Transcript Chapter Title - Mathematical sciences

14 Descriptive Statistics
14.1 Graphical Descriptions of Data
14.2 Variables
14.3 Numerical Summaries
14.4 Measures of Spread
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Excursions in Modern Mathematics, 7e: 14.1 - 2
Data Set
A data set is a collection of data values.
Statisticians often refer to the individual
data values in a data set as data points.
For the sake of simplicity, we will work with
data sets in which each data point consists
of a single number, but in more complicated
settings, a single data point can consist of
many numbers.
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Data Set
As usual, we will use the letter N to
represent the size of the data set. In reallife applications, data sets can range in size
from reasonably small (a dozen or so data
points) to very large (hundreds of millions of
data points), and the larger the data set is,
the more we need a good way to describe
and summarize it.
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Example 14.1 Stat 101 Test Scores
The day after the midterm exam in his Stat
101 class, Dr.Blackbeard has posted the
results online. The data set consists of N = 75
data points (the number of students who took
the test). Each data point (listed in the second
column) is a score between 0 and 25 (Dr.
Blackbeard gives no partial credit). Notice
that the numbers listed in the first column are
not data points–they are numerical IDs used
as substitutes for names to protect the
students’ rights of privacy.
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Excursions in Modern Mathematics, 7e: 14.1 - 5
Example 14.1 Stat 101 Test Scores
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Example 14.1 Stat 101 Test Scores
Like students everywhere, the students in the
Stat 101 class have one question foremost on
their mind when they look at the results: How
did I do? Each student can answer this
question directly from the table. It’s the next
question that is statistically much more
interesting. How did the class as a whole do?
To answer this last question, we will have to
find a way to package the results into a
compact, organized, and intelligible whole.
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Example 14.2 Stat 101 Test Scores:
Part 2
The first step in summarizing the information
in Table 14-1 is to organize the scores in a
frequency table such as Table 14-2. In this
table, the number below each score gives the
frequency of the score–that is, the number of
students getting that particular score.
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Example 14.2 Stat 101 Test Scores:
Part 2
We can readily see from Table 14-2 that there
was one student with a score of 1, one with a
score of 6, two with a score of 7, six with a
score of 8, and so on. Notice that the scores
with a frequency of zero are not listed in the
table.
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Excursions in Modern Mathematics, 7e: 14.1 - 9
Example 14.2 Stat 101 Test Scores:
Part 2
We can do even better. Figure 14-1 (next
slide) shows the same information in a much
more visual way called a bar graph, with the
test scores listed in increasing order on a
horizontal axis and the frequency of each test
score displayed by the height of the column
above that test score. Notice that in the bar
graph, even the test scores with a frequency
of zero show up–there simply is no column
above these scores.
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Excursions in Modern Mathematics, 7e: 14.1 - 10
Example 14.2 Stat 101 Test Scores:
Part 2
Figure 14-1
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Example 14.2 Stat 101 Test Scores:
Part 2
Bar graphs are easy to read, and they are a
nice way to present a good general picture of
the data. With a bar graph, for example, it is
easy to detect outliers–extreme data points
that do not fit into the overall pattern of the
data. In this example there are two obvious
outliers–the score of 24 (head and shoulders
above the rest of the class) and the score of 1
(lagging way behind the pack).
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Example 14.2 Stat 101 Test Scores:
Part 2
Sometimes it is more convenient to express
the bar graph in terms of relative frequencies
–that is, the frequencies given in terms of
percentages of the total population. Figure
14-2 shows a relative frequency bar graph for
the Stat 101 data set. Notice that we
indicated on the graph that we are dealing
with percentages rather than total counts and
that the size of the data set is N = 75.
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Excursions in Modern Mathematics, 7e: 14.1 - 13
Example 14.2 Stat 101 Test Scores:
Part 2
Figure 14-2
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Example 14.2 Stat 101 Test Scores:
Part 2
This allows anyone who wishes to do so to
compute the actual frequencies. For example,
Fig. 14-2 indicates that 12% of the 75
students scored a 12 on the exam, so the
actual frequency is given by 75  0.12 = 9
students.
The change from actual frequencies to
percentages (or vice versa) does not change
the shape of the graph–it is basically a
change of scale.
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Bar Graph versus Pictogram
Frequency charts that use icons or pictures
instead of bars to show the frequencies are
commonly referred to as pictograms. The
point of a pictogram is that a graph is often
used not only to inform but also to impress
and persuade, and, in such cases, a wellchosen icon or picture can be a more
effective tool than just a bar.
Here’s a pictogram displaying the same
data as in figure 14-2.
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Bar Graph versus Pictogram
Figure 14-3
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Example 14.3 Selling the XYZ
Corporation
This figure is a pictogram showing the growth
in yearly sales of the XYZ Corporation
between 2001 and 2006. It’s a good picture to
show at a
shareholders
meeting, but
the picture is
actually quite
misleading.
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Example 14.3 Selling the XYZ
Corporation
This figure shows a pictogram for exactly the
same data with a much more accurate and
sobering picture of how well the XYZ
Corporation
had been
doing.
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Example 14.3 Selling the XYZ
Corporation
The difference between the two pictograms
can be attributed to a couple of standard
tricks of the trade: (1) stretching the scale of
the vertical axis and (2) “cheating” on the
choice of starting value on the vertical axis.
As an educated consumer, you should always
be on the lookout for these tricks. In graphical
descriptions of data, a fine line separates
objectivity from propaganda.
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