The result is described below as "remarkable". I see nothing remarkable about it, except possibly for the idiot of a teacher.
Interesting semantic question, though: Are those properly called "z" scores? According to the usual definition to be found in most elementary statistics textbooks, a standardized score (z) is what you get when you take a raw score (x), subtract an expected value (mu) from it, and divide the difference by a standard deviation (sigma). Note that this standardization is w.r.t. the population values (parameters). While it is usual, when these values are unknown, to substitute the sample estimates thereof (x-bar, s), the resultant score cannot be really a "z" in terms of the original definition: at best it's, say, a "z-hat". The hypothetical instructor (HI) clearly had no STANDARDS of any kind in mind; HI did not standardize the students' scores against a population, but only against themselves: and there aren't enough of them to give a reliable value. Had HI has some STANDARD in mind, HI would have used the parameters of that standard. (E.g., "I think that a large number of competent students who have studied this material ought to exhibit a mean of 70 and a standard deviation of 10." The standard scores for the three tests would then be 0.0, +3.0, +0.5 for Ann, and -1.0 for everybody else, if the HI's standard be the same for all three tests. [And for these data, z-scores to two decimal places are an instance of spurious precision.]) In other words, HI defaulted on his responsibility to set reasonable standards for marks. Unfortunately, faced with incompetent teachers, there's not much one can do, except maybe complain to management (which Ann has every right to do, IMHO). To borrow a phrase from Robert Dawson: <grin, duck, & run> On Sat, 2 Aug 2003, Dennis Roberts wrote in part: > A class of 5 students took a test on statistics. The results were > Ann - 70, Belinda - 40, Cal - 40, Daniel - 40 and Erika - 40. The > teacher standardised the scores. Ann earned a z-score of 1.79 (to 2 > decimal places). > > The next week the same class took another test. Ann studied really > hard and blitzed the test. The results were Ann - 100, Belinda - 40, > Cal - 40, Daniel - 40 and Erika - 40. Ann was really proud of her > achievement. But then the teacher standardised the scores. Ann was > devastated - her z-score was still 1.79. > > Ann decided that studying hard was a waste of time so she decided to > watch TV instead of study. The results on that week's test were > predictable: Ann - 45, Belinda - 40, Cal - 40, Daniel - 40 and Erika > - 40. Ann thought that maybe she should have studied after all. But > then the teacher standardised the scores. Ann was right the first > time - studying is a waste of time! Her z-score was still 1.79. > > What gives? > [This remarkable result is from an article in the Australian Senior > Mathematics Journal, Vol 17, No 1, by Ed Staples ----------------------------------------------------------------------- Donald F. Burrill [EMAIL PROTECTED] 56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816 . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
