1. Chapter 3 Displaying Data
Stat-Slide-Show, Copyright by Quant Systems Inc.

2. Graphs advantages: Picture worth 1000 words?
They convey information immediately.
They provide a powerful means of emphasizing a message. As a presentation tool, they are unequaled.
They are persuasive. They often display the totality of the data.
They can often disclose hidden relationships, and provide the spark for creative thinking.
They engage the audience - they are more provocative, especially if they are in color.

3. Heart Rate (per min.) of 50 Students
Frequency Distribution
Heart Rate (per min.) of 50 Students
Frequency Distribution
Heart Rate
Number of
Students
57.50 to 67.5
67.51 to 77.5
77.51 to 87.5
87.51 to 97.5
97.51 to 107.5 3 13 29 4 1
3 - 3

4. Two main steps in the construction of a frequency distribution:
1. Choosing the classifications
Starting values, interval widths
2. Counting the number in each Class

5. Bar Chart
The bar chart is a simple graphical display in which the length of each bar corresponds to the number of observations in a category. (Len. prop to #)

6. The Aesthetics of Bar Chart Construction
Flat Bar Chart
3 - 6

7. Stacked Bar Chart
3 - 7

8. 3-D Bar Chart
3 - 8

9. Graphing Tricks Susan William Beth Rob 187 201 207 193
Salesperson
Total Sales (in thousands)
Susan
William
Beth
Rob
187
201
207
193
Graphing Tricks
3 - 9

10. Pie Charts
Flat Pie Chart
3-D Pie Chart

11. Frequency Distributions -- Quantitative Data

12. The Distribution of Letters in the English Language and the International Morse Code

13. Selecting the Number of Classes
Generally, fewer than four classes would be too much compression of the data, and greater than 20 classes provides too little summary information. [In my tests, I may ask 2 or 3 classes]

14. Determining the Class Width
Class endpoints with fractional values will make the graph slightly harder to digest.
If possible, try to keep the width to an integer value.
class width>=
largest value - smallest value
number of classes

15. Relative Frequency
The relative frequency of any class is the number of observations in the class divided by the total number of observations.
relative
frequency
number in class
total number of observations =

16. Relative
Frequency
Distribution
for Heart Rate Data

17. Cumulative Frequency
The cumulative frequency is the sum of the frequency of a particular class and all preceding classes. (“Less than Ogive” cumulative)

18. Cumulative Frequency Distribution for Heart Rate Data LowLim UppLim
L-thanCum
57.5
67.5 3
67.51
77.5 13 16
77.51
87.5 29 45
87.51
97.5 4 49
97.51
107.5 1 50
Total

19. Cumulative Relative Frequency
The cumulative relative frequency is the proportion of observations in a particular class and all preceding classes.
The last value is always unity (=1)

20. Cumulative Relative Frequency Distribution for Heart Rate Data

21. “Greater Than Ogive” Cumulative Frequency
The cumulative frequency is the sum of the frequency of a particular class and all succeeding classes. (“Greater than Ogive” cumulative)

22. GreaterThan Cumu-lativeFreqDistribution for Heart Rate Data
LowLim
UppLim
Freq
Gr-than Cum
57.5
67.5 3 50
67.51
77.5 13 47
77.51
87.5 29 34
87.51
97.5 4 5
97.51
107.5 1
Total

23. Histogram
A histogram is a graphical image of a frequency or relative frequency distribution in which the height of each bar corresponds to the frequency or relative frequency of the class.

24. GreaterThan Cumu-lativeFreqDistribution for Heart Rate Data

25. Histogram of Student Heart Rate Data3 13 29 4 1
Beats Per Minute

26. Frequency Polygon
A freq. polygon is usually drawn right on top of the freq. histogram. by joining the midpoints of consecutive pillars.
Take Care to include dummy intervals before drawing the freq. Polygon, find the midpoints of the dummy intervals and then join them

27. 3-D Histogram of Student Heart Rate Data
Beats Per Minute

28. Stem and Leaf Display
The stem and leaf display is a hybrid graphical method.
The display is similar to a histogram, but the data remains visible to the user.

