1. Chapter 4 Displaying and Summarizing Quantitative Data
Display: Histograms, Stem and Leaf Plots
Numerical Summaries: Median, Mean, Quartiles, Standard Deviation

2. Relative Frequency Histogram of Exam Grades
.30
.25
.20
Relative frequency
.15
.10
.05
Sample histogram;
Do not confuse with bar chart 40 50 60 70 80 90
100
Grade 6

3. Frequency Histograms
Sample histogram

4. Frequency Histograms
A histogram shows three general types of information:
It provides visual indication of where the approximate center of the data is.
We can gain an understanding of the degree of spread, or variation, in the data.
We can observe the shape of the distribution.
Center, spread, shape

5. All 200 m Races 20.2 secs or less

6. Histograms Showing Different Centers
Same shape and spread

7. Histograms - Same Center, Different Spread

8. Frequency and Relative Frequency Histograms
identify smallest and largest values in data set
divide interval between largest and smallest values into between 5 and 20 subintervals called classes
* each data value in one and only one class
* no data value is on a boundary
Outline of procedure for constructing histogram
Construction of histogram typically automated
More important to understand what histogram tells you about the data rahter than construction minutiae.

9. How Many Classes?
Approximations; not one correct answer to number of intervals (classes) to use

10. Histogram Construction (cont.)
* compute frequency or relative frequency of observations in each class
* x-axis: class boundaries;
y-axis: frequency or relative frequency scale
* over each class draw a rectangle with height corresponding to the frequency or relative frequency in that class
Tedious, that’s why we automate

11. Example. Number of daily employee absences from work
106 obs; approx. no of classes=
{2(106)}1/3 = {212}1/3 = 5.69
1+ log(106)/log(2) = = 7.73
There is no single “correct” answer for the number of classes
For example, you can choose 6, 7, 8, or 9 classes; don’t choose 15 classes
Data on p. 5 of coursepack and in Excel file

12. EXCEL Histogram
Histogram produced by Excel (after a few modifications)

13. Absences from Work (cont.)
6 classes
class width: ( )/6=37/6=
6 classes, each of width 7; classes span 6(7)=42 units
data spans =37 units
classes overlap the span of the actual data values by 42-37=5
lower boundary of 1st class: (1/2)(5) units below 121 = = 118.5

14. EXCEL histogram

15. Grades on a statistics exam
Data:

16. Frequency Distribution of Grades
Class Limits Frequency
40 up to 50
50 up to 60
60 up to 70
70 up to 80
80 up to 90
90 up to 100
Total 2 6 8 7 5 30 4

17. Relative Frequency Distribution of Grades
Class Limits Relative Frequency
40 up to 50
50 up to 60
60 up to 70
70 up to 80
80 up to 90
90 up to 100
2/30 = .067
6/30 = .200
8/30 = .267
7/30 = .233
5/30 = .167 5

18. Relative Frequency Histogram of Grades
.30
.25
.20
Relative frequency
.15
.10
.05
Frequency and relative frequency histogram of same data will have the same shape. 40 50 60 70 80 90
100
Grade 6

19. Based on the histo-gram, about what percent of the values are between 47.5 and 52.5?
50% 5%
17%
30%
Countdown 10

20. Stem and leaf displays Have the following general appearance stem leaf
6 4
Probably haven’t seen one before now; here’s what one looks like.

21. Stem and Leaf Displays
Partition each no. in data into a “stem” and “leaf”
Constructing stem and leaf display
1) deter. stem and leaf partition (5-20 stems)
2) write stems in column with smallest stem at top; include all stems in range of data
3) only 1 digit in leaves; drop digits or round off
4) record leaf for each no. in corresponding stem row; ordering the leaves in each row helps

22. Example: employee ages at a small company
; stem: 10’s digit; leaf: 1’s digit
18: stem=1; leaf=8; 18 = 1 | 8
stem leaf
6 4
Constructing display;
Order the leaves in each stem row

23. Suppose a 95 yr. old is hired
stem leaf
6 4 7 8
9 5
Include all stems in the range of data

24. Number of TD passes by NFL teams: 2010 season (stems are 10’s digit)
leaf 3
011337 2 1 9
Smallest number? Largest number?

