1. Descriptive Statistics
Chapter 2
Descriptive Statistics

2. Chapter Outline 2.1 Frequency Distributions and Their Graphs
2.2 More Graphs and Displays
2.3 Measures of Central Tendency
2.4 Measures of Variation
2.5 Measures of Position 2

3. Frequency Distributions and Their Graphs
Section 2.1
Frequency Distributions
and Their Graphs 3

4. Section 2.1 Objectives Construct frequency distributions
Construct frequency histograms, frequency polygons, relative frequency histograms, and ogives 4

5. Summarizing data/Frequency tables
When data is collected from a survey or designed experiment, they must be organized into a manageable form. Data that is not organized is referred to as raw data.
Ways to Organize Data: Tables; Graphs; and Numerical Summaries.
A frequency distribution lists classes (or categories) of values, along with frequencies (or counts) of the number of values that fall into each class. 5

6. Frequency tables: Definitions
Lower Class Limits: are the smallest numbers that can actually belong to different classes.
Upper Class Limits: are the largest numbers that can actually belong to different classes.
Class Width: is the difference between two consecutive lower class limits or two consecutive upper class limits. 6

7. Frequency Distribution
A table that shows classes or intervals of data with a count of the number of entries in each class.
The frequency, f, of a class is the number of data entries in the class.
Class
Frequency, f
1 – 5 5
6 – 10 8
11 – 15 6
16 – 20
21 – 25
26 – 30 4
Class width 6 – 1 = 5
Lower class
limits
Upper class
limits 7

8. Constructing a Frequency Distribution
Decide on the number of classes.
Usually between 5 and 20; otherwise, it may be difficult to detect any patterns.
Find the class width.
Determine the range of the data.
Divide the range by the number of classes.
Round up to the next convenient number. 8

9. Constructing a Frequency Distribution
Find the class limits.
You can use the minimum data entry as the lower limit of the first class.
Find the remaining lower limits (add the class width to the lower limit of the preceding class).
Find the upper limit of the first class. Remember that classes cannot overlap.
Find the remaining upper class limits. 9

10. Constructing a Frequency Distribution
Make a tally mark for each data entry in the row of the appropriate class.
Count the tally marks to find the total frequency f for each class. 10

11. Guidelines for Frequency tables
1. Be sure that the classes are mutually exclusive.
2. Include all classes, even if the frequency is zero.
3. Use the same width for all classes.
4. Select convenient numbers for class limits.
5. Use between 5 and 20 classes.
6. The sum of the class frequencies must equal the number of original data values. 11

12. Example: Constructing a Frequency Distribution
The following sample data set lists the number of minutes 50 Internet subscribers spent on the Internet during their most recent session. Construct a frequency distribution that has seven classes 12

13. Solution: Constructing a Frequency Distribution
Number of classes = 7 (given)
Find the class width
Round up to 12 13

14. Solution: Constructing a Frequency Distribution
Lower limit
Upper limit 7
Use 7 (minimum value) as first lower limit. Add the class width of 12 to get the lower limit of the next class.
= 19
Find the remaining lower limits.
Class width = 12 19 31 43 55 67 79 14

15. Solution: Constructing a Frequency Distribution
The upper limit of the first class is 18 (one less than the lower limit of the second class). Add the class width of 12 to get the upper limit of the next class = 30 Find the remaining upper limits.
Lower limit
Upper limit 7 19 31 43 55 67 79
Class width = 12 18 30 42 54 66 78 90 15

16. Solution: Constructing a Frequency Distribution
Make a tally mark for each data entry in the row of the appropriate class.
Count the tally marks to find the total frequency f for each class.
Class
Tally
Frequency, f
7 – 18
IIII I 6
19 – 30
IIII IIII 10
31 – 42
IIII IIII III 13
43 – 54
IIII III 8
55 – 66
IIII 5
67 – 78
79 – 90 II 2 16
Σf = 50

17. Determining the Midpoint
Midpoint of a class
Class
Midpoint
Frequency, f
7 – 18 6
19 – 30 10
31 – 42 13
Class width = 12 17

18. Determining the Relative Frequency
Relative Frequency of a class
Portion or percentage of the data that falls in a particular class.
Class
Frequency, f
Relative Frequency
7 – 18 6
19 – 30 10
31 – 42 13 18

