Section 2.1

Frequency Distributions and Their Graphs

Statistics Mrs. Spitz Fall 2008

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. 39-43 #1-32 all Larson/Farber Ch 2

Frequency Distributions

Minutes Spent on the Phone 102 71 103 105 109 124 104 116 97 99 108 112 85 107 105 86 118 122 67 99 103 82 87 95 87 100 78 125 101 92 Make a frequency distribution table with five classes.

Key values:

Larson/Farber Ch 2

Minimum value Maximum value

=

67

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. (125 - 67) / 5 = 11.6 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.

Larson/Farber Ch 2

Construct a Frequency Distribution

Minimum = 67, Maximum = 125 Number of classes = 5 Class width = 12

Class Limits 67 78 79 90 91 102 103

Larson/Farber Ch 2

114 115 126

Do all lower class limits first.

Tally 3 5 8 9 5

Frequency Histogram

Class 67 - 78 79 - 90 91 - 102 103 -114 115 -126 3 5 8 9 5 Boundaries 66.5 - 78.5

78.5 - 90.5

90.5 - 102.5

102.5 -114.5

114.5 -126.5

3 2 1 0 9 8 7 6 5 4 66.5

3 78.5

Time on Phone 9 8 5 90.5

102.5

minutes 114.5

5 126.5

Larson/Farber Ch 2

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.

Larson/Farber Ch 2

Frequency Polygon

Class 67 - 78 79 - 90 91 - 102 103 -114 115 -126 3 5 8 9 5

3 2 1 0 9 8 7 6 5 4 3 5 Time on Phone 9 8 5 72.5

84.5

96.5

108.5

minutes 120.5

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.

Larson/Farber Ch 2

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 67 - 78 79 - 90 91 - 102 103 - 114 115 - 126 3 5 8 9 5 (67 + 78)/2 72.5

84.5

96.5

108.5

120.5

Larson/Farber Ch 2

Relative Frequency 3/30 0.10

0.17

0.27

0.30

0.17

Cumulative Frequency 3 8 16 25 30

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.

Larson/Farber Ch 2

Relative Frequency Histogram

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

Larson/Farber Ch 2

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

30 30 25 20 16 10 8 3 0 0 66.5

78.5

90.5

102.5

minutes 114.5

126.5

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.

Constructing an Ogive (Cumulative Frequency Graph) 1.

2.

3.

4.

5.

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).

Larson/Farber Ch 2