3.3 Displaying Numerical Data with Frequency Distributions and Histograms and Describing Data
A. Histograms for Discrete Numerical Data
When to Use - Discrete numerical data. Works well even for large data sets
How to Construct
1. Draw a horizontal scale, and the possible values of the variable
2. Draw a vertical scale, and mark it with either frequency or relative frequency
3. Above each possible value, draw a rectangle centered at that value. The height of each
rectangle is determined by the corresponding frequency or relative frequency.
What to Look For
CENTER
SHAPE
SPREAD
DEVIATIONS
Example Ungrouped Discrete Data
Promiscuous Raccoons
1
3
2
1
1
4
2
4
1
1
1
3
1
1
1
1
2
2
1
1
4
1
1
2
1
1
1
1
3
Frequency Distribution
DISCRETE DATA:
Frequency Histogram
Relative Frequency Histogram
Fig. 3-16, p. 99
Example Grouped Discrete Data
Relative Freq Histogram
B. Histograms for Continuous Numerical Data
Histogram for Continuous Data When the Class Interval Widths are Equal
When to Use - Continuous numerical Data. Works well even for large data sets
How to Construct
1. Mark the boundaries of the class intervals on the horizontal axis
2. Use either frequency or relative frequency on the vertical axis
3. Draw a rectangle for each class directly above the corresponding interval. The height of each
rectangle is the frequency or relative frequency of the corresponding class interval
What to Look For
CENTER
SPREAD
SHAPE
DEVIATIONS
Enrollment at Public Universities - % of College Students Enrolled in Public Institutions
86 96
66
86
80
78
62
81
77
81
77
76
73 69
76
90
78
82
70
83
46
80
78
93
66 91
76
86
58
81
91
57
81
88
71
86
84 57
44
82
79
67
86
75
55
75
80
80
85 69
The difficulty in constructing tabular or graphical displays with continuous data is that there are no
natural categories.
Also we need to figure out were to place values that fall on a boundary between classes
1)
2)
The quantity
intervals
number of observations is often used as an estimate of an appropriate number of
Example - Bicycle Accidents
Ogive – Cumulative Relative Frequency Graph
C. Class Intervals of Unequal Widths
Constructing a Histogram for Continuous Data When Class Interval Widths are Unequal
In this case, frequencies or relative frequencies should not be used on the vertical axis. Instead,
the height of each rectangle, called the density for the class, is given by
density = rectangle height = relative frequency of class interval
class interval width
The vertical axis is called the density scale.
The use of the density scale to construct the histogram ensures that the area of each rectangle is
the histogram will be proportional to the corresponding relative frequency.
U.S. Census data for San Luis Obispo County, CA, were used to construct the following
frequency distribution for commute time (in minutes) of working adults.
Commute Time
Width
0 to < 5
Freq
5200
Relative
Freq
.0518
5 to < 10
18200
.1813
10 to < 15
19600
.1952
15 to < 20
15400
.1534
20 to < 25
13800
.1375
25 to < 30
5700
.0568
30 to < 35
10200
.1016
35 to < 40
2000
.0199
40 to < 45
2000
.0199
45 to < 60
4000
.0398
60 to < 90
2100
.0209
90 to < 120
2200
.0219
Density
x
D. Do Sample Histograms Resemble
Population Histograms?