1. Numerically Summarizing Data
Chapter 3
Numerically Summarizing Data
3.1
3.2
3.3
3.4
3.5

2. Measures of Central Tendency
Section
3.1
Measures of Central Tendency
3.1
3.2
3.3
3.4
3.5

3. 3.1 Objectives
Determine the arithmetic mean of a variable from raw data
Determine the median of a variable from raw data
Explain what it means for a statistic to be resistant
Determine the mode of a variable from raw data 3

4. Objective 1 Determine the Arithmetic Mean of a Variable from Raw Data
The arithmetic mean of a variable is computed by adding all the values of the variable in the data set and dividing by the number of observations. 4

5. The population mean is a parameter.
The population arithmetic mean, μ (pronounced “mew”), is computed using all the individuals in a population.
The population mean is a parameter.
The sample arithmetic mean, (pronounced “x-bar”), is computed using sample data.
The sample mean is a statistic. 5

6. EXAMPLE Computing a Population Mean and a Sample Mean
The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company.
23, 36, 23, 18, 5, 26, 43
Compute the population mean of this data. 6

7. Objective 2 Determine the Median of a Variable from Raw Data
The median of a variable is the value that lies in the middle of the data when arranged in ascending order.
We use M to represent the median. 7

8. Determine the median of this data.
EXAMPLE Computing a Median of a Data Set with an Odd Number of Observations
The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company.
23, 36, 23, 18, 5, 26, 43
Determine the median of this data.
Step 1: 5, 18, 23, 23, 26, 36, 43
Step 2: There are n = 7 observations.
5, 18, 23, 23, 26, 36, 43
M = 23 8

9. There are 8 observations.
EXAMPLE Computing a Median of a Data Set with an Even Number of Observations
Suppose the start-up company hires a new employee. The travel time of the new employee is 70 minutes. Determine the median of the “new” data set.
23, 36, 23, 18, 5, 26, 43, 70
Step 1: 5, 18, 23, 23, 26, 36, 43, 70
There are 8 observations.
5, 18, 23, 23, 26, 36, 43, 70 9

10. Objective 3 Explain What it Means for a Statistic to Be Resistant
A numerical summary of data is said to be resistant if extreme values (very large or small) relative to the data do not affect its value substantially. 10

11. 11

12. Mean before new hire: 24.9 minutes Median before new hire: 23 minutes
EXAMPLE Computing a Median of a Data Set with an Even Number of Observations
The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company.
23, 36, 23, 18, 5, 26, 43
Suppose a new employee is hired who has a 130 minute commute. How does this impact the value of the mean and median?
Mean before new hire: 24.9 minutes Median before new hire: 23 minutes
Mean after new hire: 38 minutes Median after new hire: 24.5 minutes
After the new hire, the data will be right skewed. 12

13. EXAMPLE Describing the Shape of the Distribution
The following data represent the asking price of homes for sale in Lincoln, NE.
79,995
128,950
149,900
189,900
99,899
130,950
151,350
203,950
105,200
131,800
154,900
217,500
111,000
132,300
159,900
260,000
120,000
134,950
163,300
284,900
121,700
135,500
165,000
299,900
125,950
138,500
174,850
309,900
126,900
147,500
180,000
349,900
Source: 13

14. Therefore, we would conjecture that the distribution is skewed right.
Find the mean and median. Use the mean and median to identify the shape of the distribution. Verify your result by drawing a histogram of the data.
The mean asking price is $168,320 and the median asking price is $148,700.
Therefore, we would conjecture that the distribution is skewed right. 14

15. 15

16. Objective 4 Determine the Mode of a Variable from Raw Data
The mode of a variable is the most frequent observation of the variable that occurs in the data set.
A set of data can have no mode, one mode, or more than one mode.
If no observation occurs more than once, we say the data have no mode. 16

17. EXAMPLE Finding the Mode of a Data Set
The data on the next slide represent the Vice Presidents of the United States and their state of birth. Find the mode. 17

18. Joe Biden
Pennsylvania
The mode is New York. 18

19. Tally data to determine most frequent observation19

20. Summary 3.1 Objectives
Determine the arithmetic mean of a variable from raw data
Determine the median of a variable from raw data
Explain what it means for a statistic to be resistant
Determine the mode of a variable from raw data 20

