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100 CHAPTER 2 • Modeling One-Variable Quantitative Data

Effect of Multiplying or Dividing by a Constant
Suppose that Mr. Tabor wants to convert his students’ test scores to percentages.
Because the test was worth 50 points, he multiplies each adjusted test score by 2 to
get a student’s score out of 100 points. (Note that, with the 5-point adjustment prior
to doubling, one student ended up with a final score of 106!) Figure 2.3 shows graphs
and numerical summaries for the adjusted scores and doubled scores.
FIGURE 2.3 Dotplots
and summary statistics
for the adjusted scores Adjusted

(out of 50) and doubled
scores (out of 100) on
Mr. Tabor’s statistics test.
Doubled
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20 40 60 80 100
Test score
n x s x Min Q 1 Med Q 3 Max IQR Range
Adjusted 30 40.8 8.17 17 37 42 46 53 9 36
Doubled 30 81.6 16.34 34 74 84 92 106 18 72

From the graphs and summary statistics, we can see that the measures of center,
location, and variability have all doubled, just like the individual data values. But the
shape of the two distributions is the same. Multiplying or dividing each value in a
data set by a positive constant stretches (or compresses) the distribution by that factor.

TEACHING TIP Analyzing the effect of multiplying or dividing by a constant
It’s a good idea to compare and contrast Multiplying (or dividing) each data value by the same positive number b:
the summary points here (effect of ■ Multiplies (divides) measures of center and location (mean, five-number summary)
multiplying or dividing by a constant) by b
with the summary points at the bottom ■ Multiplies (divides) measures of variability (range, standard deviation, IQR) by b
of p. 98. Students must understand how ■ Does not change the shape of the distribution
each type of transformation (adding/
subtracting and multiplying/dividing) It is not common to multiply (or divide) each data value by a negative number b.
affects a distribution. Doing so would multiply (or divide) the measures of variability by the absolute value
of b. We can’t have a negative amount of variability! Multiplying or dividing by a
negative number would also affect the shape of the distribution, as all values would
be reflected over the y axis.
AL TERNA TE EX AMPLE EXAMPLE
How tall is a Kiwi?
How far off were our guesses?
Effect of multiplying/dividing by a Effect of multiplying/dividing by a constant
constant
PROBLEM: Refer to the “How wide is this room?” example. The graph and numerical summaries here
PROBLEM: Here are a dotplot and describe the distribution of the Australian students’ guessing error (in meters) when they tried to estimate
numerical summaries of the heights (in the width of their classroom.
centimeters) of a random sample of 50
New Zealand students (“Kiwis”) who
responded to an online poll.
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125 130 135 140 145 150 155 160 165 170 175 180
Height (cm)

n x s x Min Q 1 Med Q 3 Max IQR Range
50 159.68 15.213 126 148 164 171 183 23 57

Suppose we convert the heights
of these students into inches
(1inch = 2.54 centimeters).
(a) What shape would the distribution SOLUTION:
of height in inches have? (a) The same shape as the distribution of
(b) Find the mean of the distribution of height in centimeters: skewed to the left
=
height in inches. with a peak at 169/2.54 66.5inches.
=
(c) Find the standard deviation of the (b) Mean:159.68/2.54 62.9inches
=
distribution of height in inches. (c) Standarddeviation:15.213/2.546.0 inches
100 CHAPTER 2 • Modeling One-Variable Quantitative Data

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