1. Statistics: Concepts and Controversies
Chapter 11
Displaying Distributions with Graphs
Chapter 11
Chapter 11

2. Statistics: Concepts and Controversies
Recall
Categorical Variable: Just record group membership (using pie chart or bar graph)
Quantitative Variable: takes numerical values and it may has too many values
Chapter 11
Chapter 11

3. Comparing tuitions using a histogram
Statistics: Concepts and Controversies
Comparing tuitions using a histogram
Chicago State University charges in-state students $6834 per year.
There are 121 colleges and universities in Illinois. Their tuitions range from $1536 to $30,729.
The following is a histogram of the tuitions charged by 121 Illinois colleges and universities.
Chapter 11
Chapter 11

4. Figure 11.2 Histogram of the tuition and fees charged by 121 Illinois colleges and
universities in the 2004–2005 academic year. (Data from the Web site
This figure was created using
the SPSS software package.)

5. Comparing tuitions using a stemplot
Statistics: Concepts and Controversies
Comparing tuitions using a stemplot
Another method for displaying the same data is using stemplots.
For example, round $6834 to nearest hundreds, that is to 68. Put the thousands digit to the left of the line, and hang the hundreds one by one on these stems.
Chicago State is the red 8 on the 6 stem.
The stemplot shows more detail.
We can see that Chicago State’s tuition is 58th (from bottom) out of 121 colleges.
Chapter 11
Chapter 11

6. Figure 11. 7 Stemplot of the Illinois tuition and fee data
Figure Stemplot of the Illinois tuition and fee data. (Data from the Web site
This figure was created using
the Minitab software package.)

7. Statistics: Concepts and Controversies
How to make a histogram?
Chapter 11
Chapter 11

8. Statistics: Concepts and Controversies
How to make a histogram?
Step1: Divide the range of the data into classes of equal width.
The data in this table range from 6.3 to 17.0, so we choose
6.0 ≤ percentage over 65 < 7.0
7.0 ≤ percentage over 65 < 8.0 …
17.0 ≤ percentage over 65 < 18.0
Make sure that the classes are exclusive (no individual is in more than one class), and exhaustive (every individual appears in some class)
Chapter 11
Chapter 11

9. Statistics: Concepts and Controversies
Step 2: Count the number of individuals in each class.
Class
Count
6.0 to 6.9 1
7.0 to 7.9
8.0 to 8.9
9.0 to 9.9 3
10.0 to 10.9
11.0 to 11.9 7
12.0 to 12.9 14
13.0 to 13.9 15
14.0 to 14.9 5
15.0 to 15.9 2
16.0 to 16.9
17.0 to 17.9
Chapter 11
Chapter 11

10. Statistics: Concepts and Controversies
Step 3: Draw the histogram.
Mark on the horizontal axis the scale for the variable whose distribution is to be displayed.
The vertical axis contains the scale of counts.
There is no horizontal space between bars unless a class is empty.
Chapter 11
Chapter 11

11. Statistics: Concepts and Controversies
Interpreting Histograms
Chapter 11
Chapter 11

12. Statistics: Concepts and Controversies
Outliers
Extreme values, far from the rest of the data
May occur naturally
May occur due to error in recording
May occur due to error in measuring
Chapter 11
Chapter 11

13. Number of Books Read for Pleasure
Statistics: Concepts and Controversies
Number of Books Read for Pleasure
Chapter 11
Chapter 11

14. Statistics: Concepts and Controversies
Shape of the Data
Symmetric
bell-shaped
other symmetric shapes
Asymmetric
skewed to the right
skewed to the left
Unimodal, bimodal
Chapter 11
Chapter 11

15. Statistics: Concepts and Controversies
Chapter 11
Chapter 11

16. Symmetric Distributions Bell-Shaped
Statistics: Concepts and Controversies
Symmetric Distributions Bell-Shaped
Chapter 11
Chapter 11