19. Determining the Cumulative Frequency
Cumulative frequency of a class
The sum of the frequency for that class and all previous classes.
Class
Frequency, f
Cumulative frequency
7 – 18 6
19 – 30 10
31 – 42 13 6 + 16 + 29 19

20. Expanded Frequency Distribution
Class
Frequency, f
Midpoint
Relative
frequency
Cumulative frequency
7 – 18 6
12.5
0.12
19 – 30 10
24.5
0.20 16
31 – 42 13
36.5
0.26 29
43 – 54 8
48.5
0.16 37
55 – 66 5
60.5
0.10 42
67 – 78
72.5 48
79 – 90 2
84.5
0.04 50
Σf = 50 20

21. Graphs of Frequency Distributions
Frequency Histogram
A bar graph that represents the frequency distribution.
The horizontal scale is quantitative and measures the data values.
The vertical scale measures the frequencies of the classes.
Consecutive bars must touch.
data values
frequency 21

22. Class Boundaries Half this distance is 0.5. Class boundaries
The numbers that separate classes without forming gaps between them.
Class
Boundaries
Frequency, f
7 – 18 6
19 – 30 10
31 – 42 13
The distance from the upper limit of the first class to the lower limit of the second class is 19 – 18 = 1.
Half this distance is 0.5.
6.5 – 18.5
First class lower boundary = 7 – 0.5 = 6.5
First class upper boundary = = 18.5 22

23. Class Boundaries Class Class boundaries Frequency, f 7 – 18 6.5 – 18.5
19 – 30
18.5 – 30.5 10
31 – 42
30.5 – 42.5 13
43 – 54
42.5 – 54.5 8
55 – 66
54.5 – 66.5 5
67 – 78
66.5 – 78.5
79 – 90
78.5 – 90.5 2 23

24. Example: Frequency Histogram
Construct a frequency histogram for the Internet usage frequency distribution.
Class
Class boundaries
Midpoint
Frequency, f
7 – 18
6.5 – 18.5
12.5 6
19 – 30
18.5 – 30.5
24.5 10
31 – 42
30.5 – 42.5
36.5 13
43 – 54
42.5 – 54.5
48.5 8
55 – 66
54.5 – 66.5
60.5 5
67 – 78
66.5 – 78.5
72.5
79 – 90
78.5 – 90.5
84.5 2
The heights of the bars correspond to the frequency values, and the bars are drawn adjacent to each other (without gaps). 24

25. Solution: Frequency Histogram (using Midpoints)
* Notice the use of class boundaries to label the left and right sides of each rectangle.
* It is often more practical to use class midpoint values instead of the class boundaries.
* Both axes should be clearly labeled. 25

26. Solution: Frequency Histogram (using class boundaries)
You can see that more than half of the subscribers spent between 19 and 54 minutes on the Internet during their most recent session. 26

27. Graphs of Frequency Distributions
Frequency Polygon
A line graph that emphasizes the continuous change in frequencies.
The line segments are extended to the right and left so that the graph begins and ends on the x-axis.
data values
frequency 27

28. Example: Frequency Polygon
Construct a frequency polygon for the Internet usage frequency distribution.
Class
Midpoint
Frequency, f
7 – 18
12.5 6
19 – 30
24.5 10
31 – 42
36.5 13
43 – 54
48.5 8
55 – 66
60.5 5
67 – 78
72.5
79 – 90
84.5 2 28

29. Solution: Frequency Polygon
The graph should begin and end on the horizontal axis, so extend the left side to one class width before the first class midpoint and extend the right side to one class width after the last class midpoint.
* The points are at the same heights as the bars of the histogram and are connected. The line segments are extended to the right and left so that the graph begins and ends on the x-axis.
* Notice the use of midpoints on the horizontal axis.
You can see that the frequency of subscribers increases up to 36.5 minutes and then decreases. 29

30. Graphs of Frequency Distributions
Relative Frequency Histogram
Has the same shape and the same horizontal scale as the corresponding frequency histogram.
The vertical scale measures the relative frequencies, not frequencies.
data values
relative frequency
Instead of frequency for the vertical axis, percentages are used. 30

