Algebra unit 9.3

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Warm Up
Simplify each expression.
1. 2. 102 – 53
60 49
3. Use the data below to make a stem-and-leaf
plot.
7, 8, 10, 18, 24, 15,
17, 9, 12, 20, 25,
18, 21, 12

Objectives
Describe the central tendency of a
data set.
Create box-and-whisker plots.

Vocabulary
mean quartile
median interquartile range (IQR)
mode box-and-whisker plot
range
outlier

A measure of central tendency describes how
data clusters around a value.
• The mean is the sum of the values in the set
divided by the number of values in the set.
• The median the middle value when the values
are in numerical order, or the mean of the two
middle values if there are an even number of
values.
• The mode is the value or values that occur
most often. There may be one mode or more
than one mode. If no value occurs more often
than another, we say the data set has no mode.

The range of a set of data is the difference
between the least and greatest values in the
set. The range describes the spread of the data.

Example 1A: Finding Mean, Median, Mode, and Range
of a Data Set
Find the mean, median, mode, and range of the
data set.
The number of hours students spent on a
research project: 2, 4, 10, 7, 5
median: 2, 4, 5, 7, 10
The median is 5.
mode: none
range: 10 – 2 = 8
Write the data in numerical
order.
Add all the values and divide
by the number of values.
There are an odd number of
values. Find the middle
value.
No value occurs more than
once.
mean:

Example 1B: Finding Mean, Median, Mode, and Range
of a Data Set
Find the mean, median, mode, and range of
each data set.
The weight in pounds of six members of a
basketball team: 161, 156, 150, 156, 150, 163
order.
mean:
There are an even
number of values.
Find the mean of the
two middle values.
median: 150, 150, 156, 156, 161, 163
The median is 156.

Example 1B Continued
150, 150, 156, 156, 161, 163
modes: 150 and 156 150 and 156 both occur
more often than any other
range: 163 – 150 = 13 value.

Check It Out! Example 1a
data set.
8, 8, 14, 6 Write the data in numerical
order.
mean:
median: 6, 8, 8, 14
The median is 8.
mode: 8
range: 14 – 6 = 8
There are an even number of
values. Find the mean of
the two middle values.
8 occurs more than any
other value.

Check It Out! Example 1b
data set.
1, 5, 7, 2, 3
order.
There are an odd number of
values. Find the middle
value.
No value occurs more than
once.
mean:
median: 1, 2, 3, 5, 7
The median is 3.
mode: none
range: 7 – 1 = 6

Check It Out! Example 1c
data set.
12, 18, 14, 17, 12, 18
order.
median: 12, 12, 14, 17, 18, 18
There are an even number of
values. Find the mean of
the two middle values.
The median is 15 .
mean:

Check It Out! Example 1c Continued
data set.
12, 12, 14, 17, 18, 18
mode: 12, 18 12 and 18 both occur more
often than any other value.
range: 18 – 12 = 6

A value that is very different from other values in
the set is called an outlier. In the data below, one
value is much greater than the other values. This
causes the mean to be greater than all of the
other data values. In this case, either the median
or mode would better describe the data.

Example 2: Choosing a Measure of Central Tendency
Rico scored 74, 73, 80, 75, 67, and 55 on six
history tests. Use the mean, median, and mode
of his scores to answer each question.
mean ≈ 70.7 median = 73.5 mode = none
A. Which value gives Rico’s test average?
The average of Rico’s scores is the mean, 70.7.
B. Which values best describes Rico’s scores?
Median; most of his scores are closer to 73.5
than to 70.6.
The mean is lower than most of Rico’s scores
because he scored a 55 on one test. Since there is
no mode, it is not a good description of the data.

Check It Out! Example 2
Josh scored 75, 75, 81, 84, and 85 on five
tests. Use the mean, median, and mode of
his scores to answer each question.
mean = 80 median = 81 mode = 75
a. Which value describes the score Josh received
most often?
Josh has two scores of 75 which is the mode.
b. Which value best describes Josh’s scores?
Explain.
The median best describes Josh’s scores. The
mode is his lowest score, and the mean is
lowered by the two scores of 75.