17. Symmetric Distributions: Bell-Shaped
Statistics: Concepts and Controversies
Symmetric Distributions: Bell-Shaped
Chapter 11
Chapter 11

18. Symmetric Distributions Mound-Shaped
Statistics: Concepts and Controversies
Symmetric Distributions Mound-Shaped
Chapter 11
Chapter 11

19. Symmetric Distributions Uniform
Statistics: Concepts and Controversies
Symmetric Distributions Uniform
Chapter 11
Chapter 11

20. Asymmetric Distributions Skewed to the Left
Statistics: Concepts and Controversies
Asymmetric Distributions Skewed to the Left
Chapter 11
Chapter 11

21. Asymmetric Distributions Skewed to the Right
Statistics: Concepts and Controversies
Asymmetric Distributions Skewed to the Right
Chapter 11
Chapter 11

22. Asymmetric Distributions: Skewed to the Right
Statistics: Concepts and Controversies
Asymmetric Distributions: Skewed to the Right
Chapter 11
Chapter 11

23. Statistics: Concepts and Controversies
How to make a stemplot?
Chapter 11
Chapter 11

24. Statistics: Concepts and Controversies
Chapter 11
Chapter 11

25. Statistics: Concepts and Controversies
Chapter 11
Chapter 11

26. Statistics: Concepts and Controversies
Example: Weight Data
Following is a list of weights of a sample of students
Chapter 11
Chapter 11

27. Example: Weight Data; Frequency Table
Statistics: Concepts and Controversies
Example: Weight Data; Frequency Table
* Left endpoint is included in the group, right endpoint is not.
Chapter 11
Chapter 11

28. Example: Weight Data; Histogram
Statistics: Concepts and Controversies
Example: Weight Data; Histogram
100
120
140
160
180
200
220
240
260
280
Weight
* Left endpoint is included in the group, right endpoint is not.
Chapter 11
Chapter 11

29. Weight Data: Stemplot (Stem and Leaf Plot)
Statistics: Concepts and Controversies 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Weight Data: Stemplot (Stem and Leaf Plot) 5 2
Key
20|3 means 203 pounds
Stems = 10’s Leaves = 1’s 2
Chapter 11
Chapter 11

30. Weight Data: Stemplot (Stem and Leaf Plot)
Statistics: Concepts and Controversies
11 009
14 08
16 555
19 245
20 3
21 025
22 0 23 24 25
26 0
Weight Data: Stemplot (Stem and Leaf Plot)
Key
20|3 means 203 pounds
Stems = 10’s Leaves = 1’s
Chapter 11
Chapter 11

31. Statistics: Concepts and Controversies
Key Concepts
Displays (Stemplots & Histograms)
Graph Shapes
Symmetric
Skewed to the Right
Skewed to the Left
Outliers
Chapter 11
Chapter 11

32. Example: Shakespeare’s words
Statistics: Concepts and Controversies
Example: Shakespeare’s words
The following figure shows the distributions of lengths of words used in Shakespeare’s plays.
Notice that the vertical scale is not the count of words but the percentage of all words that have each length.
The curve is skewed to the right which is natural because short words are common and long ones are rare.
Chapter 11
Chapter 11

33. Statistics: Concepts and Controversies
Chapter 11
Chapter 11

34. Statistics: Concepts and Controversies
Exercise 11.3
Lightning strikes. The following figure comes from a study of lightning storms in Colorado. It shows the distribution of the hour of the day during which the first lightning flash for that day occurred.
Describe the shape, center, and spread of this distribution. Are there any outliers?
Chapter 11
Chapter 11

35. Statistics: Concepts and Controversies
Chapter 11
Chapter 11

36. Statistics: Concepts and Controversies
Exercise 11.4
Where do the young live? The following figure is a stemplot of the percentage of the residents aged under 18 in each of the 50 states. The stems are whole percentages and the leaves are tenths of a percent.
Chapter 11
Chapter 11

37. Figure 11.10 Stemplot of the percentage of each state’s residents
who are under 18 years old. (This figure was created using the
Minitab software package.)

