CHAPTER 6 PART 1

THE STANDARD DEVIATION AS

A RULER

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?

How would you answer the following question?

Standardizing values is useful in comparing values from different sets of data. We give the values z-scores (standardized scores) for comparisons.𝑧=

𝑥−𝑥𝑠

Recall: = data value, = the mean of the set of data, = 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.𝑧=

95−903

=1.67

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 secDownhill: time = 101.42 sec; mean = 101.807 sec; SD = 1.8356 sec𝑧 h𝑑𝑜𝑤𝑛 𝑖𝑙=

101.42−101.8071.8356

=−0.21

𝑧 𝑠𝑙𝑎𝑙𝑜𝑚=87.93−94.2714

5.2844=−1.2

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.

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

Practice:

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?𝑧1=

90−884

=0.5 𝑧 2=80−755

=1.0

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?1.5=

𝑥−884

1.5 (4 )=𝑥−88

94=𝑥

Today’s Assignment: Read Chapter 6 Homework: page 129 #8-16