frequency distributions and their graphs section 2.1 statistics mrs. spitz fall 2008
TRANSCRIPT
Frequency Distributions and
Their Graphs
Section 2.1
StatisticsMrs. SpitzFall 2008
Larson/Farber Ch 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. 39-43 #1-32 all
Larson/Farber Ch 2
Frequency Distributions
102 124 108 86 103 82
71 104 112 118 87 95
103 116 85 122 87 100
105 97 107 67 78 125
109 99 105 99 101 92
Make a frequency distribution table with five classes.
Minutes Spent on the Phone
Key values: Minimum value =Maximum value =
67125
Larson/Farber Ch 2
4. Mark a tally | in appropriate class for each data value.
Steps to Construct aFrequency Distribution
1. Choose the number of classes
2. Calculate the Class Width
3. Determine Class Limits
Should be between 5 and 15. (For this problem use 5)
Find the range = maximum value – minimum. Then dividethis by the number of classes. Finally, round up to aconvenient number. (125 - 67) / 5 = 11.6 Round up to 12.
The lower class limit is the lowest data value that belongs ina class and the upper class limit is the highest. Use theminimum value as the lower class limit in the first class. (67)
After all data values are tallied, count the tallies in eachclass for the class frequencies.
Larson/Farber Ch 2
78
90
102
114
126
3
5
8
9
5
67
79
91
103
115Do all lower class limits first.
Class Limits Tally
Construct a Frequency Distribution
Minimum = 67, Maximum = 125Number of classes = 5Class width = 12
Larson/Farber Ch 2
126.5114.5102.590.578.566.5
9
8
7
6
5
4
3
2
1
0
5
9
8
5
3
Boundaries
66.5 - 78.5
78.5 - 90.5
90.5 - 102.5
102.5 -114.5
114.5 -126.5
Frequency Histogram
Time on Phone
minutes
Class
67 - 78
79 - 90
91 - 102
103 -114
115 -126
3
5
8
9
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
9
8
7
6
5
4
3
2
1
0
5
98
5
3
Time on Phone
minutes
Class
67 - 78
79 - 90
91 - 102
103 -114
115 -126
3
5
8
9
5
72.5 84.5 96.5 108.5 120.5
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
67 - 78
79 - 90
91 - 102
103 - 114
115 - 126
3
5
8
9
5
Midpoint: (lower limit + upper limit) / 2
Relative frequency: class frequency/total frequency
Cumulative frequency: number of values in that class or in lower
Midpoint Relative Frequency Class
72.5
84.5
96.5
108.5
120.5
0.10
0.17
0.27
0.30
0.17
3
8
16
25
30
Other Information
CumulativeFrequency
(67 + 78)/2 3/30
Larson/Farber Ch 2
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.
Rel
ativ
e fr
eque
ncy
Larson/Farber Ch 2
Ogive
03
8
16
25
30
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.
66.5 78.5 90.5 102.5 114.5 126.50
10
20
30
Cum
ulat
ive
Fre
quen
cy
minutes
Minutes on Phone
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.
Larson/Farber Ch 2
Constructing an Ogive (Cumulative Frequency Graph)
1. Construct a frequency table that includes cumulative frequencies.
2. Specify the horizontal and vertical scales. The horizontal scale consists of upper class boundaries and the vertical scale measures cumulative frequencies.
3. Plot points that represent the upper class boundaries and their corresponding cumulative frequencies.
4. Connect the points in order from left to right.5. 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).