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Chapter 12
Describing Distributions with
Numbers
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Chapter 12
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Median
• Example 1 data: 2 4 6
Median (M) = 4
• Example 2 data: 2 4 6 8
Median = 5 (avg. of 4 and 6)
• Example 3 data: 6 2 4
Median  2
(order the values: 2 4 6 , so Median = 4)
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Example: Finding the median
• Consider the following set of data:
16 19 24 25 25 33 33 34 34 37 37 40 42 45 45 46 46 49 73
• There are 19 observations, hence the median
is the one in the middle, the 10th one. Hence
the median of these data is 37.
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Example: Finding the median
• Now consider the following data where there
is an even number of observations:
3 9 9 22 29 32 32 33 39 39 42 49 52 58 65 70
• The ones in the middle are 33 and 39, and
hence the median is the average of these
values, 36.
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Quartiles
• Three numbers that divide the ordered data
into four equal-sized groups.
• Q1 has 25% of the data below it.
• Q2 has 50% of the data below it. (Median)
• Q3 has 75% of the data below it.
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Example: Finding the quartiles
• Consider the following set of data:
16 19 24 25 25 33 33 34 34 37 37 40 42 45 45 46 46 49 73
• We already know that the median is 37. To find the first quartile we
consider the first half of this data, namely,
16 19 24 25 25 33 33 34 34
• There are 9 observations here, so the median is the 5th one, namely, 25.
This is the first quartile.
• To find the third quartile we consider the second half of the original data,
that is,
37 40 42 45 45 46 46 49 73
• There are again 9 observations and the median is the one in the middle
which is 45, this is the third quartile.
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Example: Finding the quartiles
• Next, consider the following set of data:
3 9 9 22 29 32 32 33 39 39 42 49 52 58 65 70
• We already know that the median is 36. To find the first quartile we
consider the first half of this data, namely,
3 9 9 22 29 32 32 33
• There are 8 observations here, so the median is the average of the 4th and
5th one, namely, 25.5. This is the first quartile.
• To find the third quartile we consider the second half of the original data,
that is,
39 39 42 49 52 58 65 70
• There are again 8 observations and the median is again the average of 4th
and 5th observations, that is 50.5, this is the third quartile.
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Example: Five-number summary and boxplot
• Here are Roger Maris’ home run counts for his 12 years in the Major
Leagues, arranged in order:
5 8 9 13 14 16 23 26 28 33 39 61
• Since there are an even number of observations, the median is the
average of two middle values, that is, (16+23)/2 = 19.5
• The first quartile is the median of
5 8 9 13 14 16
that is (9+13)/2 = 11
• The third quartile is the median of the second half,
23 26 28 33 39 61
that is (28+33)/2 = 30.5
• The minimum and maximum values are obviously 5 and 61, respectively.
• Hence a five-number summary of these data is
5 11 19.5 30.5 61
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Example: Five-number summary and boxplot
•
How does the boxplot of this distribution compare with those of the data
corresponding to Barry Bonds and Mark McGwire given in the following figure:
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Example: Education and Income
•
Interpret the following figure which contains the boxplots of income among adults
with different levels of education.
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Average or Mean
n
1
1
x   x 1  x 2  xn    xi
n
n i 1
Variance
n
1
2
2
s 
( xi  x )

( n  1) i 1
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Comparing the Mean & Median
• The mean and median of data from a
symmetric distribution should be close
together. The actual (true) mean and
median of a symmetric distribution are
exactly the same.
• In a skewed distribution, the mean is
farther out in the long tail than is the
median [the mean is “pulled” in the
direction of the possible outlier(s)].
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Example: Computing the mean and standard deviation
• Consider the following data
16 25 24 19 33 25 34 46 37 33 42 40 37 34 49 73 46 45 45
• The mean is
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Example: Computing the mean and standard deviation
• The previous figure shows Barry Bonds’s home run counts,
with their mean and distance of one observation from the
mean indicated.
• The idea behind the standard deviation is to average these 19
distances. To find the standard deviation we can use the
following table
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Example: Computing the mean and standard deviation
The average of these distances (the variance) is:
Notice that we “average” by dividing by one less than
the number of observations.
Finally, the standard deviation is computed by taking
the square root of the variation:
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Key Concepts
• Numerical Summaries
– Center (mean, median)
– Spread (variance, standard deviation, quartiles)
– Five-number summary & Boxplots
• Choosing mean versus median
• Choosing standard deviation versus fivenumber summary
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Exercise 12.7
• College tuitions. The following figure is a
stemplot of the tuition charged by 121
colleges in Illinois. The stems are thousands of
dollars and the leaves are hundreds of dollars.
• Find the five-number summary of Illinois
college tuitions.
• Would the mean tuition be clearly smaller
than the median, about the same, or larger?
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Figure 11.7 Stemplot of the Illinois tuition and fee data. (Data from the Web site
www.collegeillinois.com/en/collegefunding/costs.htm. This figure was created using
the Minitab software package.)
Exercise 12.11
• The richest 1%. The distribution of individual
incomes in the U.S. is strongly skewed to the
right. In 2004, the mean and median incomes
of the top 1% of Americans were $315,000
and $1,259,000.
• Which of these numbers is the mean?
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Exercise 12.31
• Raising Pay. A school system employs teachers at salaries
between $30,000 and $60,000. The teachers’ union and the
school board are negotiating the form of next year’s increase
in the salary schedule. Suppose that every teacher is given a
flat $1,000 raise.
• How much will the mean salary increase? The median salary?
• Will a flat $1,000 raise increase the spread as measured by
the distance between the quartiles?
• Will a flat $1,000 raise increase the spread as measured by
the standard deviation of the salaries?
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