31. Example: Relative Frequency Histogram
Construct a relative frequency histogram for the Internet usage frequency distribution.
Class
Class boundaries
Frequency, f
Relative
frequency
7 – 18
6.5 – 18.5 6
0.12
19 – 30
18.5 – 30.5 10
0.20
31 – 42
30.5 – 42.5 13
0.26
43 – 54
42.5 – 54.5 8
0.16
55 – 66
54.5 – 66.5 5
0.10
67 – 78
66.5 – 78.5
79 – 90
78.5 – 90.5 2
0.04 31

32. Solution: Relative Frequency Histogram
A histogram and relative frequency histogram will have the same shape.
From this graph you can see that 20% of Internet subscribers spent between 18.5 minutes and 30.5 minutes online. 32

33. Graphs of Frequency Distributions
Cumulative Frequency Graph or Ogive
A line graph that displays the cumulative frequency of each class at its upper class boundary.
The upper boundaries are marked on the horizontal axis.
The cumulative frequencies are marked on the vertical axis.
Ogives are useful for determining the number of values less than some particular class boundary.
data values
cumulative frequency 33

34. Constructing an Ogive
Construct a frequency distribution that includes cumulative frequencies as one of the columns.
Specify the horizontal and vertical scales.
The horizontal scale consists of the upper class boundaries.
The vertical scale measures cumulative frequencies.
Plot points that represent the upper class boundaries and their corresponding cumulative frequencies. 34

35. Constructing an Ogive Connect the points in order from left to right.
The graph should start at the lower boundary of the first class (cumulative frequency is zero) and should end at the upper boundary of the last class (cumulative frequency is equal to the sample size). 35

36. Example: Ogive
Construct an ogive for the Internet usage frequency distribution.
Class
Class boundaries
Frequency, f
Cumulative frequency
7 – 18
6.5 – 18.5 6
19 – 30
18.5 – 30.5 10 16
31 – 42
30.5 – 42.5 13 29
43 – 54
42.5 – 54.5 8 37
55 – 66
54.5 – 66.5 5 42
67 – 78
66.5 – 78.5 48
79 – 90
78.5 – 90.5 2 50 36

37. Solution: Ogive
* The points are the frequency values of each class in a cumulative frequency table.
* Notice the use of boundaries on the horizontal scale and that the graph begins with the lower boundary of the
first class and ends with the upper boundary of the last class.
* Ogives are useful for determining the number of values less than some particular class boundary.
From the ogive, you can see that about 40 subscribers spent 60 minutes or less online during their last session. The greatest increase in usage occurs between 30.5 minutes and 42.5 minutes. 37

38. Practice Questions
Q. (2.1) The average quantitative GRE scores for the top 30 graduate schools of engineering are listed below. Construct a frequency distribution with six classes.
767
770
761
760
771
768
776
756
763
747
766
754
778
762
780
750
746
764
769
759
757
753
758
The answers of these practice questions are in chapter 2 practice questions slides. 38

39. Practice Questions
Q. (2.2)
The ages of the signers of the Declaration of Independence
are shown below (age is approximate since only the birth
year appeared in the source and one has been omitted since
his birth year is unknown). Construct a frequency
distribution for the data using seven classes. 41 54 47 40 39 35 50 37 49 42 70 32 44 52 30 34 69 45 33 63 60 27 38 36 43 48 46 31 55 62 53 39

40. Practice Questions
Q. (2.3) For 108 randomly selected college applicants, the following frequency distribution for entrance exam scores was obtained. Construct a histogram, frequency polygon, and ogive for the data.
Class limits
126 – 134 9 28
117 – 125 43
108 – 116 22
99 – 107 6
90 – 98
Frequency 40

41. Practice Questions
Q. (2.4) The number of calories per serving for selected ready – to - eat cereals is listed here. Construct a frequency distribution using seven classes. Draw a histogram, frequency polygon, and ogive for the data, using relative frequencies. Describe the shape of the histogram.
130
190
140 80
100
120
220
110
210 90
200
180
260
270
160
240
115
225 41

42. Section 2.1 Summary Constructed frequency distributions
Constructed frequency histograms, frequency polygons, relative frequency histograms and ogives 42