Organizing data using frequency distributions. Illustrating data ...fmalam.kau.edu.sa/Files/0007085/Files/157188_STAT_110_CH...Overview of Chapter 2 Sec. # Title Page(s) 2 - 1 Organizing

Chapter 2: Frequency Distributions and Graphs

Objectives:

❑ Organizing data using frequency

distributions.

❑ Illustrating data using graphs.

❑ Interpreting graphs.

© FAROUQ MOHAMMAD ALAM, DEPT OF STAT, KAU, 2019

Overview of Chapter 2

Sec. # Title Page(s)

2 - 1 Organizing data 42 - 56

2 - 2Histogram, frequency polygons, and

ogives57 - 74

2 - 3 Other types of graphs 74 - 108


2 – 1: Organizing data

Inductee Blood

type

1 A

2 B

3 B

4 AB

⋮ ⋮

25 A

These data are called

raw data ! (i.e., they

are in their original

form)


2 – 1: Organizing data (cont.)

By using descriptive statistical methods, we

transform raw data into …


1. Frequency Distributions

Inductee Blood

type

1 A

2 B

3 B

4 AB

⋮ ⋮

25 A

Blood type Number of

inductees

A 5

B 7

O 9

AB 4

Descriptive

Statistics


2. Graphs

A20%

B28%

O36%

AB16%

A B O AB

Inductee Blood

type

1 A

2 B

3 B

4 AB

⋮ ⋮

25 A

Descriptive

Statistics



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

A class is a quantitative or qualitative aspect in which data are accordingly distributed.

A frequency of a class is the number of data values placed in this class.



Types of frequency distributions:

Categorical frequency distribution is used

when in the cases of nominal-level or ordinal-

level data.

Grouped frequency distribution is used in the

case of quantitative data with large range.

Ungrouped frequency distribution is used in

the case of quantitative data with relatively

small range or when the data are discrete.


Example 2 – 1: Distribution of Blood Types

(Categorical frequency distribution, page 43)

Twenty-five army inductees were given a

blood test to determine their blood type.

The data set is as shown in page 43.

This is called the

sample size!


Example 2 – 1 (cont.)

Step 1. determine the classes.

We have four blood types, therefore, there are four classes are they are A, B, O, and AB.

Step 2. Create a table with three columns, the first column is for blood types (classes), the second column for counting, and the third column is for the number (#) of inductees (frequencies).

Step 3. Tally data and then delete column 2.



A AB

AB

A

A AB

AB A A

Class Tally Frequency

A |||| 5

B 7

O 9

AB |||| 4

Total 25

Raw data Frequency distribution



Blood type

(Class)

# of inductees

(Frequency)

A 5 = 𝒇𝟏

B 7 = 𝒇𝟐

O 9 = 𝒇𝟑

AB 4 = 𝒇𝟒

Total ∑𝒇 = 𝒇𝟏 + 𝒇𝟐 + 𝒇𝟑 + 𝒇𝟒 = 𝟐𝟓


Important Rules!

Regardless of the type of the frequency

distribution, if the sample size is represented by 𝒏, then

∑𝒇 = 𝒏

Regardless of the type of the frequency

distribution, the (cumulative) percentage of class

number 𝒊 is defined as

𝑷𝒊 =𝒇𝒊𝒏× 𝟏𝟎𝟎%


Important Rules! (cont.)

When manually drawing the pie-chart, we need to

calculate the degree of class number 𝒊 which is defined as

𝑫𝒊 =𝒇𝒊𝒏× 𝟑𝟔𝟎°


Example 2 – 1 (Revisited)

Blood type

(Class)

# of inductees

(Frequency)

Percentage of

inductees (%)

Degree of

inductees

A 5 20% 72˚

B 7 28% 100.8˚

O 9 36% 129.6˚

AB 4 16% 57.6˚

Total 25 100% 360˚


Example 2 – 2: Record High Temperatures

(Groped frequency distribution, page 47)

The data in page 47 represent the record

high temperatures in degrees Fahrenheit (˚F)

for each of the 50 (= 𝒏) states. Construct a grouped frequency distribution for the data

using 7 classes.



Step 1. determine the classes as follows:

Calculate the range (𝑹) which is the difference between the highest value (𝑯) and the lowest value (𝑳), i.e.

𝑹 = 𝑯− 𝑳 = 𝟏𝟑𝟒 − 𝟏𝟎𝟎 = 𝟑𝟒

Given the number of classes, the class width is defined as:

𝐜𝐥𝐚𝐬𝐬 𝐰𝐢𝐝𝐭𝐡 = 𝒉 =𝑹

𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐜𝐥𝐚𝐬𝐬𝐞𝐬 = 𝒌

=𝟑𝟒

𝟕= 𝟒. 𝟗 ≈ 𝟓



Step 1. (cont.)

A class of a grouped frequency distribution consists of

class limits (boundaries); a lower limit, which is the

smallest data value that can be included in the class, and

an upper limit, which represents the largest data value

that can be included in the class.



