Describing Data:
Frequency Tables, Frequency
Distributions, and Graphic Presentation
Chapter 2
McGraw-Hill/Irwin
©The McGraw-Hill Companies, Inc. 2008
GOALS
•Organize qualitative data into a frequency table.
•Present a frequency table as a “bar chart” (Excel
they are called column chart) or a pie chart.
•Organize quantitative data into a frequency
distribution.
•Present a frequency distribution for quantitative data
using histograms, frequency polygons, and
cumulative frequency polygons.
2
Mutually Exclusive
 An
individual, object, or measurement is
included in only one category
It can’t be in two categories
– Example: A particular phone call cannot
originate with both AT&T and MCI
–
3
Frequency Table

Frequency Table: A grouping of qualitative data into
mutually exclusive classes (categories) showing the
number of observations in each class
Sales data from Auto Dealership
P rice
4
P rice ($000) Age of Custome r ForD
24,624.00
24.624
50 Domestic
Ca r T ype
GM
23,032.00
27,556.00
20,384.00
20,953.00
37,270.00
21,006.00
27,594.00
29,636.00
26,357.00
38,262.00
38,910.00
23,947.00
50 Domestic
Ford
59 Domestic
GM
32 Domestic
GM
29 Domestic
Ford
35 Foreign
Mercedes
57 Domestic
GM
43 Domestic
GM
51 Domestic
GM
31 Foreign
Honda
39 Foreign
Mercedes
25 Foreign
Honda
43 Domestic
Ford
23.032
27.556
20.384
20.953
27.270
21.006
27.594
29.636
26.357
28.262
38.910
23.947
We would like a Frequecy Table that shows
how many of each Car Type we sold last
month from the Auto Dealership data
(counting).
Count of Car Type
Car Type
Ford
GM
Honda
Mercedes
Toyota
Grand Total
Total
28
22
13
10
7
80
Relative Class Frequencies


Class frequencies can be converted to relative class
frequencies to show the fraction of the total number of
observations in each class.
A relative frequency captures the relationship between
a class total and the total number of observations.
Car Type
GM
Ford
Mercedes
Honda
Toyota
5
Frequency
Relative Frequency
22
28
10
13
7
80
27.50%
35.00%
12.50%
16.25%
8.75%
100.00%
Textbook: Bar Charts 
Excel: Column Chart
Number of Autos Sold
30
28
25
22
F
r 20
e
q
u 15
e
n
10
c
y
13
10
7
5
0
Ford
6
GM
Honda
Mercedes
Toyota
 In Excel,
this is a
Column chart.
Column charts
are good for
Nominal Level
Data. Notice
that the
columns do not
touch.
Pie Charts
Percentage of Autos Sold
9%
13%
35%
Ford
GM
Honda
Mercedes
16%
Toyota
27%
7
Frequency Distribution
A Frequency distribution is a
grouping of data into mutually
exclusive categories showing
the number of observations in
each class.
•The raw data are more easily interpreted if organized into a frequency
distribution
•The resulting frequency distribution helps a person to quickly see the
“shape” of the data
•Although the frequency distribution will result in the loss of some detail,
seeing patterns in the data can help a person to make better decisions
8
5 Steps To Organize Raw Data Into A
Frequency Distribution





9
Step 1: Decide on Number of Classes
Step 2: Determine The Class Interval
Step 3: Set The Individual Class Limits
Step 4: Tally The Data Into Classes
Step 5: Count The Tallies in Each Class & Present
the Frequency Distribution
Step 1: Determining The Number Of Classes



Goal is to use just enough classes so you
can see the “shape” of the data.
You must use professional judgment.
Useful recipe to determine the number
of classes:
2k ≥ n
n = total observations
k = number of classes

10
Best to use 5 < k < 15
General
guidelines that
are not always
possible to
follow. Thus,
making
Frequency
Distributions is
often refer to
as an “art”.

Definitions

Class Interval
–
–

Distance between lower limit of class and lower limit of
the next class
The class interval is obtained by subtracting the lower
limit of a class from the lower limit of the next class (also
midpoint to midpoint)
Class Midpoint (Class Mark)
–
–
The midpoint can be thought of as the “typical value” for
the class
This is the average of the upper and lower class limits:
(Lower class limit + upper class limit)/2
11
Step 2: Determine The Class Interval Or Width

12
Class interval should be
the same for every
interval
– If they are not equal
graphs may be
misleading, &
calculations may be
problematic
– In some cases, where
there is a potential for
many empty classes,
unequal class interval
may be necessary

The classes all taken
together must cover at
least the distance from
the lowest value in the
raw data up to the highest
value:
Determine Class Interval
H-L
i
≥
k
i =
Class Interval
H =
Highest Value
L =
Lowest Value
k = Number of Classes
EXAMPLE – Creating a Frequency
Distribution Table
Ms. Kathryn Ball of AutoUSA
wants to develop tables, charts,
and graphs to show the typical
selling price on various dealer
lots. The table on the right
reports only the price of the 80
vehicles sold last month at
Whitner Autoplex.
13
Constructing a Frequency Table Example

Step 1: Decide on the number of classes.
A useful recipe to determine the number of classes (k) is
the “2 to the k rule.” such that 2k > n.
There were 80 vehicles sold. So n = 80. If we try k = 6, which
means we would use 6 classes, then 26 = 64, somewhat less
than 80. Hence, 6 is not enough classes. If we let k = 7, then 27
128, which is greater than 80. So the recommended number of
classes is 7.

