1. Numerically Summarizing Data
Chapter 3
Numerically Summarizing Data

2. Measures of Central Tendency
Section
Measures of Central Tendency
3.1

3. The population mean is a parameter.
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.
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. 3

4. If x1, x2, …, xN are the N observations of a variable from a population, then the population mean, µ, is
If x1, x2, …, xn are the n observations of a variable from a sample, then the sample mean, , is 4

5. Compute the population mean of this data.
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. 5

6. EXAMPLE Computing a Population Mean and a Sample Mean6

7. The following data represent the exams scores of different students.
EXAMPLE Computing a Population Mean and a Sample Mean
The following data represent the exams scores of different students.
82, 77, 90, 71, 61, 68, 74, 84, 94, 88
Compute the population mean of this data. 7

8. We use M to represent the median.
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.
Steps in Finding the Median of a Data Set
Step 1 Arrange the data in ascending order.
Step 2 Determine the number of observations, n.
Step 3 Determine the observation in the middle of the data set. 8

9. Steps in Finding the Median of a Data Set
If the number of observations is odd, then the median is the data value exactly in the middle of the data set. That is, the median is the observation that lies in then (n + 1)/2 position.
If the number of observations is even, then the median is the mean of the two middle observations in the data set. That is, the median is the mean of the observations that lie in the n/2 position and the n/2 + 1 position. 9

10. 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.
M = 23
Step 3:
5, 18, 23, 23, 26, 36, 43 10

11. Step 2: There are n = 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
Step 2: There are n = 8 observations.
Step 3:
5, 18, 23, 23, 26, 36, 43, 70 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 12

13. 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. 13

14. 14

15. 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: 15

16. Find the mean and median
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. 16

17. 17

18. A set of data can have no mode, one mode, or more than one mode.
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. 18

19. Find the mode of each set: (a). 1, 2, 3, 4, 5, 5, 6, 6, 7, 8, 9, 10
EXAMPLE Finding the Mode of a Data Set
Find the mode of each set:
(a). 1, 2, 3, 4, 5, 5, 6, 6, 7, 8, 9, 10
(b). 11, 11, 12, 12, 13, 13, 13, 15, 15, 15, 15
(c). 1, 5, 10, 15, 16, 20, 21 19

20. Tally data to determine most frequent observation20

21. Measures of Dispersion
Section
Measures of Dispersion
3.2

22. Dispersion is the degree to which the data are spread out.
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

23. (b)
Percent 20
Percent 20 23

24. 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 24

25. The population standard deviation of a variable is the square root of the sum of squared deviations about the population mean divided by the number of observations in the population, N. That is, it is the square root of the mean of the squared deviations about the population mean.
The population standard deviation is symbolically represented by σ (lowercase Greek sigma). 25

26. where x1, x2, . . . , xN are the N observations in the population and μ is the population mean.26

27. A formula that is equivalent to the one on the previous slide, called the computational formula, for determining the population standard deviation is 27

28. 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. 28

29. xi μ
xi – μ
(xi – μ)2 23 36 18 5 26 43 29

30. xi (xi )2 Using the computational formula, yields the same result. 23
529 36
1296 18
324 5 25 26
676 43
1849
Σ xi = 174
Σ (xi)2 = 5228 30

31. The sample standard deviation, s, of a variable is the square root of the sum of squared deviations about the sample mean divided by n – 1, where n is the sample size.
where x1, x2, , xn are the n observations in the sample and is the sample mean. 31

32. A formula that is equivalent to the one on the previous slide, called the computational formula, for determining the sample standard deviation is 32

33. We call n – 1 the degrees of freedom because the first n – 1 observations have freedom to be whatever value they wish, but the nth value has no freedom. It must be whatever value forces the sum of the deviations about the mean to equal zero. 33

34. 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 seven employees of a start-up web development company:
5, 26, 36
Find the sample standard deviation. 34

35. xi 5 26
3.667 36
13.667 35

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

37. Basically the larger the standard deviation, the more dispersion the distribution has.
The variance of a variable is the square of the standard deviation. The population variance is σ2 and the sample variance is s2.
(NOTE: to find the variance, all you have to do is square the standard deviation)

38. EXAMPLE Computing a Population Variance
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 and sample variance of this data. 38

39. EXAMPLE Computing a Population Variance
Recall that the population standard deviation (from slide #49) is σ = so the population variance is σ2 = minutes
and that the sample standard deviation is s = 12.26, so the sample variance is s2 = minutes 39

