Worksheet 8 Shifting and Scaling Data, Z-Scores
BUS/ST 350 Lecture Worksheet #8 Reiland Name
Shifting Data, Rescaling Data, z-scores
1. Suppose the class took a 40-point quiz. Results show a mean score of 30, median 32, IQR 8, standard deviation 6, min 12, and Q1 27. (Suppose YOU got a 35.) What happens to each of the statistics if… • I decide to weight the quiz as 50 points, and will add 10 points to every score. Your score is now 45. • I decide to weight the quiz as 80 points, and double each score. Your score is now 70. • I decide to count the quiz as 100 points; I’ll double each score and add 20 points. Your score is now 90. Statistic original (y)y+10 2y 2y+20 mean 30 %! '! )! median 32 %# '% )% IQR 8 )"'"' stan dev 6 '"#"# minimum 12 ## #% %% Q27" $( &% (% your score 35 %& (! *!
2. Let’s talk about scoring the decathlon. Silly example, but suppose two competitors tie in each of the first eight events. In the ninth event, the high jump, one clears the bar 1 in. higher. Then in the 1500-meter run the other one runs 5 seconds faster. Who wins?
It depends on knowing whether it is harder to jump an inch higher or run 5 seconds faster. We have to be able to compare two fundamentally different activities involving different units.
Standard deviations to the rescue! If we knew the mean and standard deviation of performances by world-class athletes in each event, we could compute how far each performance was from the mean in standard deviation units, that is, we could compute the z-scores. The z-scores enable us to compare “apples” and “oranges”.
So consider the three athletes’ performances shown below in a three event competition. Note that each competitor placed first, second, and third in an event. Who gets the gold medal? Who turned in the most remarkable performance of the competition? To begin to answer this question we'll calculate the z-scores for each competitor in each event.
Event Competitor 100 m dash shot put long jump "!Þ""! '''! #'#' A 10.1 sec DœœÞ&Dœœ#Dœœ!!Þ# 66ft $ 26ft Þ& * *Þ*"! '!'! #(#' B 9.9 secDœ!Þ# œ Þ& 60ft Dœ $Þ&œ!27ft Dœ œ#* "!Þ$"! '$'! #(Þ#&#' C 10.3 secDœ!Þ# œ"Þ& 63ft Dœ $ œ" 27ft 3in Dœ Þ& œ#Þ&* Mean (in all events by 10 sec 60ft 26ft world-class athletes) St. Dev. (in all events 0.2 sec 3ft 6 inches by world-class athletes)
* Note that in the calculation of the z-scores for the long jump, the standard deviation of 6 inches is changed to .5 feet so that we are using the same units, feet, in both the numerator and denominator. ST 350 Worksheet #8 page 2 IMPORTANT: Note that for the shot put and long jump, large positive z-scores mean better performance; for the 100 m dash large negative z-scores mean better performance (that is, it is better when a competitor runs faster).
We will determine who wins the gold medal by adding the shot put and long jump z-scores and the negative of the 100 m dash z-score for each competitor. The competitor with the largest sum wins the gold medal.
Who wins? Competitor AÀ ÐÞ&Ñ#!œ"Þ& Competitor B: ÐÞ&Ñ!#œ#Þ& Competitor C: Ð"Þ&Ñ"#Þ&œ#
Competitor B wins!!