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Name: ___________________________________ Date: ______________ Period: _______
4.2 Describing Variability and Comparing Distributions (Module 2 Lessons 4-8)
Lesson 4: Summarizing Deviations from the Mean
Essential Question: How are deviations from the mean calculated and interpreted?
Exercises 1–4
A consumers’ organization is planning a study of the various brands of batteries that are available.
As part of its planning, it measures lifetime (i.e., how long a battery can be used before it must be
replaced) for each of six batteries of Brand A and eight batteries of Brand B. Dot plots showing
the battery lives for each brand are shown below.
1. Does one brand of battery tend to last longer, or are they roughly the same? What
calculations could you do in order to compare the battery lives of the two brands?
2. Do the battery lives tend to differ more from battery to battery for Brand A or for
Brand B?
3. Would you prefer a battery brand that has battery lives that do not vary much from
battery to battery? Why or why not?
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The table below shows the lives (in hours) of the Brand A batteries.
Life (Hours)
83
94
96
106
113
114
Deviation from the Mean
4. Calculate the deviations from the mean for the remaining values, and write your answers in
the appropriate places in the table.
The table below shows the battery lives and the deviations from the mean for Brand B.
Life (Hours)
73
76
92
94
110
117
118
124
Deviation from the
Mean
−27.5
−24.5
−8.5
−6.5
9.5
16.5
17.5
23.5
Exercises 5–10
The lives of five batteries of a third brand, Brand C, were determined. The dot plot below shows
the lives of the Brand A and Brand C batteries.
5. Which brand has the greater mean battery life? (You should be able to answer this
question without doing any calculations.)
6. Which brand shows greater variability?
7. Which brand would you expect to have the greater deviations from the mean (ignoring the
signs of the deviations)?
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The table below shows the lives for the Brand C batteries.
Life (Hours)
115
119
112
98
106
Deviation from the
Mean
8. Calculate the mean battery life for Brand C. (Be sure to include a unit in your answer.)
9. Write the deviations from the mean in the empty cells of the table for Brand C.
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Lesson 6: Interpreting the Standard Deviation
Essential Question: How is the standard deviation calculated using a calculator?
Example 1
Using your calculator, find the mean and standard deviation for the following set of data.
A set of eight men have heights (in inches) as shown below.
67.0 70.9 67.6 69.8 69.7 70.9 68.7 67.2
Indicate the mean and standard deviation you obtained from your calculator to the nearest
hundredth.
Mean: ____________________
Standard Deviation: ____________________
Exercise 1
The heights (in inches) of nine women are as shown below.
68.4 70.9 67.4 67.7 67.1 69.2 66.0 70.3 67.6
Use the statistical features of your calculator or computer software to find the mean and the
standard deviation of these heights to the nearest hundredth.
Mean: ____________________
Standard Deviation: ____________________
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Exploratory Challenge/Exercises 2–5
2. A group of people attended a talk at a conference. At the end of the talk, ten of the
attendees were given a questionnaire that consisted of four questions. The questions were
optional, so it was possible that some attendees might answer none of the questions, while
others might answer 1, 2, 3, or all 4 of the questions (so, the possible numbers of questions
answered are 0, 1, 2, 3, and 4).
Suppose that the numbers of questions answered by each of the ten people were as shown in
the dot plot below.
Use the statistical features of your calculator to find the mean and the standard deviation of
the data set.
Mean: ____________________
Standard Deviation: ____________________
3. Suppose the dot plot looked like this:
a. Use your calculator to find the mean and the standard deviation of this distribution.
b. Remember that the size of the standard deviation is related to the size of the deviations
from the mean. Explain why the standard deviation of this distribution is greater than
the standard deviation in Exercise 2.
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4. Suppose that all ten people questioned answered all four questions on the questionnaire.
a.
What would the dot plot look like?
b. What is the mean number of questions answered?
c.
What is the standard deviation?
5. Continue to think about the situation previously described where the numbers of questions
answered by each of ten people was recorded.
a.
Draw the dot plot of the distribution of possible data values that has the largest possible
standard deviation. (There were ten people at the talk, so there should be ten dots in
your dot plot.) Use the scale given below.
b. Explain why the distribution you have drawn has a larger standard deviation than the
distribution in Exercise 4.
Lesson Summary

The mean and the standard deviation of a data set can be found directly using the statistical features of a
calculator.

The size of the standard deviation is related to the sizes of the deviations from the mean. Therefore, the
standard deviation is minimized when all the numbers in the data set are the same and is maximized
when the deviations from the mean are made as large as possible.
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BOX-PLOTS
Measures of central tendency describe how data tends toward one value. You may also
need to know how data is spread across several values.
QUARTILES: Divide data into FOUR equal parts. Each quartile contains one-fourth (25%)
of the values in the set.
A BOX-PLOT can be used to show how the values in a data set are distributed. The
minimum is the least value that is not an outlier. The maximum is the greatest value that
is not an outlier. You need five values to make a box-and-whisker plot:
MINIMUM
LOWER QUARTILE (Q1)
MEDIAN
UPPERQUARTILE (Q3)
MAXIMUM
FOLLOW THESE STEPS TO CREATE A BOX-PLOT
STEP #1: On Calculator, Select “1: New Document”, then “4: Add Lists…”.
STEP #2: Enter your data in column “A”.
STEP #3: Press “menu”, then “4: Statistics”, then “1: stat calcs…”, then “1: one variable…”
STEP #4: Select “OK” twice.
STEP #5 Scroll down to find minX, Q1. Med, Q3 & maxX, then copy down the values to
create your box plot.
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PRACTICE:
(1) The numbers of runs scored by a softball team in 19 games are given. Use the data to
make a box-plot
2, 3, 3, 3, 4, 4, 4, 5, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 14
a) What is the interquartile range?
INTERQUARTILE RANGE: The difference between the upper and lower quartiles. _____
- _____
b) What is the median?
c) What percent of the data does the box part of the box plot capture?
d) What percent of the data falls between the minimum value and Q1?
e) What percent of the data falls between Q3 and the maximum value?
f) What is the range?
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(2) Use the data to make a box-plot
11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 17, 18, 18, 19, 22, 23
a) What is the interquartile range?
b) What is the median?
c) What percent of the data does the box part of the box plot capture?
d) What percent of the data falls between the minimum value and Q1?
e) What percent of the data falls between Q3 and the maximum value?
f) What is the range?
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(3) In the 2004 Olympic Games in Athens, the following results occurred for the men’s pole
vault finals:
5.95, 5.90, 5.85, 5.80, 5.75, 5.75, 5.65, 5.64, 5.55, 5.55, 5.55, 5.50
a.) Find the mean, median, mode and range of this data set.
b.) The gold medal was won by Timothy Mack of the United States. What was his height in
the pole vault event?
c.) Create a box-plot from this data
d) What is the interquartile range?
e) What percent of the data does the box part of the box plot capture?
f) What percent of the data falls between the minimum value and Q1?