25. Pulse Rates n = 138 Ignore the circles showing in the graphic;
*Stem rows have leaves o through 4
Note that leaves in each row are ordered

26. Advantages/Disadvantages of Stem-and-Leaf Displays
1) each measurement displayed
2) ascending order in each stem row
3) relatively simple (data set not too large)
Disadvantages
display becomes unwieldy for large data sets

27. Population of 185 US cities with between 100,000 and 500,000
Multiply stems by 100,000
Each leaf should be just 1 digit to keep display simple; sometimes data has to be rounded or truncated
Note four rows for each stem value so display is not too wide.
3|6 = 360,000

28. Back-to-back stem-and-leaf displays
Back-to-back stem-and-leaf displays. TD passes by NFL teams: 1999, 2009 multiply stems by 10
1999
2009 2 4 6 3
0444
6655
011113 1
421
0122

29. Below is a stem-and-leaf display for the pulse rates of 24 women at a health clinic. How many pulses are between 67 and 77?
Stems are 10’s digits 4 6 8 10 12
Countdown 10

30. Interpreting Graphical Displays: Shape
Symmetric distribution
A distribution is symmetric if the right and left sides of the histogram are approximately mirror images of each other.
Skewed distribution
A distribution is skewed to the right if the right side of the histogram (side with larger values) extends much farther out than the left side. It is skewed to the left if the left side of the histogram extends much farther out than the right side.
Complex, multimodal distribution
Not all distributions have a simple overall shape, especially when there are few observations.

31. Shape (cont.)Female heart attack patients in New York state
Age: left-skewed
Cost: right-skewed

32. Shape (cont.): Outliers
An important kind of deviation is an outlier. Outliers are observations that lie outside the overall pattern of a distribution. Always look for outliers and try to explain them.
The overall pattern is fairly symmetrical except for 2 states clearly not belonging to the main trend. Alaska and Florida have unusual representation of the elderly in their population.
A large gap in the distribution is typically a sign of an outlier.
This is from the book. Imagine you are doing a study of health care in the 50 US states, and need to know how they differ in terms of their elderly population.
This is a histogram of the number of states grouped by the percentage of their residents that are 65 or over.
You can see there is one very small number and one very large number, with a gap between them and the rest of the distribution.
Values that fall outside of the overall pattern are called outliers. They might be interesting, they might be mistakes - I get those in my data from typos in entering RNA sequence data into the computer.
They might only indicate that you need more samples. Will be paying a lot of attention to them throughout class both for what we can learn about biology and also because they can cause trouble with your statistics.
Guess which states they are (florida and alaska).
Alaska
Florida

33. Center: typical value of frozen personal pizza? ~$2.65

34. Spread: fuel efficiency 4, 8 cylinders
4 cylinders: more spread
8 cylinders: less spread

35. Other Graphical Methods for Economic Data
Time plots
plot observations in time order, with time on the horizontal axis and the vari-able on the vertical axis
** Time series
measurements are taken at regular intervals (monthly unemployment, quarterly GDP, weather records, electricity demand, etc.)

36. Unemployment Rate, by Educational Attainment

37. Water Use During Super Bowl

38. Winning Times 100 M Dash

39. Annual Mean Temperature

40. End of Histograms, Stem and Leaf plots

41. Describing Distributions Numerically: Medians and Quartiles

42. 2 characteristics of a data set to measure
center
measures where the “middle” of the data is located
variability
measures how “spread out” the data is

43. The median: a measure of center
Given a set of n measurements arranged in order of magnitude,
Median= middle value n odd
mean of 2 middle values, n even
Ex. 2, 4, 6, 8, 10; n=5; median=6
Ex. 2, 4, 6, 8; n=4; median=(4+6)/2=5

44. Student Pulse Rates (n=62)
38, 59, 60, 60, 62, 62, 63, 63, 64, 64, 65, 67, 68, 70, 70, 70, 70, 70, 70, 70, 71, 71, 72, 72, 73, 74, 74, 75, 75, 75, 75, 76, 77, 77, 77, 77, 78, 78, 79, 79, 80, 80, 80, 84, 84, 85, 85, 87, 90, 90, 91, 92, 93, 94, 94, 95, 96, 96, 96, 98, 98, 103
Median = (75+76)/2 = 75.5

