Download 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
mean
median
IQR
stan dev
minimum
Q"
your score
original (y)
30
32
8
6
12
27
35
y+10
%!
%#
)
'
##
$(
%&
2y
'!
'%
"'
"#
#%
&%
(!
2y+20
)!
)%
"'
"#
%%
(%
*!
2. Let’s talk about scoring the decathlon. Silly example, but suppose two competitors tie in
each of the first eight events. In the ninth event, the high jump, one clears the bar 1 in.
higher. Then in the 1500-meter run the other one runs 5 seconds faster. Who wins?
It depends on knowing whether it is harder to jump an inch higher or run 5 seconds
faster. We have to be able to compare two fundamentally different activities involving
different units.
Standard deviations to the rescue! If we knew the mean and standard deviation of
performances by world-class athletes in each event, we could compute how far each
performance was from the mean in standard deviation units, that is, we could compute
the z-scores. The z-scores enable us to compare “apples” and “oranges”.
So consider the three athletes’ performances shown below in a three event competition.
Note that each competitor placed first, second, and third in an event. Who gets the gold
medal? Who turned in the most remarkable performance of the competition? To begin to
answer this question we'll calculate the z-scores for each competitor in each event.
Competitor
A
B
C
Mean (in all events by
world-class athletes)
St. Dev. (in all events
by world-class athletes)
"!Þ""!
œ Þ&
!Þ#
*Þ*"!
9.9 sec D œ !Þ# œ  Þ&
10.3 sec D œ "!Þ$"!
œ "Þ&
!Þ#
Event
shot put
66ft D œ '''!
œ#
$
'!'!
60ft D œ $ œ !
63ft D œ '$'!
œ"
$
long jump
26ft D œ #'#'
œ !*
Þ&
#(#'
27ft D œ Þ& œ #*
27ft 3in D œ #(Þ#&#'
œ #Þ&*
Þ&
10 sec
60ft
26ft
0.2 sec
3ft
6 inches
100 m dash
10.1 sec D œ
* 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 A À  ÐÞ&Ñ  #  ! œ "Þ&
Competitor B:  Ð  Þ&Ñ  !  # œ #Þ&
Competitor C:  Ð"Þ&Ñ  "  #Þ& œ #
Competitor B wins!!