Step 2: Determine the class interval or width.
The formula is: i  (H-L)/k where i is the class interval, H is
the highest observed value, L is the lowest observed value,
and k is the number of classes.
($35,925 - $15,546)/7 = $2,911
Round up to some convenient number, such as a multiple of 10
or 100. Use a class width of $3,000
14
Step 3: Set The Individual Class Limits


Classes must be mutually exclusive
Avoid overlapping or unclear class limits:
–
–

Include lower limit
Exclude upper limit
Example of class limits:
–
$12,000 up to $15,000 and $15,000 up to
$18,000



Avoid open ended classes (problems with

The lower limit of the first class should be a
multiple of the class interval (not always
possible)
Convenient multiples of ten are useful
You must compare the actual range to the
range implied by the number of classes &
class interval


15
$12,000 & $14,999 belong in the first class
$15,000 belongs in the second class
graphing)
General
guidelines that
are not always
possible to
follow. Thus,
making
Frequency
Distributions
is often refer
to as an “art”.
Constructing a Frequency Table Example

16
Step 3: Set the individual class limits
Constructing a Frequency Table
17

Step 4: Tally the vehicle
selling prices into the
classes.

Step 5: Count the number
of items in each class.
Observed Patterns:



Range: about $15,000 to
about $36,000
Concentration between
$18,000 & $27,000
Largest concentration is
in $18,000 - $21,000
class
–


18
Typical Value = (18+21)/2 = 19.5
K.
Two sold for $33,000 or
more
8 sold for less than
$18,000
Relative Frequency Distribution
To convert a frequency distribution to a relative frequency
distribution, each of the class frequencies is divided by the
total number of observations.
19
Graphic Presentation of a
Frequency Distribution
The three commonly used graphic forms
are:
 Histograms
 Frequency
polygons
 Cumulative frequency distributions
20
Histogram
Histogram for a frequency distribution based on quantitative data is
very similar to the column charts (book says: bar chart) showing the
distribution of qualitative data. The classes are marked on the
horizontal axis and the class frequencies on the vertical axis. The
class frequencies are represented by the heights of the bars. The
columns must touch in order to visually articulate that the class
interval spans from lower class limit to upper class limit.
21
Other Notes About Histogram


Histograms constructed from Relative Frequency
Distributions look the same (have the same shape), but
instead, the vertical axis would show percentages
Histograms must have the columns touching:
–
–
22
The columns must touch in order to visually articulate that the class
interval spans from lower class limit to upper class limit (a
continuous variable)
For nominal or ordinal level data, the columns are not drawn
adjacent to each other
 The category labels are usually words
Frequency Polygon
23

A frequency polygon
also shows the shape
of a distribution and is
similar to a histogram.

It consists of line
segments connecting
the points formed by
the intersections of the
class midpoints and the
class frequencies.
Cumulative Frequency Distribution
24
Cumulative Frequency Distribution
25
Second Example of a Cumulative Frequency
Distribution (prices of vehicles are lower)
Selling Prices
($ thousands)
12
up to
15
15
up to
18
18
up to
21
21
up to
24
24
up to
27
27
up to
30
30
up to
33
Total
26
Cumulative Frequency Distribution for
Vehicles Selling Price
Number of
Vehicles Sold
Cumulative
(Frequency)
Frequency
8
8
23
31
17
48
18
66
8
74
4
78
2
80
80
=
=
=
=
=
=
8 + 23
31 + 17
48 + 18
66 + 8
74 + 4
78 + 2
Cumulative Frequency Polygon
80
100%
70
Number of Vehicles Sold
60
75%
50
50%
40
30
25%
20
10
0
9
12
15
18
21
24
Selling Price ($000)
27
27
30
33
Cumulative Frequency Polygon





Plot line on coordinate
system
X-axis = Upper limit of
class
Y-axis (Left) =
Cumulative Frequency
Y-axis (Right) = %
First point on graph is:
(lower limit of first class, 0)
28
x
12
15
18
21
24
27
30
33
y (left)
0
8
31
48
66
74
78
80
Cumulative Frequency Polygon
80
100%
70
Number of Vehicles Sold
60
75%
50
50%
40
30
25%
20
10
0
9
12
15
18
21
24
Selling Price ($000)
29
27
30
33
x
12
15
18
21
24
27
30
33
y (left)
0
8
31
48
66
74
78
80
Cumulative Frequency Polygon
80
100%
70
Number of Vehicles Sold
60
75%
50
50%
40
30
25%
20
10
0
9
12
15
18
21
24
Selling Price ($000)
30
27
30
33
50% of the
vehicles
sold for
less than
about
$19,500
Cumulative Frequency Polygon
80
100%
70
Number of Vehicles Sold
60
75%
50
50%
40
30
25%
20
10
0
9
12
15
18
21
24
Selling Price ($000)
31
27
30
33
25 of the
vehicles
sold for
less than
about
$17,500
Cumulative Frequency Polygon
80
100%
70
Number of Vehicles Sold
60
75%
50
50%
40
30
25%
20
10
0
9
12
15
18
21
24
Selling Price ($000)
32
27
30
33
80% of the
vehicles
sold for
less than
about
$24,000
End of Chapter 2
33