14
Descriptive Statistics
What a Data Set Tells Us
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section
Section14.2,
1.1, Slide
Slide11
14.2 Measures of Central
Tendency
• Compute the mean, median, and
mode of distributions.
• Find the five-number summary of
a distribution.
• Apply measures of central
tendency to compare data.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 14.2, Slide 2
The Mean and the Median
We use the Greek letter Σ (capital sigma) to indicate a
sum. For example, we will write the sum of the data
values 7, 2, 9, 4, and 10 by Σx = 7 + 2 + 9 + 4 + 10.
We represent the mean of a sample of a population by
x (read as “x bar”), and we will use the Greek letter μ
(lowercase mu) to represent the mean of the whole
population.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 14.2, Slide 3
The Mean and the Median
• Example: A car company has been studying its
safety record at a factory and found that the
number of accidents over the past 5 years was
25, 23, 27, 22, and 26. Find the mean annual
number of accidents for this 5-year period.
• Solution: We add the number of accidents and
divide by 5.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 14.2, Slide 4
The Mean and the Median
• Example: The water temperature at a point
downstream from a plant for the last 30 days is
summarized in the table. What is the mean
temperature for this distribution?
(continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 14.2, Slide 5
The Mean and the Median
• Solution: A third column is added to the table
that contains the products of the raw scores and
their frequencies.
The mean is
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 14.2, Slide 6
The Mean and the Median
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 14.2, Slide 7
The Mean and the Median
• Example: Listed are
the yearly earnings of
some celebrities.
a) What is the mean of
the earnings of the
celebrities on this list?
b) Is this mean an accurate measure of the
“average” earnings for these celebrities?
(continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 14.2, Slide 8
The Mean and the Median
• Solution (a): Summing the salaries and dividing
by 10 gives us
Solution (b): Eight of the celebrities have
earnings below the mean, whereas only two have
earnings above the mean. The mean in this
example does not give an accurate sense of
what is “average” in this set of data because it
was unduly influenced by higher earnings.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 14.2, Slide 9
The Mean and the Median
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 14.2, Slide 10
The Mean and the Median
• Example: The table lists the ages
at inauguration of the presidents who
assumed office between 1901 and
1993. Find the median age for this
distribution.
• Solution: We first arrange the ages
in order to get
There are 17 ages. The middle age is
the ninth, which is 55.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 14.2, Slide 11
The Mean and the Median
• Example: Fifty 32-ounce quarts of a particular
brand of milk were purchased and the actual
volume determined. The results of this survey
are reported in the table. What is the median for
this distribution?
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 14.2, Slide 12
The Mean and the Median
• Solution: Because the 50 scores are in
increasing order, the two middle scores are in
positions 25 and 26. We see that 29 ounces is in
position 25 and 30 ounces is in position 26. The
median for this distribution is
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 14.2, Slide 13
The Five Number Summary
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 14.2, Slide 14
The Five Number Summary
• Example: Consider the list of ages of the
presidents from a previous example:
42, 43, 46, 51, 51, 51, 52, 54, 55,
55, 56, 56, 60, 61, 61, 64, 69.
Find the following for this data set:
a) the lower and upper halves
b) the first and third quartiles
c) the five-number summary
(continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 14.2, Slide 15
The Five Number Summary
• Solution:
(a): Finding the median, we can identify the lower
and upper halves.
(b): The median of the lower half is
The median of the upper half is
(continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 14.2, Slide 16
The Five Number Summary
(c): The five number summary is
We represent the five-number summary by a
graph called a box-and-whisker plot.
(continued on next slide)
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Section 14.2, Slide 17
The Five Number Summary
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 14.2, Slide 18
The Five Number Summary
• Example: Find the mode for each data set.
a) 5, 5, 68, 69, 70
b) 3, 3, 3, 2, 1, 4, 4, 9, 9, 9
c) 98, 99, 100, 101, 102
d) 2, 3, 4, 2, 3, 4, 5
• Solution: a) The mode is 5.
b) There are two modes: 3 and 9.
In c) and d) there is no mode.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 14.2, Slide 19
Comparing Measures of Central
Tendency
• Example: Assume that you are negotiating the
contract for your union. You have gathered
annual wage data and found that three workers
earn $30,000, five workers earn $32,000, three
workers earn $44,000, and one worker earns
$50,000. In your negotiations, which measure of
central tendency should you emphasize?
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 14.2, Slide 20
Comparing Measures of Central
Tendency
• Solution:
Mode: $32,000
Median: $32,000
Mean:
The mean is $36,000.
To make the salaries appear as low as possible,
you would want to use the mode and median.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 14.2, Slide 21