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

41. 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. 41

42. (Source: http://www.publichealth.pitt.edu/interior.php?pageID=101.)
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. 42

43. EXAMPLE Interpret a Percentile
In general, the 70th percentile is the score such that 70% of the individuals 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%. 43

44. 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, the 1st quartile is equivalent to the 25th percentile.
The 2nd quartile divides the bottom 50% of the data from the top 50% of the data, so that the 2nd quartile is equivalent to the 50th percentile, which is equivalent to the median.
The 3rd quartile divides the bottom 75% of the data from the top 25% of the data, so that the 3rd quartile is equivalent to the 75th percentile. 44

45. Step 1 Arrange the data in ascending order.
Finding Quartiles
Step 1 Arrange the data in ascending order.
Step 2 Determine the median, M, or second quartile, Q2 .
Step 3 Divide the data set into halves: the observations below (to the left of) M and the observations above M. The first quartile, Q1 , is the median of the bottom half, and the third quartile, Q3 , is the median of the top half. 45

46. Find and interpret the quartiles for speed in the construction zone.
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. 46

47. Step 1: The data is already in ascending order.
EXAMPLE Finding and Interpreting Quartiles
Step 1: The data is already in ascending order.
Step 2: There are n = 14 observations, so the median, or second quartile, Q2, is the mean of the 7th and 8th observations. Therefore, M = 32.5.
Step 3: The median of the bottom half of the data is the first quartile, Q1.
20, 24, 27, 28, 29, 30, 32
The median of these seven observations is 28. Therefore, Q1 = 28. The median of the top half of the data is the third quartile, Q3. Therefore, Q3 = 38. 47

48. Interpretation:
25% of the speeds are less than or equal to the first quartile, 28 miles per hour, and 75% of the speeds are greater than 28 miles per hour.
50% of the speeds are less than or equal to the second quartile, 32.5 miles per hour, and 50% of the speeds are greater than 32.5 miles per hour.
75% of the speeds are less than or equal to the third quartile, 38 miles per hour, and 25% of the speeds are greater than 38 miles per hour. 48

49. The interquartile range, IQR, is the range of the middle 50% of the observations in a data set. That is, the IQR is the difference between the third and first quartiles and is found using the formula
IQR = Q3 – Q1 49

50. 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.
Q1 = Q3 = 38
The range of the middle 50% of the speed of cars traveling through the construction zone is 10 miles per hour. 50

51. 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 51

52. Example
Find the quartiles and interquartile range of the following numbers on collision coverage claims.
$6751
$9908
$3461
$2336
$21,147
$2332

53. Outliers To find outliers, determine the first and third quartile.
Compute the interquartile range IQR.
Determine the fences.
Lower Fence = Q1 – 1.5(IQR)
Upper Fence = Q (IQR)
If a data value is outside of the fence, it is an outlier.

54. Example
Find any outliers in the collision data:
$6751
$9908
$3461
$2336
$21,147
$2332

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

56. The five-number summary of a set of data consists of the smallest data value, Q1, the median, Q3, and the largest data value. We organize the five-number summary as follows: 56

57. 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. 57

58. 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%
Source: 58

59. 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%
The smallest number is 6.5%. The largest number is 14.5%. The first quartile is 12.0%. The second quartile is 13.6%. The third quartile is 14.4%.
Five-number Summary:
6.5% % % % % 59

60. Step 1 Determine the lower and upper fences.
Drawing a Boxplot
Step 1 Determine the lower and upper fences.
Lower Fence = Q1 – 1.5(IQR)
Upper Fence = Q (IQR)
where IQR = Q3 – Q1
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. 60

61. Drawing a Boxplot
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 (*). 61

62. EXAMPLE Constructing a Boxplot
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. Construct a boxplot of the data. 62

63. Pulaski Bank and Trust Company 6.5% Rainier Pacific Savings Bank 12.0%
EXAMPLE Constructing a Boxplot
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%
Source: 63

64. Lower Fence = Q1 – 1.5(IQR) = 12 – 1.5(2.4) = 8.4%
Step 1: 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% * [ ] 64

65. Example
Construct a boxplot of the following data about the amount of time in minutes it took men aged to finish a 5-km race:
19.95
23.25
23.32
25.55
25.83
26.28
42.47
28.58
28.72
30.18
30.35
30.95
32.13
49.17
33.23
33.53
36.68
37.05
37.43
41.42
54.63

66. Use a boxplot and quartiles to describe the shape of a distribution.
The interest rate boxplot indicates that the distribution is skewed left. 66