STATISTICS
       
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Z-Scores You
may have discovered now that it is possible to describe the quality of
a bolt in the sample from Screws and Bolts Inc. by knowinghow many standard
deviations are between its length and the mean of the sample. That would
tell us how good a particular bolt is compared to the other bolts in the
sample. Or, to use another example, you could judge how good a player's
performance in a basketball game is by comparing his score from the game
to his average score. So, when a player's performance is measured, it may
make more sense to say how many standard deviations his score is from his
mean performance, than just to say how many points he scored in the game.
Let's look at the scores of two basketball players (you are probably tired of bolts anyway!):
You will notice that in game 6, both players scored 10 points. Does that mean that their personal performance was equally good? When you look at how many points they usually score you see that Kareem probably has a better average than Michael. So, wasn't Michael's personal performance better? You will find that Kareem's mean score is 15 points, and Michael's mean score is 8 points. Moreover, you will find that the standard deviation of Kareem's scores is approximately 3 points (exactly 2.9), and the standard deviation of Michael's scores is approximately 4 points (exactly 3.9). So, in game 6, Kareem's score was 5/3=1.67 standard deviations below his average, and Michael's score was 2/4=.5 standard deviations above his average. Therefore, we must conclude that Michael's personal performance was better than Kareem's. When individual bolts in a sample from Screws and Bolts Inc., or scores from individual games for the basketball players, are described in this way, we call them z-scores. The Kareem's z-score in game 6 was -1.67, but Michael's z-score in game 6 was +.5
Area 10 Mathematics and Technology Professional Development Center Permission is granted to duplicate these materials for classroom use.
Last updated on 1/30/1999 |
