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How would you answer the following question?
The men’s combined skiing event in the winter Olympics consists of two races: a
downhill and a slalom. Times for the two events are added together and the skier
with the lowest total time wins. In the 2006 Winter Olympics, the mean slalom time
was 94.2714 seconds with a standard deviation of 5.2844 seconds. The mean
downhill time was 101.807 seconds with a standard deviation of 1.8356 seconds. Ted
Ligety of the United States, who won the gold medal with a combined time of 189.35
seconds, skied the slalom in 87.93 seconds and the downhill in 101.42 seconds. On
which race did he do better compared with the competition?
Standardizing values is useful in comparing values
from different sets of data. We give the values zscores (standardized scores) for comparisons.
𝑥 − 𝑥
𝑧=
𝑠
Recall: 𝑥 = data value, 𝑥 = the mean of the set of
data, 𝑠 = standard deviation of the set of data
Example:
Find the z-score for 95 if the data has a mean
of 90 and a standard deviation of 3.
95 − 90
𝑧=
= 1.67
3
This means that 95 is 1.67 standard deviations
away from the mean.
 A positive z-score means the value is
greater than the mean.
 A negative z-score means the value is
less than the mean.
 A z-score of 0 means the value is
equal to the mean.
Now, back to the question… On which race did Ted Ligety do better?
Slalom: time = 87.93 sec; mean = 94.2714 sec; SD = 5.2844 sec
𝑧𝑠𝑙𝑎𝑙𝑜𝑚
87.93 − 94.2714
=
= −1.2
5.2844
Downhill: time = 101.42 sec; mean = 101.807 sec; SD = 1.8356 sec
𝑧𝑑𝑜𝑤𝑛ℎ𝑖𝑙
101.42 − 101.807
=
= −0.21
1.8356
Which z-score is better? z = -1.2 or z = -0.21?
Context is important. Since we are looking at
race times, the lower the better. The time for the
slalom was 1.2 standard deviations below the
average, while the time for the downhill was
only 0.21 standard deviations below the average.
So the better performance was in the slalom.
Practice:
In the 2006 winter Olympics men’s combined event, Ivica
Kostelic of Croatia skied the slalom in 89.44 seconds and the
downhill in 100.44 seconds. He beat Ted Ligety in the downhill,
but not in the slalom. Who should have won the gold medal?
Calculate the z-scores for Ivica Kostelic
(slalom mean = 94.2714 and SD = 5.2844 sec; downhill
mean = 101.807 and SD = 1.8356 sec)
Compare the sum of the z-scores
Ted Ligety had a z-score sum of -1.41 compared to
Ivica Kostelic with a z-score sum of -1.65.
Kostelic’s total is better.
Because the standard deviation of the downhill
times is so much smaller, Kostelic’s better
performance there means that he would have won
the event if standardized scores were used. Ligety
won the gold, but perhaps Kostelic should have.
Your Statistics teacher has announced that the lower of your
two tests will be dropped. You got a 90 on test 1 and an 80 on
test 2. You’re all set to drop the 80 until she announces that she
grades on a curve. She standardized the scores in order to
decide which is the lower one. If the mean on the first test was
88 with a standard deviation of 4 and the mean on the second
was 75 with a standard deviation of 5, which one will be
dropped?
90 − 88
𝑧1 =
= 0.5
4
80 − 75
𝑧2 =
= 1.0
5
If the mean on the first test was 88 and the standard
deviation was 4, which test score would be needed to
have a z-score of 1.5?
𝑥 − 88
1.5 =
4
1.5 4 = 𝑥 − 88
94 = 𝑥
Today’s Assignment:
 Read Chapter 6
 Homework: page 129 #8-16