Measures of central tendency describe how data
tends toward one value. You may also need to
know how data is spread out across several
values.
Quartiles divide a data set into four equal parts.
Each quartile contains one-fourth of the values in
the set. The interquartile range (IQR) is the
difference between the upper and lower
quartiles. The IQR represents the middle half of
the data.

A box-and-whisker 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: the minimum, first quartile, median,
third quartile, and maximum.

Helpful Hint
Mathematically, any value that is 1.5(IQR) less
than the first quartile or 1.5(IQR) greater than
the third quartile is an outier.

Example 3: Sports Application
The number of runs scored by a softball team
at 19 games is given. Use the data to make a
box-and-whisker plot.
3, 8, 10, 12, 4, 9, 13, 20, 12, 15, 10, 5, 11,
5, 10, 6, 7, 6, 11
Step 1 Order the data from least to greatest.
3, 4, 5, 5, 6, 6, 7, 8, 9, 10, 10, 10, 11, 11, 12,
12, 13, 15, 20
Step 2 Identify the five needed values and
determine whether there are any outliers.

Example 3 Continued
3, 4, 5, 5, 6, 6, 7, 8, 9, 10, 10, 10, 11, 11, 12, 12, 13, 15, 20
Q1
6
Q3
12
Q2
10
Minimum
3
Maximum
20
IQR: 12 – 6 = 6 1.5(6) = 9
6 – 9 = –3 12 + 9 = 21
No values are less than –3 or greater than 21, so
there are no outliers.

Example 3 Continued
First quartile Third quartile
Minimum Maximum
Median
● ● ● ● ●
0 8 16 24
Half of the scores are between 6 and 12 runs per
game. One-fourth of the scores are between 3 and
6. The greatest score earned by this team is 20.

Writing Math
An outlier is represented on a box-and-whisker
plot by a point that is not connected to the box
by whiskers.

Check It Out! Example 3
Use the data to make a box-and-whisker plot.
13, 14, 18, 13, 12, 17, 15, 12, 13, 19, 11, 14,
14, 18, 22, 23
Step 1 Order the data from least to greatest.
11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 17, 18, 18,
19, 22, 23
Step 2 Identify the five needed values and
determine whether there are any outliers.

11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 17, 18, 18, 19, 22, 23
Q1
13
Q3
18
Q2
14
Minimum
11
Maximum
23
Check It Out! Example 3 Continued
IQR: 18 – 13 = 5 1.5(5) = 7.5
13 – 7.5 = 5.5 18 + 7.5 = 25.5
No values are less than 5.5 or greater than 25.5, so
there are no outliers.

Check It Out! Example 3 Continued
First quartile Third quartile
Minimum Maximum
8 16 24
Half of the data are between 13 and 18. One-fourth
of the data are between 11 and 13. The
greatest value is 23.
Median
• • • • ●

1. Find the mean, median, mode, and range of the
data set.
Lesson Quiz: Part I
The number of hours Gerald mowed lawns in
one week: 7, 3, 5, 4, 5
mean: 4.8; median: 5; mode: 5; range: 4

Lesson Quiz: Part II
The following list gives times of Tara’s one-way
ride to school (in minutes) for one week:
12, 23, 13, 14, 13. Use the mean, median, and
mode of her times to answer each question.
mean = 15 median = 13 mode = 13
2. Which value describes the time that occurred
most often? mode, 13
3. Which value best describes Tara’s ride time?
Explain.
Median or mode: 13; 13 occurred twice,
and most times are near this value.

Lesson Quiz: Part III
4. The number of inches of snow that fell during
the last 8 winters in one city are given. Use the
data to make a box-and-whisker plot.
25, 17, 14, 27, 20, 11, 29, 32
11
15.5 22.5 28
32

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Algebra unit 9.3

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Algebra unit 9.3