Integration: Statistics

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7-7 Integration Statistics Box-and-Whisker
Plots
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In 1996, the site of the Summer Games of the XXVI
Olympiad was Atlanta, Georgia. The table shows
the number of gold medals won by the top 10
medal-winning teams.
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We can describe these data using the mean,
median, and mode. We can also use the median,
along with the quartiles and interquartile range,
to obtain a graphic representation of the data.
A type of diagram, of graph, that shows quartiles
and extreme values of data is called a
Box-and-Whisker Plot.
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?
Suppose we wanted to make a box-and-whisker plot
of the numbers of gold medals won by each of the
nations in the table. First, arrange the data in
numerical order. Next, compute the median and
quartiles. Also, identify the extreme values.
7 9 9 9 13 15 16 20 26 44
Median (Q2)
The median for this set of data is the average of
the eighth and ninth values.
Median 13 15 or 14 2
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? ?
Recall that the Lower quartile (Q1) is the
median of the lower half of the distribution of
values. The upper quartile (Q3) is the median
of the upper half of the data.
7 9 9 9 13 15 16 20 26 44

Q1
Q3
Q19 Q320
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The extreme values are the least value (LV),7,
and the greatest value (GV), 44.
Now we have the information we need to draw a
box-and-whisker plot.
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Step 1 Draw a number line. Assign a scale to
the number line that includes the extreme values.
Plot dots to represent the extreme values (LV
and GV) the upper and lower quartile points (Q3
and Q1) and the median (Q2).
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14
22
30
38
44
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Step 2 Draw a box to designate the data falling
between the upper and lower quartiles. Draw a
vertical line through the point representing the
median. Draw a segment from the lower quartile
to the least value and one from the upper
quartile to the greatest value. These segments
are the whiskers of the plot.
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14
22
30
38
44
Even though the whiskers are different lengths,
each whisker contains at least one fourth of the
data while the box contains one half of the data.
Compound inequalities can be used to describe
the data in each fourth. Assume that the
replacement set for x is the set of data.
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Step 3 Before finishing the box-and-whisker
plot, check for outliers. In Lesson 5-7 you
learned that an outlier is any element of the set
of data that is more than 1.5 interquartile
ranges above the upper quartile of below the
lower quartile. Recall that the interquartile
range (IQR) is the difference between the upper
and lower quartiles, or in this case, 20-9 or 11.
6
14
22
30
38
44
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Step 4 If x is an outlier in this set of data,
then the outliers can be described as (x / lt
-7.5or xgt36.5). In this case, there are no data
less than 7.5. However, 44 is greater than
36.5, so it is an outlier. We now need to revise
the box-and-whisker plot Outliers are plotted as
isolated points, and the right whisker is
shortened to stop at 26.
6
14
22
30
38
44
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EXAMPLE

• Refer to the application at the beginning of the
  lesson. Use the box-and-whisker plot for the
  gold medals to answer each question.
• What percent of the teams won between 14 and20
  gold medals?
• What does the box-and-whisker plot tell us about
  the upper half of the data compared to the lower
  half?
• a. The right half of the box in the plot
  indicates 25 of the values in the distribution.
  Since the box goes from 14 to 20, we know that
  25 of the teams won between 14 and 20 gold
  medals.
• b. The upper half of the data is spread out
  while the lower half is fairly clustered together.

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The End