1) A frequency distribution organizes raw data into classes and frequencies in a table. It should have between 5-20 classes of equal width. 2) A histogram uses bars of different heights to represent class frequencies from a frequency distribution. The tallest bar shows the category with the largest concentration of data values. 3) A frequency polygon connects the midpoints of the bars on a histogram. An ogive graph shows the cumulative frequencies on the y-axis to illustrate what percentage of data is accounted for at each class.

1. Frequency Distributions
and Graphs
Chapter 2

2. A frequency distribution is the organization
of raw data in table from, using classes and
frequency.

3. 1 2 6 7 12 13 2 6 9 5
18 7 3 15 15 4 17 1 14 5
4 16 4 5 8 6 5 18 5 2
9 11 12 1 9 2 10 11 4 10
9 18 8 8 4 14 7 3 2 6
The number of miles that the employees of a large
department store traveled to work each day

4. Class Limits
Frequency
(in miles)
1-3 10
4-6 14
7-9 10
10-12 6
13-15 5
16-18 5
Total 50
How to construct a grouped frequency
Distribution?

5.  Number of classes
It should be between 5 and 20.
Some Statisticians use “ 2k “ rule.
k 1 2 3 4 5 6 7 8 9 10
2k 2 4 8 16 32 64 128 256 512 1,024

6. 2 to k rule
Essentially we would look to construct k
classes for our frequency distribution, when
the value of 2k first exceeds the number of
observations in our sample. So, if we had a
sample with 39 observations, we would first
consider constructing 6 classes, because 26
= 64, the first power of 2 with a value larger
than the sample size of 39.

7. A guide, not a dictator.
Strictly speaking the 2k rule is a guide, not a
rule. If the 2k rule suggests you need 6
classes, also consider using 5 or 7 classes
... but certainly not 3 or 9.

8.  Class interval or class width
H : the highest value, L: the smallest value
 Class interval can also be estimated
based on # of observations
H L
w
k
H L
1 3.322lnn
w

9.  Select the lower limit of the first class and
set the limits of each class
It could be L or any value smaller than L.
It should be an even multiple of the class
interval.

10.  There should be between 5 and 20 classes.
 The classes must be continuous.
 The classes must be exhaustive.
 The classes must be mutually exclusive.
 The classes must be equal in width.

11. Relative frequency
o Relative frequency of a class is the
frequency of that class divided by to total
number of frequency.
f
RF
n

12.  Example
These data represent the record high
temperatures for each of the 50 states.
Construct a grouped frequency distribution for
the data using 7 classes.
112 100 127 120 134 118 105 110 109 112
110 118 117 116 118 122 114 114 105 109
107 112 114 115 118 117 118 122 106 110
116 108 110 121 113 120 119 111 104 111
120 113 120 117 105 110 118 112 114 114

13. Class limits
Class
boundaries Frequency
Relative
frequency
Cumulative
frequency
100-104 99.5-104.5 2 0.04 2
105-109 104.5-109.5 8 0.16 10
110-114 109.5-114.5 18 0.36 28
115-119 114.5-119.5 13 0.26 41
120-124 119.5-124.5 7 0.14 48
125-129 124.5-129.5 1 0.02 49
130-134 129.5-134.5 1 0.02 50

14. Histogram
A histogram is a graph that displays the data
by using contiguous vertical bars (unless the
frequency of a class is 0) of various heights to
represent the frequencies of the classes.

15. Example
Construct a histogram to represent the data
shown below for the record high temperature:
Class
boundaries Frequency
99.5-104.5 2
104.5-109.5 8
109.5-114.5 18
114.5-119.5 13
119.5-124.5 7
124.5-129.5 1
129.5-134.5 1

16. 99.5
Histogram
104.5 109.5 114.5 119.5 124.5 129.5
18
15
12
9
6
3
oThe largest concentration is in the class
109.5 – 114.5.

