seconds). Have you tried an example of running speed when comparing say four-year olds (slower speeds, less variability) to the same kids when they're 8 years old (faster speeds and greater variability) to show that, even though he runs faster now and is 5 seconds faster than the average at each age, Person X might be a slower runner _relative to other kids_ as an 8 y.o. compared to when he was a 4 y.o.? -Max On Mon, 24 Feb 2003, Hetzel, Rod wrote: > Hi everyone: > > I need your help with something. I have a student who just does not > understand z-scores. I have met with him for at least two hours outside > of class and he still doesn't understand the concept. In particular, he > doesn't seem to understand why you need to include standard deviation in > the calculation of z-scores. "Why can't you just compare the raw > scores?" is his frequent question. I explained to him in various ways > that the z-score is a transformed score that can take scores from two > different distributions and put them on a common metric, that it gives > you a summary statistic that tells you an individual's score in relation > to the mean and standard deviation, that it provides a way to compare > scores from two different distributions, etc. > > Here is the example that my student keeps coming back to: "Jack and > Jill are intense competitors, but they never competed against each > other. Jack specialized in long-distance running and Jill was an > excellent sprint swimmer. As you can see from the distributions in each > table, each was best in their event. Take the analysis one step farther > and use z-scores to determine who is the more outstanding competitor." > > LONG-DISTANCE RUNNING > Jack: 37 min > Bob: 39 min > Joe: 40 min > Ron: 42 min > > SPRING SWIMMING > Jill: 24 sec > Sue: 26 sec > Peg: 27 sec > Ann: 28 sec > > Here are the relevant statistics: > RUNNING MEAN: 39.5 > RUNNING SD: 1.803 > JACK'S ZSCORE: -1.39 > > SWIMMING MEAN: 26.25 > SWIMMING SD: 1.479 > JILL'S ZSCORE: -1.52 > > When I have met with the student, he has not understood how Jill is the > more outstanding competitor. He makes the comment that Jack is > obviously the better competitor because Jack scored an entire 3 minutes > faster than the next finisher whereas Jill scored only 2 seconds faster > than her runner-up. "Why do you have to even look at the other scores > in the distribution to tell that Jack is the better competitor? He > finished a full three minutes ahead of his competitors and Jill just > barely finished ahead of her competitors." I have drawn some diagrams > of normal distributions to show how Jill's score on the distribution is > further away from the mean and closer to the tail, but my student thinks > that I am somehow changing the scores and cheating the system when I > transform the raw scores to z-scores. Even after I show him how the > position of the score remains unchanged, he cannot grasp in this case > how Jill is the more outstanding competitor. I've tried switching > examples with him (e.g., distributions of test scores, changing C > temperature to F temperature, etc.), but nothing seems to be sinking in. > He has a fairly high level of anxiety about statistics but tends to > cover it up with humor and sarcasm. He took statistics with another > professor last semester and told me that all statistics is a bunch of > bull**** that serves no useful purpose other than obscuring the > painfully-obvious truth. > > So, I have two questions for all of you out there in TIPS land... > > 1. Given what I've told you about the student's struggles with > z-scores, does anyone have any specific ideas on how to present this > information to him? I think I'm in a rut with him and need a fresh way > to explain this. > > 2. Would anyone be willing to share with me any z-score examples that > you use for your own assignments and exams? I am running out of new > examples to use with this student and was hoping that perhaps you would > be willing to share some of your own examples. This would give my > student some more opportunities to calculate z-scores > > 3. How do you work with students who just don't seem to get statistics? > Everyone else in the class seems to understand z-scores well, but I'm > struggling a bit in trying to reach this student. I find that I am > hardly ever at a loss for words when teaching clinical courses, but I'm > reaching my limit with this student. This is certainly not my area of > expertise, so I'm hoping that some of you stats-people can help out with > this! > > Thanks for your assistance with this problem! > > Rod > > --- > You are currently subscribed to tips as: [EMAIL PROTECTED] > To unsubscribe send a blank email to [EMAIL PROTECTED] > Maxwell Gwynn, PhD [EMAIL PROTECTED] Department of Psychology (519) 884-0710 ext 3854 Wilfrid Laurier University Waterloo, Ontario N2L 3C5 Canada --- You are currently subscribed to tips as: [EMAIL PROTECTED] To unsubscribe send a blank email to [EMAIL PROTECTED]
