1. Frequency Distributions and Their Graphs
Section 2.1
Frequency Distributions and Their Graphs
Statistics
Mrs. Spitz
Fall 2008

2. Objectives
How to construct a frequency distribution including midpoints, relative frequencies, and cumulative frequencies.
How to construct frequency histograms, frequency polygons, relative frequency histograms and ogives.
Assignment: pp #1-32 all

3. Frequency Distributions
Minutes Spent on the Phone
This data will be used for several examples. You might want to duplicate it for use with slides that follow.
Make a frequency distribution table with five classes.
Minimum value =
Maximum value = 67
Key values:
125

4. Steps to Construct a Frequency Distribution
1. Choose the number of classes
Should be between 5 and 15. (For this problem use 5)
2. Calculate the Class Width
Find the range = maximum value – minimum. Then divide
this by the number of classes. Finally, round up to a
convenient number. ( ) / 5 = Round up to 12.
3. Determine Class Limits
The lower class limit is the lowest data value that belongs in
a class and the upper class limit is the highest. Use the
minimum value as the lower class limit in the first class. (67)
4. Mark a tally | in appropriate class for each data value.
After all data values are tallied, count the tallies in each
class for the class frequencies.

5. Construct a Frequency Distribution
Minimum = 67, Maximum = 125
Number of classes = 5
Class width = 12 3 5 8 9
Class Limits Tally 78 90
102
114
126 67 79 91
103
115
Do all lower class limits first.

6. Frequency Histogram Boundaries Class 3 66.5 - 78.5 67 - 78 78.5 - 90.5
Class 3 5 8 9
Time on Phone 9 9 8 8 7 6 5 5 5
A histogram is a bar graph for which the bars touch. To form boundaries, find the distance between consecutive classes. Add half that distance to the lower limits and half to the upper limits. In this case the distance is 1 unit so add .5 to all upper limits and subtract .5 from all lower ones. The data must be quantitative. This histogram is labeled at the class boundaries. Explain that midpoints could have been labeled instead. 4 3 3 2 1 6 6 . 5 7 8 . 5 9 . 5 1 2 . 5 1 1 4 . 5 1 2 6 . 5
minutes

7. What is a histogram?
A histogram is a bar graph for which the bars touch. To form boundaries, find the distance between consecutive classes. Add half that distance to the lower limits and half to the upper limits. In this case the distance is 1 unit so add .5 to all upper limits and subtract .5 from all lower ones. The data must be quantitative. This histogram is labeled at the class boundaries. Midpoints could have been labeled instead.

8. Frequency Polygon Time on Phone minutes Class 3 67 - 78 79 - 90 5 3 5 8 9
Time on Phone 9 9 8 8 7 6 5 5 5 4 3 3 2 1
The frequency polygon is labeled at midpoints. Explain that midpoints could have been labeled instead.
72.5
84.5
96.5
108.5
120.5
minutes
Mark the midpoint at the top of each bar. Connect consecutive midpoints. Extend the frequency polygon to the axis. The frequency polygon is labeled at midpoints. Midpoints could have been labeled instead.

9. Other Information Midpoint: (lower limit + upper limit) / 2
Relative frequency: class frequency/total frequency
Cumulative frequency: number of values in that class or in lower
Class
Midpoint
Relative
Frequency
Cumulative
Frequency
( )/2
3/30 3 5 8 9
72.5
84.5
96.5
108.5
120.5
0.10
0.17
0.27
0.30 3 8 16 25 30
The first two columns reflect the work done in previous slides. Once the first midpoint is calculated, the others can be found by adding the class width to the previous midpoint. Notice the last entry in the cumulative frequency column is equal to the total frequency.

10. Explanation of previous slide
The first two columns reflect the work done in previous slides. Once the first midpoint is calculated, the others can be found by adding the class width to the previous midpoint. Notice the last entry in the cumulative frequency column is equal to the total frequency.

11. Relative Frequency Histogram
Time on Phone
Relative frequency
A relative frequency histogram will have the same shape as a frequency histogram.
minutes
Relative frequency on vertical scale. A relative frequency histogram will have the same shape as a frequency histogram.

12. Ogive
An ogive (pronounced o-jive) reports the number of values in the data set that are less than or equal to the given value, x.
Minutes on Phone 3 8 16 25 30
66.5
78.5
90.5
102.5
114.5
126.5 10 20 30
Cumulative Frequency
Label each boundary on the horizontal axis. Start with 0 for the lower boundary of the first class. Then mark points corresponding to cumulative frequencies at the upper boundaries. Connect with line segments.
minutes
Label each boundary on the horizontal axis. Start with 0 for the lower boundary of the first class. Then mark points corresponding to cumulative frequencies at the upper boundaries. Connect with line segments.

13. Constructing an Ogive (Cumulative Frequency Graph)
Construct a frequency table that includes cumulative frequencies.
Specify the horizontal and vertical scales. The horizontal scale consists of upper class boundaries and the vertical scale measures cumulative frequencies.
Plot points that represent the upper class boundaries and their corresponding cumulative frequencies.
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).