Hi everyone: Thanks to everyone for their helpful guidance on the z-score situation. Actually, I ended using an combination of all of your suggestions. I used David's examples of focusing on two particular distributions of tests scores (see below for the specifics). I framed this situation in the context of Max's example of a wager between two roommates. I also took Mike's approach to discussing the differences between various standard scores and then discussing the formula. This was the third hour outside of the classroom that I have devoted to helping this student understand z-scores, but I think that he is finally beginning to understand the concept. Now, on to T-scores.... :)
Thanks again for all of your help! Rod ______________________________________________ Roderick D. Hetzel, Ph.D. Department of Psychology LeTourneau University Post Office Box 7001 2100 South Mobberly Avenue Longview, Texas 75607-7001 Office: Education Center 218 Phone: 903-233-3893 Fax: 903-233-3851 Email: [EMAIL PROTECTED] Homepage: http://www.letu.edu/people/rodhetzel > -----Original Message----- > From: David L. Carpenter [mailto:[EMAIL PROTECTED] > Sent: Tuesday, February 25, 2003 4:02 PM > To: Teaching in the Psychological Sciences > Subject: Re: z-score woes > > > Rod, > > Maybe an example closer to home would help. Give him as an > example his score on two hypothetical exams in the same > class. Set it up so his numerical score on the first exam is > lower than his numerical score on the second exam, but on the > first exam is above the mean of the class and below it on the > second exam. Ask him which exam he did better on. > Obviously he did higher on the second, but relatively (and on > the grading > curve) he did poorer on the second exam. Maybe that will get > him thinking a little outside his box. > > Dave Carpenter > Dept. of Psychology > St. Bonaventure University > St. Bonaventure, NY 14778 > > [EMAIL PROTECTED] > > > > 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] > --- You are currently subscribed to tips as: [EMAIL PROTECTED] To unsubscribe send a blank email to [EMAIL PROTECTED]