21. Measures of Dispersion
Section
3.2
Measures of Dispersion

22. 3.2 Objectives Determine the range of a variable from raw data
Determine the standard deviation and variance of a variable from raw data
Use the Empirical Rule to describe data that are bell shaped
Use Chebyshev’s Inequality to describe any data set 22

23. (a) What was the mean wait time?
To order food at a McDonald’s restaurant, one must choose from multiple lines, while at Wendy’s Restaurant, one enters a single line. The following data represent the wait time (in minutes) in line for a simple random sample of 30 customers at each restaurant during the lunch hour. For each sample, answer the following:
(a) What was the mean wait time?
(b) Draw a histogram of each restaurant’s wait time.
(c ) Which restaurant’s wait time appears more dispersed? Which line would you prefer to wait in? Why? 23

24. Which restaurant’s wait time appears more dispersed?
Wait Time at Wendy’s
Wait Time at McDonald’s
(a) The mean wait time in each line is 1.39 minutes.
Which restaurant’s wait time appears more dispersed? 24

25. Objective 1 Determine the Range of a Variable from Raw Data
The range, R, of a variable is the difference between the largest data value and the smallest data values. That is,
Range = R = Largest Data Value – Smallest Data Value 25

26. EXAMPLE Finding the Range of a Set of Data
The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company.
23, 36, 23, 18, 5, 26, 43
Find the range.
Range = 43 – 5
= 38 minutes 26

27. Objective 2
Determine the Standard Deviation and Variance of a Variable from Raw Data
The standard deviation of a variable is the square root of the mean of the squared deviations about the population mean.
The population standard deviation is represented by σ (lowercase Greek sigma).
The sample standard deviation is represented by s.
The variance of a variable is the square of the standard deviation.
The population variance is σ2 and the sample variance is s2. 27

28. Standard Deviation Formulas
Sample
Population
population variance = σ2
sample variance = s2

29. Compute the population standard deviation of this data.
EXAMPLE Computing a Population Standard Deviation
The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company.
23, 36, 23, 18, 5, 26, 43
Compute the population standard deviation of this data. 29

30. xi μ
xi – μ
(xi – μ)2 23 36 18 5 26 43
Variance = 30

31. Using the computational formula, yields the same standard deviation.xi
(xi )2 23
529 36
1296 18
324 5 25 26
676 43
1849
Σ xi = 174
Σ (xi)2 = 5228
Variance = 31

32. Find the sample standard deviation.
EXAMPLE Computing a Sample Standard Deviation
Here are the results of a random sample taken from the travel times (in minutes) to work for all three employees of a start-up web development company:
5, 26, 36
Find the sample standard deviation. 32

33. xi 5 26
3.667 36
13.667
Variance = 33

34. xi (xi )2 Using the computational formula, yields the same result. 525 26
676 36
1296
Σ xi = 67
Σ (xi)2 = 1997
Variance = 34

35. Reminder of Wendy’s and McDonald’s Histogram
Which do we suspect will have a larger standard deviation?
Wait Time at Wendy’s
Wait Time at McDonald’s 35

36. EXAMPLE Comparing Standard Deviations
Determine the standard deviation waiting time for Wendy’s and McDonald’s. Which is larger? Why?
Don’t actually calculate the standard deviation
Wait Time at Wendy’s
Wait Time at McDonald’s
Sample standard deviation for Wendy’s: minutes
Sample standard deviation for McDonald’s: minutes
Recall from earlier that the data is more dispersed for McDonald’s resulting in a larger standard deviation. 36

37. Objective 3
Use the Empirical Rule to Describe Data That Are Bell Shaped
The Empirical Rule
If a distribution is roughly bell shaped, then
• Approximately 68% of the data will lie within 1 standard deviation of the mean.
• Approximately 95% of the data will lie within 2 standard deviations of the mean.
Approximately 99.7% of the data will lie within 3 standard deviations of the mean. 37

38. 34%
34%
0.15%
2.35%
2.35%
0.15%
13.5%
13.5% 38

39. EXAMPLE Using the Empirical Rule
The following data represent the serum HDL cholesterol of the 54 female patients of a family doctor. 39

40.
Determine the percentage of all patients that have serum HDL within 3 standard deviations of the mean according to the Empirical Rule.
(b) Determine the percentage of all patients that have serum HDL between 34 and 69.1 according to the Empirical Rule.
(c) Determine the actual percentage of patients that have serum HDL between 34 and 69.1.
According to the Empirical Rule, 99.7%
13.5% + 34% + 34% = 81.5%
45 out of the 54 or 83.3%
For part c) go back a slide and count how many out of 45 40