Step 1. (cont.) (textbook)

Lower limits: Consider the lowest value to be the starting point for the lowest class limits, i.e., first use 100 as the lower limit of the first class then repeatedly add the class width to get the lower limit of the next six classes, i.e., 105, 110, 115, 120, 125, 130.

Upper limits: Subtract one unit from the lower limits of the second class until the seventh class to get the upper limit of the first class until the sixth class. Finally, use the largest value as the upper limit of the final class.



Step 1. (cont.)

The limits of the classes are:

𝑳 = 100 – 104

105 – 109

𝒍𝟑 =110 – 114 = 𝒖𝟑115 – 119

120 – 124

125 – 129

130 – 134 = 𝑯



Step 2. Create a table with three columns, the first

column is for temperature (classes), the second

column for counting, and the third column is for the

number (#) of states (frequencies).




Temperature

(Class)

# of states

(Frequency)

100 – 104 2

105 – 109 8

110 – 114 18

115 – 119 13

120 – 124 7

125 – 129 1

130 – 134 1

Total 50


Important Rules!

Sometimes, we need to calculate the midpoints

of each class which is given by

𝐜𝐥𝐚𝐬𝐬 #𝒊 𝐦𝐢𝐝𝐩𝐨𝐢𝐧𝐭 =𝒍𝒊 + 𝒖𝒊𝟐



Grouped (and also ungrouped) frequency

distributions use class boundaries so that there are

no gaps in the frequency distribution. They are given

by

𝒍𝒊 − 𝟎. 𝟓, 𝒖𝒊 + 𝟎. 𝟓

in the case of grouped frequency distributions.



Usually, grouped frequency distributions consist of

equal class widths. The class width based on the

limits of any class 𝒊 is given by

𝐜𝐥𝐚𝐬𝐬 𝐰𝐢𝐝𝐭𝐡 = 𝒍𝒊+𝟏 − 𝒍𝒊or

𝐜𝐥𝐚𝐬𝐬 𝐰𝐢𝐝𝐭𝐡 = 𝒖𝒊+𝟏 − 𝒖𝒊



Temperature

(Class)

Class

boundaries

Class

Midpoints

# of states

(Frequency)

100 – 104 99.5 – 104.5 102 2

105 – 109 104.5 – 109.5 107 8

110 – 114 109.5 – 114.5 112 18

115 – 119 114.5 – 119.5 117 13

120 – 124 119.5 – 124.5 122 7

125 – 129 124.5 – 129.5 127 1

130 – 134 129.5 – 134.5 132 1

Total 50


Example 2 – 3: MPGs for SUVs

(Ungrouped frequency distribution, page 49)

The data shown in page 49 represent the

number of miles per gallon (mpg) that 30 (=𝒏) selected four-wheel-drive sports utility vehicles obtained in city driving. Construct a

frequency distribution, and analyze the

distribution.



Step 1. determine the classes.

Notice that the range of data is small (𝑅 = 19 − 12 = 7). The classes are 12, 13, 14, 15, 16, 17, 18, and 19.

Step 2. Create a table with three columns, the first column is for

MPG (classes), the second column for counting, and the third

column is for the # of SUVs (frequencies).




MPG

(Class)

# of SUVs

(Frequency)

12 6

13 1

14 3

15 6

16 8

17 2

18 3

19 1

Total 30


Remark!

Notice that the class boundaries calculated for the

ungrouped frequency distribution since MPG is an

example of a continuous variable.

Only in the case of continuous data, we can

obtain class boundaries by subtracting 0.5 from

each class value to get the lower class boundary,

and adding 0.5 to each class value to get the

upper class boundary.



MPG

(Class)Class boundaries

# of SUVs

(Frequency)

12 11.5 – 12.5 6

13 12.5 – 13.5 1

14 13.5 – 14.5 3

15 14.5 – 15.5 6

16 15.5 – 16.5 8

17 16.5 – 17.5 2

18 17.5 – 18.5 3

19 18.5 – 19.5 1

Total 30


2 – 2: Histograms, frequency polygons,

and ogives

Histogram:

Definition is in page 57, and the corresponding

illustrative Example 2 – 4 is in pages 57-58.

Frequency polygon:



Ogive:




Histogram


Frequency Polygon


Cumulative Frequency Distribution

A cumulative frequency distribution is a

distribution that shows the number of data values

less than or equal to each upper boundary.

The values are found by adding the frequencies of

the classes less than or equal to the upper class

boundary of a specific class. This gives an

ascending cumulative frequency. The last

cumulative frequency must be equal to 𝒏.



Temp.

(Class)

Class

boundariesFrequency

Cumulative

Frequency

Less than 99.5 0

100 – 104 99.5 – 104.5 2 Less than 104.5

105 – 109 104.5 – 109.5 8 Less than 109.5

110 – 114 109.5 – 114.5 18 Less than 114.5

115 – 119 114.5 – 119.5 13 Less than 119.5

120 – 124 119.5 – 124.5 7 Less than 124.5

125 – 129 124.5 – 129.5 1 Less than 129.5

130 – 134 129.5 – 134.5 1 Less than 134.5© FAROUQ MOHAMMAD ALAM, DEPT OF STAT, KAU, 2019


Temp.

