Describing Data with Tables and Graphs

A frequency distribution is a collection of
observations produced by sorting
observations into classes and showing their
frequency (f) of occurrence in each class.


Ungrouped data
Grouped data
◦ Group data
◦ Lose detail
◦ Gain simplified picture of the data



1. Each observation should be included in
one, and only one, class.
2. List all classes, even those with zero
frequencies.
3. All classes should have equal intervals.
Optional
 4. All classes should have both an upper
boundary and a lower boundary
 5. Select the class interval from convenient
numbers, such as 1, 2, 3, … 10, particularly 5
and 10 or multiples of 5 and 10.
 6. The lower boundary of each class interval
should be a multiple of the class interval.
 7. Aim for a total of approximately 10
classes.
1.
2.
3.
4.
5.
6.
7.
8.
9.
Find the range.
Find the class interval required to span the range.
Round off to the nearest convenient interval.
Determine where the lowest class should begin.
Determine where the lowest class should end.
Working upward, list as many equivalent classes as
are required to include the largest observation.
Indicate with a tally the class in which each
observation falls.
Replace the tally count with each class with a
frequency (f) and show the total of all frequencies.
Supply the headings for both columns and a title for
the table.

Highest – Lowest data value.

Range/desired number of classes

Keep the intervals equal

The lowest score should be a multiple of the
class interval.

Add the interval value to the lowest value and
subtract 1
 Working
upward, list as many equivalent
classes as are required to include the largest
observation

Indicate with a tally the class in which each
observation falls

Replace the tally count with each class with a
frequency (f) and show the total of all
frequencies.

Supply the headings for both columns and a
title for the table

Progress Check 2.2
(page 32)
◦ Construct a frequency distribution for grouped
data.
91
85
84
79
80
87
96
75
86
104
95
71
105
90
77
123
80
100
93
108
98
69
99
95
90
110
109
94
100
103
112
90
90
98
89

Relative frequency distribution (page 35)
◦ Relative frequency distributions show the frequency
of each class as a part or fraction of the total
frequency for the entire distribution
GRE
f
725-749
1
700-724
3
675-699
14
650-674
30
625-649
34
600-624
42
575-599
30
550-574
27
525-549
13
500-524
4
475-499
2
Total
200

Cumulative frequency distribution (page 36)
◦ Cumulative frequency distributions show the
number of observations in each class and in all
lower-ranked classes.
◦ Use the data from progress check 2.5 and create a
cumulative frequency distribution. See table 2.6
page 37.

Percentile ranks
(page 38)
◦ Percentile rank of a score indicates the percentage
of scores in the entire distribution with similar or
smaller values than that score.

Examples of Frequency distributions for
qualitative data (p 38)
Figure 1. Pie chart of iMac
purchases illustrating frequencies
of previous computer ownership.
Figure 2. Bar chart of iMac purchases as a
function of previous computer ownership.
Figure 5. A line graph of the number of
people playing different card games on
Sunday and Wednesday.
◦ A line graph for quantitative data which also
emphasizes the continuity of continuous variables.

Histograms
(page 41)
Equal units along the horizontal axis (X)
Equal units along the vertical axis (Y)
Intersection of axes define the origin (0)
Numerical scales increase from left to right, bottom
to top.
◦ Body reflects frequency for the classes reflected by
height of bars.
◦
◦
◦
◦

Stem and leaf display
◦ A device for sorting quantitative data on the basis
of leading and trailing digits.
 (progress
check 2.10 page 45)
 Construct a stem and leaf display for the
following IQ scores obtained from a group of
four-year-old children.
120
126
108
102
98
85
141
132
118
88
123
109
117
124
137
106
99
104
78
143
111
113
96
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Frequency polygons
◦ Common shapes




Normal
Bimodal
Positively skewed
Negatively skewed
 Decide
on the appropriate type of graph.
 Draw the horizontal axis, then the vertical axis.
 Identify the string of class intervals for horizontal
axis.
 Superimpose the string of class intervals (with
gaps for bar graphs) along the horizontal axis.
 Superimpose progression of numbers along
vertical axis.
 Construct bars to reflect the frequency.
 Supply labels for both axes and a title.
2.9%
3.0%
3.0%
3.0%
Q1
Q2
Q3
2.6%
0.1%
Q2
Q3
2007
Q4
2008
This link to a histogram applet shows the
duration, in minutes, for geyser eruptions of
Old Faithful in Yellowstone National Park. To
see how varying bin (bar) widths affect the
shape of the data, change the width by using
your mouse to drag the arrow underneath the
bin width scale at the bottom of the
histogram.

Progress Check 2.13 (p49)