41. Objective 4 Use Chebyshev’s Inequality to Describe Any Set of Data
For any data set or distribution, at least
of the observations lie within k standard deviations of the mean, where k is any number greater than 1.
Note: We can also use Chebyshev’s Inequality based on sample data. 41

42. Chebychev’s Theorem If k = 2, what would Chebychev’s Theorem imply?
And, k = 3?
k = 3
k = 2:
In any data set, at least 75% of the data lie within 2 standard deviations of the mean.
In any data set, at least 88.9% of the data lie within 3 standard deviations of the mean.
Larson/Farber 4th ed.

43. Example 1: Using Chebychev’s Theorem
The age distribution for Florida is shown in the histogram. Apply Chebychev’s Theorem to the data using k = 2. What can you conclude?
k = 2
= 0.75
-10.4
14.4
39.2 64
88.8
At least 75% of the population of Florida is between 0 and 88.8 years old.
Larson/Farber 4th ed.

44. Example2 : Using Chebychev’s Theorem
The mean time in a women’s 400 meter dash is seconds, with a standard deviation of Apply Chebychev’s Theorem to estimate how many out of 52 runners will have a time between and seconds.
Finally: Ask, how many standard deviations from the mean?
k = 4
52.87
53.92
54.97
56.02
57.07
58.12
59.17
60.22
61.27
How many women of the 52 runners have a time between and sec?
15/16 of 52 = ≈ 48 runners (they will tell you to round down)
Larson/Farber 4th ed.

45. Summary 3.2 Objectives Determine the range of a variable from raw data
Determine the standard deviation and variance of a variable from raw data
Use the Empirical Rule to describe data that are bell shaped 68% - 95% %
Use Chebyshev’s Inequality to describe any data set 45

46. Measures of Central Tendency and Dispersion from Grouped Data
Section
3.3
Measures of Central Tendency and Dispersion from Grouped Data

47. 3.3 Objectives Approximate the mean of a variable from grouped data
Compute the weighted mean of a variable from grouped data
Approximate the standard deviation of a variable from grouped data 47

48. We have discussed how to compute descriptive statistics from raw data, but often the only available data have already been summarized in frequency distributions (grouped data). Although we cannot find exact values of the mean or standard deviation without raw data, we can approximate these measures using the techniques discussed in this section. 48

49. Objective 1 Approximate the Mean of a Variable from Grouped Data
Approximate the Mean of a Variable from a Frequency Distribution
Population Mean
Sample Mean
where xi is the midpoint or value of the ith class
fi is the frequency of the ith class 49

50. EXAMPLE Approximating the Mean from a Relative Frequency Distribution
The National Survey of Student Engagement is a survey that (among other things) asked first year students at liberal arts colleges how much time they spend preparing for class each week. The results from the 2007 survey are summarized below. Approximate the mean number of hours spent preparing for class each week.
Hours
1-5
6-10
11-15
16-20
21-25
26-30
31-35
Frequency
130
250
230
180
100 60 50
Source: 50

51. Time Frequency xi xi fi 1 - 5 130 3 390 6 - 10 250 8 2000 11 - 15 230
1 - 5
130 3
390
6 - 10
250 8
2000
230 13
2990
180 18
3240
100 23
2300
26 – 30 60 28
1680
31 – 35 50 33
1650 51

52. Objective 2
Compute the Weighted Mean 52

53. EXAMPLE Computed a Weighted Mean
Bob goes to the “Buy the Weigh” Nut store and creates his own bridge mix. He combines 1 pound of raisins, 2 pounds of chocolate covered peanuts, and 1.5 pounds of cashews. The raisins cost $1.25 per pound, the chocolate covered peanuts cost $3.25 per pound, and the cashews cost $5.40 per pound. What is the cost per pound of this mix. 53

54. Objective 3
Approximate the Standard Deviation of a Variable from Grouped Data 54

55. Population Standard Deviation Sample Standard Deviation
Approximate the Standard Deviation of a Variable from a Frequency Distribution
Population Standard Deviation
Sample
Standard Deviation
where xi is the midpoint or value of the ith class
fi is the frequency of the ith class 55