38. Statistics: Concepts and Controversies
Utah has the largest percentage of young adults. What is the percentage for this state?
Ignoring Utah, describe the shape, center, and the spread of this distribution.
Is the distribution for young adults more or less spread out than the distribution in the figure for older adults?
Chapter 11
Chapter 11

39. Figure 11. 6 Making a stemplot of the data in Table 11. 1
Figure Making a stemplot of the data in Table Whole percents form the
stems, and tenths of a percent form the leaves. (This figure was created using the
Minitab software package.)

40. Answer for 11.4 Utah has 31.0% young adults
Without Utah , the distribution is roughly symmetric, centered at about 24.2%, spread from 21.2% to 27.6%
The distribution of young adult is less spread out than the distribution of older adults.
Chapter 11

41. Statistics: Concepts and Controversies
Exercise 11.5
Minority students in engineering. The following figure is a histogram of the number of minority students (black, Hispanic, Native American) who earned doctorate degrees in engineering from each of the 152 universities in the years 2000 through 2002.
Briefly describe the shape, center, and spread of this distribution.
Chapter 11
Chapter 11

42. Figure 11.11 The distribution of number of engineering doctorates earned by minority
students at 152 universities, 2000 to (Data from the 2003 National Science
Foundation Survey of Earned Doctorates, found at the Web site webcaspar.nsf.gov/.
This figure was created using the SPSS software package.)

43. Statistics: Concepts and Controversies
Exercise 11.6
Returns on common stocks. The total return on a stock is the change in its market price plus any dividend payments made. Total return is usually expressed as a percentage of the beginning price. The following figure is a histogram of the distribution of total returns for all 1528 common stocks listed on the New York Stock Exchange in one year.
Chapter 11
Chapter 11

44. Figure 11.12 The distribution of total returns for all New York Stock Exchange
common stocks in one year. (Based on J. K. Ford, “Diversification: how many
stocks will suffice?” American Association of Individual Investors Journal,
January 1990, pp. 14–16.)

45. Statistics: Concepts and Controversies
Describe the overall shape of the distribution of total returns.
What is the approximate center of this distribution? Approximately what were the smallest and largest total returns? (This describes the spread of the distribution.)
A return less than zero means that owners of stock lost money. About what percentage of all stocks lost money?
Chapter 11
Chapter 11

46. Statistics: Concepts and Controversies
Answer for 11.6
The distribution is roughly symmetric
The center is about 15%
The smallest return was between -70% and -60%, while the largest was between 100% and 110%
About ( =23%) of stocks lost money.
Chapter 11
Chapter 11

47. Statistics: Concepts and Controversies
Exercise 11.8
Automobile fuel economy. Government regulations require automakers to give the city and highway gas mileages for each model of car. The following table gives the highway mileage (miles per gallon) for 31 model year 2004 midsize cars. Make a stemplot of the highway gas mileages of these cars.
What can you say about the overall shape of the distribution?
Where is the center (the value such that half the cars have better gas mileage and half have worse gas mileage)?
Two of these cars are subject to the “gas guzzler tax” because of their low gas mileage. Which two?
Chapter 11
Chapter 11

48. Model MPG Acura RL 24 Lexus ES350 27 Bentley Arnage 15 Lexus GS460
BMW 535i 26
Lincoln Town Car 23
Buick Lacrosse
Maybach 57 16
Cadillac CTS
Maybach 62
Cadillac STS
Mazda 6
Chevy Malibu 30
Benz E350
Chrysler Sebring
Benz E550 22
Dodge Avenger
Nissan Maxima 25
Honda Accord 31
Pontiac Grand Prix 28
Hyundai Sonata
Rolls Royce Phantom 18
Infiniti G35
Saturn Aura
Infiniti M35
Toyota Camry
Jaguar S-Type R
Volkswagen Passat
Kia Optima
Volvo S80 AWD
Kia Spectra 32
Chapter 11