Slide 6 - 1Copyright © 2010 Pearson Education, Inc.

The boxplots shown above summarize two data sets, I and II. Based on the boxplots, which of the following statements about these two data sets CANNOT be justified?

a. The range of data set I is equal to the range of data set II.b. The interquartile range of data set I is equal to the interquartile range

of data set II.c. The median of data set I is less than the median of data set II.d. Data set I and data set II have the same number of data points.e. About 75% of the values in data set II are greater than or equal to

about 50% of the values in data set I.

Slide 6 - 2Copyright © 2010 Pearson Education, Inc.

The boxplots shown above summarize two data sets, I and II. Based on the boxplots, which of the following statements about these two data sets CANNOT be justified?

a. The range of data set I is equal to the range of data set II.b. The interquartile range of data set I is equal to the interquartile range

of data set II.c. The median of data set I is less than the median of data set II.d. Data set I and data set II have the same number of data points.e. About 75% of the values in data set II are greater than or equal to

about 50% of the values in data set I.

Copyright © 2010 Pearson Education, Inc.

Chapter 6The Standard Deviationas a Ruler and the Normal Model

Slide 6 - 4Copyright © 2010 Pearson Education, Inc.

The Standard Deviation as a Ruler

The trick in comparing very different-looking values is to use standard deviations as our rulers.

The standard deviation tells us how the whole collection of values varies, so it’s a natural ruler for comparing an individual to a group.

Slide 6 - 5Copyright © 2010 Pearson Education, Inc.

Standardizing with z-scores

We compare individual data values to their mean, relative to their standard deviation using the following formula:

We call the resulting values standardized values, denoted as z. They can also be called z-scores.

y yz

s

Slide 6 - 6Copyright © 2010 Pearson Education, Inc.

Standardizing with z-scores (cont.)

Standardized values have no units. z-scores measure the distance of each data

value from the mean in standard deviations. A negative z-score tells us that the data value is

below the mean, while a positive z-score tells us that the data value is above the mean.

Slide 6 - 7Copyright © 2010 Pearson Education, Inc.

Benefits of Standardizing

Standardized values have been converted from their original units to the standard statistical unit of standard deviations from the mean.

Thus, we can compare values that are measured on different scales, with different units, or from different populations.

Slide 6 - 8Copyright © 2010 Pearson Education, Inc.

Example: 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 time wins. In the 2006 Olympics, the mean slalom time was 94.2714 seconds with s = 5.2844 seconds. The mean downhill time was 101.807 sec. with s=1.8356 sec. Ted Ligety of the US, who won the gold with a combined time of 189.35 sec. skied the slalom in 87.93 sec and the downhill in 101.42 sec. On which race did he do better compared with the competition?

Slide 6 - 9Copyright © 2010 Pearson Education, Inc.

Example: In the 2006 Olympics men’s combined event, Ivica Kostelic of Croatia skied the slalom in 89.44 sec and the downhill in 100.44 sec. He thus beat Ted Ligety in the downhill, but not the slalom. Maybe he should have won the gold medal. Considered in terms of standardized scores, which skier did better?

Slide 6 - 10Copyright © 2010 Pearson Education, Inc.

Shifting Data

Sometimes it is necessary to add or subtract a number from all of your data values. How does this affect the mean, median, IQR, standard deviation and range? Example: Find the mean, median IQR,

standard deviation and range of 10, 15, 20, 25, 30. Then add 5 to each number and find the mean, median IQR and range of the new numbers. Which measures change when adding (or subtracting) a constant?

Slide 6 - 11Copyright © 2010 Pearson Education, Inc.

Shifting Data

Sometimes it is necessary to multiply or divide a number from all of your data values. How does this affect the mean, median, IQR,standard deviation and range? Example: Find the mean, median IQR and

range of 10, 15, 20, 25, 30. Then multiply each number by 2 and find the mean, median IQR, standard deviation and range of the new numbers. Which measures change when multiplying (or dividing) by a constant?

Slide 6 - 12Copyright © 2010 Pearson Education, Inc.

Shifting Data (cont.)

The following histograms show a shift from men’s actual weights to kilograms above recommended weight:

Slide 6 - 13Copyright © 2010 Pearson Education, Inc.

Rescaling Data (cont.)

The men’s weight data set measured weights in kilograms. If we want to think about these weights in pounds, we would rescale (multiply by a constant) the data:

Slide 6 - 14Copyright © 2010 Pearson Education, Inc.

Homework Pg. 129 1-23(odd)