29. Stem and Leaf Display Population
Consider the following data: 97, 99, 108, 110, and 111.
Table 1
data stem leaf
Stem and Leaf Display
Stem Leaves
09 | 7 9
10 | 8
11 | 0 1

30. Stem and Leaf Display
Consider the following data: 97, 99, 108, 110, and 111.
Table 2
data stem leaf
Stem and Leaf Display
Stem Leaves
0 |
1 |

31. Stem and Leaf Display
6 |
7 |
8 |
9 |
Stem
Leaves

32. Frequency distribution in social sciences
Largest value=98
Smallest Value=62
Desired Classes =4 (given to you in an exam, remind me if I forget to give it to you)
Starting Value for classes =60 (given to you).
Width  [98-62]/4=36/4=9
Width should be at least 9. Let us pick a round number a bit larger than 9, say 10 and try if it works.

33. Unclassified Data are Lists of numbers

34. Less-than and Greater-than ogives
classes
TallyMarks
Frequency
Cumulative
Greater Than
LessThan
Ogive
60-70
IIII/ 5 50
70-80
5,5,5,I 16 21 45
80-90
5,5,5,5,5 25 46 29
90 100
IIII 4
Total

35. Classification is OK No orphan points and no orphan intervals
No points are assigned to more than one interval.
All intervals have some nonzero frequency and all points do belong to some one interval or other.

36. Ambiguity in upper limits of class intervals
The ambiguity is resolved by convention that The actual upper limit of any interval is understood to be a notch lower than what is shown. (printing convenience in Govt. data)
Thus the interval 60 to 70 has upper limit of 70, The convention says that this upper limit is actually
If the number is 70 it always goes to the next interval

37. Trial and Error in Classification
A classification is said to work if we have included all data points in at least one of the intervals. (No orphan points, no orphan intervals)
60 to 70 is our first interval, All numbers are included in the chosen intervals, hence no problem.
If there are orphan points go back to interval width selection. Make the common width bigger. If orphan intervals, make width shorter.

38. Frequency Polygon (starting –ending dummy intervals)
Need to add a Dummy interval at the start and at the end with zero Frequency for freq.poly.
Find the starting value of first dummy interval by subtracting the width
Similarly find the Ending value of the last dummy interval by adding the width to the upper limit of the last interval.
E.g.,If you start with 4 intervals, after adding the dummy intervals end up with 6

39. Frequency Polygon joins histogram midpoints sequentially
Find the mid-point of the lower dummy interval as the simple average of its upper and lower limits
Join it to the midpoint on the top of the pillar of the next interval and end in midpoint of the upper dummy interval.
If we did not have the dummy intervals, it will not really be a polygon since it will have a line hanging above the horizontal axis.

40. Ordered Array
An ordered array is a listing of all the data in either increasing or decreasing magnitude.
Data listed in increasing order is said to be listed in rank order.
Data listed in decreasing order is said to be listed in reverse rank order.

41. Ordered Array (used for medians etc)
Example:
The personnel records for a clothing department store located in the local mall were examined, and the current ages for all employees were noted.
Ages (raw)
Ages (ordered)

42. Time Sequence Plot
A time sequence plot graphs data using time as the horizontal axis.

43. United States Population

44. Time Sequence Plot - Bar Graph
Population of the United States (in millions) p o u l a t i n
Year 1 2 3
1790
1800
1810
1820
1830
1840
1850
1860
1870
1880
1890
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990

45. Time Sequence Plot LineGraph
Population of the United States (in millions) P o p u l a t i n
Year
1790
1800
1810
1820
1830
1840
1850
1860
1870
1880
1890
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
300
250
200
150
100 50

46. Time Sequence / Ribbon Plot
Population of the United States (in millions)

47. A Look at World Population
Population Size (in millions)
Year
1000
2000
3000
4000
5000
6000
1600
1700
1800
1900 P o p u l a t i n