17. 99.5 104.5 109.5 114.5 119.5 124.5 129.5
9
6
3
12
15
18
Frequency
Polygone

18. The Ogive is a graph that represents the
cumulative frequencies for the classes in a
frequency distribution.

19. Class
boundaries Frequency
Cumulative
Frequency
99.5-104.5 2 2
104.5-109.5 8 10
109.5-114.5 18 28
114.5-119.5 13 41
119.5-124.5 7 48
124.5-129.5 1 49
129.5-134.5 1 50

20. 99.5 104.5 109.5 114.5 119.5 124.5 129.5
50
40
30
20
10
Cumulative Frequency Polygone

21. Other types of Graphs
Bar Chart
Bar Chart is use to represent a frequency
distribution for a categorical variable, and the
frequencies are displayed by the heights of
vertical bars.

22. Example
The table shown here displays the number of
crimes investigated by law enforcement
officers in U.S. national parks during 1995.
Construct a Bar chart for the data.
Type Number
Homicide 13
Rape 34
Robbery 29
Assault 164

23. 50
150
100
Homicide
Rape
Robbery
Assault
13
34 29
164
Total number of crime: 234

24. Pie Graph
A pie graph is a circle that is divided into
sections or wedges according to the
percentage of frequencies in each category of
the distribution.

25. Example
This frequency distribution shows the number
of pounds of each snack food eaten during
the 1998 Super Bowl. Construct a pie graph
for the data.
Snack
Million
pounds
Potato Chips 11.2
Tortilla Chips 8.2
Pretzels 4.3
Popcorn 3.8
Snack nuts 2.5

26. We need to find percentages for each
category and then compute the
corresponding sectors so that we divide the
circle proportionally.
Snack
Million
pounds percentage Degree
Potato Chips 11.2 37.33% ≈134º
Tortilla Chips 8.2 27.33% ≈98º
Pretzels 4.3 14.33% ≈41º
Popcorn 3.8 12.67% ≈46º
Snack nuts 2.5 8.33% ≈30º

27. Potato
Chips
37%
T
ortilla
Chips
28%
Snack
nuts
Popcorn
8%
13%
Pretzels
14%
Potato Chips
Tortilla Chips
Pretzels
Popcorn
Snack nuts

28. Stem and Leaf Plots
A stem and leaf plot is a data plot that uses
part of the data value as the stem and part
of the data value as the leaf to form groups
or classes.

29. Example
At an outpatient testing center, the number
of cardiograms performed each day for 20
days is shown. Construct a tem and leaf plot
for the data.
25 31 20 32 13
14 43 02 57 23
36 32 33 32 44
32 52 44 51 45

30. It is helpful to arrange the data in order but it
is not required.
02, 13, 14, 20, 23, 25, 31, 32, 32, 32, 32, 33,
36, 43, 44, 44, 45, 51, 52, 57
Leading
digit (Stem)
Trailing
digit (Leaf)
0 2
1 3 4
2 0 3 5
3 1 2 2 2 2 3 6
4 3 4 4 5
5 1 2 7

31. EXERCISES 1
The following data represent the color of men’s
dress shirts purchased in the men’s department
of a large department store. Construct a
categorical frequency distribution, bar chart and
pie chart for the data (W= white, BL= blue,
BR= brown, Y= yellow, G= gray).

32. EXERCISES 1(Cont.)
W W BR Y BL BL W W Y G
W W BL BR BL BR BL BL BR Y
BL G W BL W W BL W BL BR
Y BL G BR G BR W W BR Y
W BL Y W W BL W BR G G

33. EXERCISES 2
The ages of the signers of the Declaration of
Independence of the US are shown below.
41 54 47 40 39 35 50 37 49 42 70 32
44 52 39 50 40 30 34 69 39 45 33 42
44 63 60 27 42 34 50 42 52 38 36 45
35 43 48 46 31 27 55 63 46 33 60 62
35 46 45 34 53 50 50

34. EXERCISES 2 (Cont.)
1) Construct a frequency distribution using
seven classes. Include relative frequency,
percentage and Cumulative frequency.
2) Construct a histogram, frequency poly-
gone, and Ogive.
3) Develop a stem-and-leaf plot for the data.

35. Thank You for your
attention!
Good Luck!