56. EXAMPLE Approximating the Mean from a Relative Frequency Distribution
The National Survey of Student Engagement is a survey that (among other things) asked first year students at liberal arts colleges how much time they spend preparing for class each week. The results from the 2007 survey are summarized below. Approximate the standard deviation number of hours spent preparing for class each week.
Hours
1-5
6-10
11-15
16-20
21-25
26-30
31-35
Frequency
130
250
230
180
100 60 50
Source: 56

57. L2 L1 L3
Time
Freq xi
1 - 5
130 3
–11.25
16,
6 - 10
250 8
–6.25
230 13
–1.25
180 18
3.75
100 23
8.75
26 – 30 60 28
13.75
11,343.75
31 – 35 50 33
18.75
17, 57

58. 3.3 Objectives Approximate the mean of a variable from grouped data
Compute the weighted mean
Approximate the standard deviation of a variable from grouped data 58

59. Measures of Position and Outliers
Section
3.4
Measures of Position and Outliers

60. 3.4 Objectives Determine and interpret z-scores Interpret percentiles
Determine and interpret quartiles
Determine and interpret the interquartile range
Check a set of data for outliers 60

61. Objective 1 Determine and Interpret z-scores
The z-score represents the distance that a data value is from the mean in terms of the number of standard deviations.
Population z-score
Sample z-score 61

62. EXAMPLE Using Z-Scores
The mean height of males 20 years or older is 69.1 inches with a standard deviation of 2.8 inches. The mean height of females 20 years or older is 63.7 inches with a standard deviation of 2.7 inches. Data is based on information obtained from National Health and Examination Survey. Who is relatively taller?
Kevin Garnett whose height is 83 inches or
Candace Parker whose height is 76 inches
Kevin Garnett’s height is 4.96 standard deviations above the mean. Candace Parker’s height is 4.56 standard deviations above the mean. Kevin Garnett is relatively taller. 62

63. Objective 2 Interpret Percentiles
The kth percentile, denoted, Pk , of a set of data is a value such that k percent of the observations are less than or equal to the value. 63

64. Interpret this admissions requirement.
EXAMPLE Interpret a Percentile
The Graduate Record Examination (GRE) is a test required for admission to many U.S. graduate schools. The University of Pittsburgh Graduate School of Public Health requires a GRE score no less than the 70th percentile for admission into their Human Genetics MPH or MS program. (Source:
Interpret this admissions requirement.
In general, the 70th percentile is the score such that 70% of the those who took the exam scored worse, and 30% of the individuals scores better. In order to be admitted to this program, an applicant must score as high or higher than 70% of the people who take the GRE. Put another way, the individual’s score must be in the top 30%. 64

65. Objective 3 Determine and Interpret Quartiles
Quartiles divide data sets into fourths, or four equal parts.
The 1st quartile, denoted Q1, divides the bottom 25% the data from the top 75%. Therefore, Q1 = P25.
The 2nd quartile divides the bottom 50% of the data from the top 50% of the data. Q2 = P50 = M
The 3rd quartile divides the bottom 75% of the data from the top 25% of the data. Q3 = P75 65

66. EXAMPLE Finding and Interpreting Quartiles
A group of Brigham Young University—Idaho students (Matthew Herring, Nathan Spencer, Mark Walker, and Mark Steiner) collected data on the speed of vehicles traveling through a construction zone on a state highway, where the posted speed was 25 mph. The recorded speed of 14 randomly selected vehicles is given below:
20, 24, 27, 28, 29, 30, 32, 33, 34, 36, 38, 39, 40, 40
Find and interpret the quartiles for speed in the construction zone.
The data values are already in the correct order
Q1 = 28
= P25
Q2 = 32.5
= M = P50
Q3 = 38
= P75 66

67. Objective 4 Determine and Interpret the Interquartile Range
The interquartile range, IQR, is the range of the middle 50% of the observations in a data set.
IQR = Q3 – Q1 67

68. Determine and interpret the interquartile range of the speed data.
EXAMPLE Determining and Interpreting the Interquartile Range
Determine and interpret the interquartile range of the speed data.
20, 24, 27, 28, 29, 30, 32, 33, 34, 36, 38, 39, 40, 40
The range of the middle 50% of the speed of cars traveling through the construction zone is 10 miles per hour. 68

69. Suppose a 15th car travels through the construction zone at 100 miles per hour. How does this value impact the mean, median, standard deviation, and interquartile range?
Without 15th car
With 15th car
Mean
32.1 mph
36.7 mph
Median
32.5 mph
33 mph
Standard deviation
6.2 mph
18.5 mph
IQR
10 mph
11 mph 69