(Class)

Class

boundariesFrequency

Cumulative

Frequency

Less than 99.5 0

100 – 104 99.5 – 104.5 2 Less than 104.5 2

105 – 109 104.5 – 109.5 8 Less than 109.5 10

110 – 114 109.5 – 114.5 18 Less than 114.5 28

115 – 119 114.5 – 119.5 13 Less than 119.5 41

120 – 124 119.5 – 124.5 7 Less than 124.5 48

125 – 129 124.5 – 129.5 1 Less than 129.5 49

130 – 134 129.5 – 134.5 1 Less than 134.5 𝟓𝟎 = 𝒏

Keep Adding!


Ogive


Why using ogives?

The ogive is mainly used to visually represent how

many cumulative frequency (percentage) are

approximately below a certain upper class

boundary and vice versa.


Case 1: Getting the cumulative frequency based on

an upper class boundary.

cumulative frequency ≈ 𝟒𝟓

Upper (limit) boundary= 𝟏𝟐𝟐


Case 2: Getting the upper class boundary based on a

cumulative frequency.

Upper (limit) boundary≈ 𝟏𝟏𝟕. 𝟓

cumulative frequency = 𝟑𝟓


Case 3: Getting the cumulative frequencies based on

two upper class boundaries.

cumulative frequency = 𝟒𝟓

𝐑𝐞𝐪𝐮𝐢𝐫𝐞𝐝 cumulative frequency = 𝟒𝟓 − 𝟑𝟓 = 𝟏𝟎

cumulative frequency = 𝟑𝟓


Remarks!

If the sample size and the (cumulative)

percentage of a specific class are given, then the

corresponding (cumulative) frequency of this class

can be calculated by

𝒇𝒊 =𝑷𝒊

𝟏𝟎𝟎%× 𝒏


Remarks! (cont.)

If both the (cumulative) frequency and the

(cumulative) percentage of a specific class are

given, then the sample size of this class can be

calculated by

𝒏 =𝒇𝒊𝑷𝒊

× 𝟏𝟎𝟎%


2 – 2: Other types of graphs

Bar graph:


illustrative Example 2 – 8 is in pages 77.

Time series chart:




2 – 2: Other Types of Graphs

Pie chart:

Definition is in page 80, with Example 2 – 11 in

pages 80-81, and Example 2 – 12 in pages 82.

Stem-and-leaf plot:

Definition is in page 84, with Example 2 – 14 in

pages 84, and Example 2 – 15 in pages 85.


Summary of graphs

Bar graph is obtained based on a categorical frequency

distribution, i.e., it is typically used with qualitative data.

Note that this graph is often preferred when the data are

ordinal-level qualitative. Also, note that this figure can

be also be used when the data are

Time series charts are used when the quantitative data

are observed over a period of time (e.g., minutes, hours,

etc.). Here, the independent variable is the time and the

variable observed over time is the dependent variable.


Summary of graphs (cont.)

Like the bar graph, the Pie chart is obtained based

on a categorical frequency distribution, and it is

highly recommended for nominal-level qualitative

data.

Stem-and-leaf plot is only used when the data are

quantitative.


Bar graph

0

2

4

6

8

10

12

14

16

18

A+ A B+ B C+ C D+ D F DN

Grade


Time series chart

0

100

200

300

400

500

600

700

800

2003 2004 2005 2006 2007

Number of homicides


Pie Chart

Remember: the degree of a class in a pie chart is defined as

𝒇

𝒏× 𝟑𝟔𝟎°

A20%

B28%

O36%

AB16%

A B O AB


Stem and leaf plot

The stem and leaf plot is similar to a horizontally

flipped histogram!


Stem and leaf plot

Consider the following numbers 1403 and 102.

The stem of 1403 is 140 and the leaf is 3.

The stem of 102 is 10 and the leaf is 2.


Stem and leaf plot

By joining the stems and the leaves, we notice that the

minimum is 02, while the maximum is 57.


Summary

Nominal Ordinal Discrete Continuous

Categorical

frequency

distribution

✓ ✓

Grouped

frequency

distribution

✓

(large range)

Ungrouped

frequency

distribution

✓✓

(small range)


Summary (cont.)

Nominal Ordinal Discrete Continuous

Bar Chart ✓ ✓ ✓

Time Series

Independent variable:

time

Dependent variable:

discrete or continuous

Pie Chart ✓ ✓

Stem-and-

leaf plot✓ ✓


Documents

Organizing data using frequency distributions. Illustrating data ...fmalam.kau.edu.sa/Files/0007085/Files/157188_STAT_110_CH...Overview of Chapter 2 Sec. # Title Page(s) 2 - 1 Organizing