45. Medians are used often Year 2011 baseball salaries
Median $1,450,000 (max=$32,000,000 Alex Rodriguez; min=$414,000)
Median fan age: MLB 45; NFL 43; NBA 41; NHL 39
Median existing home sales price: May 2011 $166,500; May 2010 $174,600
Median household income (2008 dollars) 2009 $50,221; 2008 $52,029

46. The median splits the histogram into 2 halves of equal area

47. Examples
Example: n = 7
Example n = 7 (ordered):
Example: n = 8
Example n =8 (ordered)
m = 14.1
m = ( )/2 = 15.8

48. Below are the annual tuition charges at 7 public universities
Below are the annual tuition charges at 7 public universities. What is the median tuition?
4429
4960
4971
5245
5546
7586
5245
4965.5
4960
4971 5

49. Below are the annual tuition charges at 7 public universities
Below are the annual tuition charges at 7 public universities. What is the median tuition?
4429
4960
5245
5546
4971
5587
7586
5245
4965.5
5546
4971 6

50. The range and interquartile range
Measures of Spread
The range and interquartile range

51. Ways to measure variability
range=largest-smallest
OK sometimes; in general, too crude; sensitive to one large or small data value
The range measures spread by examining the ends of the data
A better way to measure spread is to examine the middle portion of the data

52. Quartiles: Measuring spread by examining the middle
The first quartile, Q1, is the value in the sample that has 25% of the data at or below it (Q1 is the median of the lower half of the sorted data).
The third quartile, Q3, is the value in the sample that has 75% of the data at or below it (Q3 is the median of the upper half of the sorted data).
Q1= first quartile = 2.3
m = median = 3.4
We are going to start out with a very general way to describe the spread that doesn’t matter whether it is symmetric or not - quartiles.
Just as the word suggests - quartiles is like quarters or quartets, it involves dividing up the distribution into 4 parts.
Now, to get the median, we divided it up into two parts. To get the quartiles we do the exact same thing to the two halves.
Use same rules as for median if you have even or odd number of observations.
Now, what an we do with these that helps us understand the biology of these diseases?
Q3= third quartile = 4.2

53. Quartiles and median divide data into 4 pieces
1/4
1/4
1/4
1/4
Q M Q3

54. Quartiles are common measures of spread
University of Southern California
UNC-CH

55. Rules for Calculating Quartiles
Step 1: find the median of all the data (the median divides the data in half)
Step 2a: find the median of the lower half; this median is Q1;
Step 2b: find the median of the upper half; this median is Q3.
Important:
when n is odd include the overall median in both halves;
when n is even do not include the overall median in either half.

56. Example 11
n = 10
Median
m = (10+12)/2 = 22/2 = 11
Q1 : median of lower half
Q1 = 6
Q3 : median of upper half
Q3 = 16

57. Pulse Rates n = 138
Median: mean of pulses in locations 69 & 70: median= (70+70)/2=70
Q1: median of lower half (lower half = 69 smallest pulses); Q1 = pulse in ordered position 35;
Q1 = 63
Q3 median of upper half (upper half = 69 largest pulses); Q3= pulse in position 35 from the high end; Q3=78

58. Below are the weights of 31 linemen on the NCSU football team
Below are the weights of 31 linemen on the NCSU football team. What is the value of the first quartile Q1? #
stem
leaf 2 22 55 4 23 57 6 24 26 7 25 10
257 12 27 59
(4) 28
1567 15 29
35599 30
333 31 45 5 32
155 33 1 34
287
257.5
263.5
262.5
Countdown 10

59. Interquartile range lower quartile Q1 middle quartile: median
upper quartile Q3
interquartile range (IQR)
IQR = Q3 – Q1
measures spread of middle 50% of the data

60. Example: beginning pulse rates
Q3 = 78; Q1 = 63
IQR = 78 – 63 = 15

61. Below are the weights of 31 linemen on the NCSU football team
Below are the weights of 31 linemen on the NCSU football team. The first quartile Q1 is What is the value of the IQR? #
stem
leaf 2 22 55 4 23 57 6 24 26 7 25 10
257 12 27 59
(4) 28
1567 15 29
35599 30
333 31 45 5 32
155 33 1 34
23.5
39.5 46
69.5
Countdown 10