70. Objective 5 Check a Set of Data for Outliers
Checking for Outliers by Using Quartiles
Step 1 Determine Q1, Q3, and IQR.
Step 2 Determine the fences. Fences serve as cutoff points for determining outliers.
Lower Fence = Q1 – 1.5(IQR)
Upper Fence = Q (IQR)
Step 3 If a data value is less than the lower fence or greater than the upper fence, it is considered an outlier. 70

71. Check the speed data for outliers.
EXAMPLE Determining and Interpreting the Interquartile Range
Check the speed data for outliers.
20, 24, 27, 28, 29, 30, 32, 33, 34, 36, 38, 39, 40, 40
Step 1: Q1 = 28 mph, Q3 = 38 mph, and IQR = 10.
Step 2: The fences are
Lower Fence = Q1 – 1.5(IQR) = 28 – 1.5(10) = 13 mph
Upper Fence = Q (IQR) = (10) = 53 mph
Step 3: There are no values less than 13 mph or greater than 53 mph. Therefore, there are no outliers. 71

72. 3.4 Objectives Determine and interpret z-scores Interpret percentiles
Determine and interpret quartiles
Determine and interpret the interquartile range
Check a set of data for outliers 72

73. The Five-Number Summary and Boxplots
Section
3.5
The Five-Number Summary and Boxplots

74. 3.5 Objectives Compute the five-number summary
Draw and interpret boxplots 74

75. Objective 1 Compute the Five-Number Summary
The five-number summary of a set of data consists of the minimum value, Q1, the median, Q3, and the maximum data value. 75

76. EXAMPLE Obtaining the Five-Number Summary
Every six months, the United States Federal Reserve Board conducts a survey of credit card plans in the U.S. The following data are the interest rates charged by 10 credit card issuers randomly selected for the July 2005 survey. Determine the five-number summary of the data. 76

77. Pulaski Bank and Trust Company 6.5% Rainier Pacific Savings Bank 12.0%
EXAMPLE Obtaining the Five-Number Summary
Institution
Rate
Pulaski Bank and Trust Company
6.5%
Rainier Pacific Savings Bank
12.0%
Wells Fargo Bank NA
14.4%
Firstbank of Colorado
Lafayette Ambassador Bank
14.3%
Infibank
13.0%
United Bank, Inc.
13.3%
First National Bank of The Mid-Cities
13.9%
Bank of Louisiana
9.9%
Bar Harbor Bank and Trust Company
14.5%
Notice: the data is NOT in order
Source: 77

78. First, we write the data in ascending order:
EXAMPLE Obtaining the Five-Number Summary
First, we write the data in ascending order:
6.5%, 9.9%, 12.0%, 13.0%, 13.3%, 13.9%, 14.3%, 14.4%, 14.4%, 14.5%
Five-number Summary:
6.5% % % % % 78

79. Objective 2
Draw and Interpret Boxplots 79

80. Step 1 Determine the lower and upper fences.
Drawing a Boxplot
Step 1 Determine the lower and upper fences.
Step 2 Draw a number line long enough to include the maximum and minimum values. Insert vertical lines at Q1, M, and Q3. Enclose these vertical lines in a box.
Step 3 Label the lower and upper fences.
Step 4 Draw a line from Q1 to the smallest data value that is larger than the lower fence. Draw a line from Q3 to the largest data value that is smaller than the upper fence. These lines are called whiskers.
Step 5 Any data values less than the lower fence or greater than the upper fence are outliers and are marked with an asterisk (*). 80

81. * Construct a boxplot of the data from the last example.
EXAMPLE Obtaining the Five-Number Summary
Construct a boxplot of the data from the last example.
First, we write the data in ascending order:
6.5%, 9.9%, 12.0%, 13.0%, 13.3%, 13.9%, 14.3%, 14.4%, 14.4%, 14.5%
Five-number Summary: 6.5% % % % %
The interquartile range (IQR) is 14.4% - 12% = 2.4%.
The lower and upper fences are:
Lower Fence = Q1 – 1.5(IQR) = 12 – 1.5(2.4) = 8.4%
Upper Fence = Q (IQR) = (2.4) = 18.0%
Skewed:
Left
Discuss percentiles from the graph
Outliers are 6.5% *
–––––––– – 81

82. Use a boxplot and quartiles to describe the shape of a distribution.82

83. 3.5 Objectives Compute the five-number summary
Draw and interpret boxplots 83