Chapter 12

Describing Distributions with Numbers

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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 of5 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

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xnx x x

nxn i

i

n

1 1

1 2

1

Variance

sn

x xii

n2

1

12

1

( )( )

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:

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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 five-

number 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 sitewww.collegeillinois.com/en/collegefunding/costs.htm. This figure was created usingthe 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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