62. 5-number summary of data
Minimum Q1 median Q3 maximum
Pulse data

63. End of Medians and Quartiles

64. Numerical Summaries of Symmetric Data.
Measure of Center: Mean
Measure of Variability: Standard Deviation

65. Symmetric Data Body temp. of 93 adults

66. Recall: 2 characteristics of a data set to measure
center
measures where the “middle” of the data is located
variability
measures how “spread out” the data is

67. Measure of Center When Data Approx. Symmetric
mean (arithmetic mean)
notation

69. Connection Between Mean and Histogram
A histogram balances when supported at the mean. Mean x = 140.6

70. Mean: balance point Median: 50% area each half right histo: mean 55
Mean: balance point Median: 50% area each half right histo: mean yrs, median 57.7yrs

71. Properties of Mean, Median
1. The mean and median are unique; that is, a data set has only 1 mean and 1 median (the mean and median are not necessarily equal).
2. The mean uses the value of every number in the data set; the median does not.

72. Example: class pulse rates

73. 2010, 2011 baseball salaries 2010 n = 845  = $3,297,828
median = $1,330,000
max = $33,000,000
2011
n = 848
 = $3,305,393
median = $1,450,000
max = $32,000,000

74. Disadvantage of the mean
Can be greatly influenced by just a few observations that are much greater or much smaller than the rest of the data

75. Mean, Median, Maximum BB Salaries

76. Skewness: comparing the mean, and median
Skewed to the right (positively skewed)
mean>median

77. Skewed to the left; negatively skewed
Mean < median
mean=78; median=87;

78. Symmetric data
mean, median approx. equal

79. Describing Variability of symmetric data

80. Describing Symmetric Data (cont.)
Measure of center for symmetric data:
Measure of variability for symmetric data?

81. Example
2 data sets:
x1=49, x2=51 x=50
y1=0, y2=100 y=50

82. On average, they’re both comfortable
0 100

83. Ways to measure variability
1. range=largest-smallest
ok sometimes; in general, too crude; sensitive to one large or small obs.

84. Previous Example

85. The Sample Standard Deviation, a measure of spread around the mean
Square the deviation of each observation from the mean; find the square root of the “average” of these squared deviations

86. Calculations … Mean = 63.4 Sum of squared deviations from mean = 85.2
Women height (inches)
Mean = 63.4
Sum of squared deviations from mean = 85.2
(n − 1) = 13; (n − 1) is called degrees freedom (df)
s2 = variance = 85.2/13 = 6.55 inches squared
s = standard deviation = √6.55 = 2.56 inches

87. 2. Then take the square root to get the standard deviation s.
We’ll never calculate these by hand, so make sure to know how to get the standard deviation using your calculator, Excel, or other software.
Mean
± 1 s.d.
2. Then take the square root to get the standard deviation s.
1. First calculate the variance s2.

88. Population Standard Deviation

89. Remarks
1. The standard deviation of a set of measurements is an estimate of the likely size of the chance error in a single measurement

90. Remarks (cont.)
2. Note that s and s are always greater than or equal to zero.
3. The larger the value of s (or s ), the greater the spread of the data.
When does s=0? When does s =0?
When all data values are the same.

91. Remarks (cont.)
4. The standard deviation is the most commonly used measure of risk in finance and business
Stocks, Mutual Funds, etc.
5. Variance
s2 sample variance
 2 population variance
Units are squared units of the original data
square $, square gallons ??

92. Remarks 6):Why divide by n-1 instead of n?
degrees of freedom
each observation has 1 degree of freedom
however, when estimate unknown population parameter like m, you lose 1 degree of freedom

93. Remarks 6) (cont.):Why divide by n-1 instead of n? Example
Suppose we have 3 numbers whose average is 9
x1= x2=
then x3 must be
once we selected x1 and x2, x3 was determined since the average was 9
3 numbers but only 2 “degrees of freedom”
Choose ANY values for x1 and x2
Since the average (mean) is 9, x1 + x2 + x3 must equal 9*3 = 27, so x3 = 27 – (x1 + x2)

94. Computational Example

95. class pulse rates

96. Review: Properties of s and s
s and s are always greater than or equal to 0
when does s = 0? s = 0?
The larger the value of s (or s), the greater the spread of the data
the standard deviation of a set of measurements is an estimate of the likely size of the chance error in a single measurement

97. Summary of Notation

98. End of Chapter 4