FREQUENCY DISTRIBUTIONS AND GRAPHS

Chapter 2:

Section 1 – Organizing DataLearning Target- I will be able to organize data using a

frequency distribution.

Frequency Distribution

The organization of raw data in table form, using classes and frequencies.

Class – a qualitative or quantitative category

Frequency – the number of data values contained in a specific class

Example: 49 57 38 73 81 74 59 76 65 6954 56 69 68 78 65 85 49 69 6148 81 68 37 43 78 82 43 64 6752 56 81 77 79 85 40 85 59 8060 71 57 61 69 61 83 90 87 74

Class Limits

Tally Frequency

35 – 41 /// 342 – 48 /// 349 – 55 //// 456 – 62 //// //// 1063 – 69 //// //// 1070 – 76 //// 577 – 83 //// //// 1084 - 90 //// 5

Total: 50

Categorical Frequency Distribution Used for data that can be placed in

specific categories such as nominal and ordinal level data

Example:

Raw Data – Blood Types

Steps to Make a Categorical Frequency Distribution

A B B AB OO O B AB BB B O A OA O O O AB

AB A O B A

Step 1: Make a Table

Class Tally Frequency

Percent

A

B

O

AB

Steps continued Step 2: Tally the data Step 3: Count the

tallies and place results under frequencies.

Step 4: Find the percentage of each value by using the following formula % = f/n x 100%

Step 5: Find the totals for the frequency and percent column.

Class

Tally Frequency

Percent

A //// 5 20

B //// // 7 28

O //// ////

9 36

AB //// 4 16

Total: 25

Total: 100

Grouped Frequency Distribution

Use when the range of data is large and must be grouped into classes that are more than one unit in width

Example

Class Limits Class Boundaries

Tally Frequency

24 – 30 23.5 – 30.5 /// 331 – 37 30.5 – 37.5 / 138 – 44 37.5 – 44.5 //// 545 – 51 44.5 – 51.5 //// //// 952 – 58 51.5 – 58.5 //// / 659 – 65 58.5 – 65.5 / 1

Total: 25

Vocabulary for Grouped Frequency Distributions

Class Limits Lower class limits – the smallest data value

that can be included in the class Upper class limits – the largest data value

that can be included in the class Class Boundaries – numbers to separate

the classes so there are no gaps Class Width – subtract the lower class

limit of the first class from the lower class limit of the second class

Steps to Make a Grouped Frequency Distribution Step 1: Determine the classes

Find the highest and lowest values Find the range Select number of classes desired (5 – 20) Find the width by dividing the range by the

number of classes and rounding up Select a starting point (usually the lowest value

or any convenient number less than the lowest value); add the width to get the lower limits

Find the upper class limits Find the boundaries

Steps (cont’)

Step 2: Tally the data Step 3: Find the numerical

frequencies from the tallies, and find the cumulative frequencies

Reasons for constructing

1. To organize the data in a meaningful, intelligible way.

2. To enable the reader to determine the nature or shape of the distribution

3. To facilitate computational procedures for measures of average and spread

4. To enable the researcher to draw charts and graphs for the presentation of data.

5. To enable the reader to make comparisons among different data sets.

Applying the Concepts 2-1

Section 2-2

Histograms, Frequency Polygons, and Ogives

Learning Target

I will be able to represent data in a frequency distribution graphically using histograms, frequency polygons, and ogives.

Histograms

A graph that displays data with contiguous (touching) vertical bars of various heights to represent the frequencies of the classes.

Miles0123456

.5-3.5 hours3.5-5.5 hours5.5-7.5 hours7.5-9.5 hours9.5-11.5 hours

Histograms (cont’)

How to create a histogram

Step 1: Draw and label the x and y axes.

Step 2: Represent the frequency on the y-axis and class boundaries on the x-axis. Give the graph a title.

Step 3:Using the frequencies as heights, draw vertical bars for each class.

Example - HistogramsClass

BoundariesFrequency

99.5 – 104.5 2104.5 – 109.5 8109.5 – 114.5 18114.5 – 119.5 13119.5 – 124.5 7124.5 – 129.5 1129.5 – 134.5 1

99.5-

104.5

109.5

-114.5

119.5

-124.5

129.5

-134.5

048

121620

Record High Temper-atures

Frequency

Frequency Polygons

A graph that displays the data by using lines that connect the points plotted for the frequencies at the midpoint of the classes

3 8 13 18 23 28 33 38 430123456

Frequency

Frequency

Steps to Create a Frequency Polygon Step 1: Find the midpoints of each

class Add the upper and lower boundaries and

divide the sum by 2. Step 2: Draw the x and y axes. Label

the x with the midpoint and use a suitable scale on the y axis for the frequency.

Step 3: Using the midpoints for the x values and the frequencies as y values, plot the points.

Steps (cont’)

Step 4: Connect adjacent points with line segments. Draw a line back to the x axis at the beginning and end of the graph, at the same distance that the previous and next midpoints would be located. To do this, add a class to the beginning

and end of the frequency distribution with a frequency of zero for both.

Example – Frequency Polygon

97 102

107

112

117

122

127

132

137

0

5

10

15

20Record High Temperatures

Frequency

Class Boundaries

Midpoints

Frequency

99.5 – 104.5 102 2104.5 – 109.5 107 8109.5 – 114.5 112 18114.5 – 119.5 117 13119.5 – 124.5 122 7124.5 – 129.5 127 1129.5 – 134.5 132 1

A histogram and frequency polygon are two ways to represent the same data set. The choice of which one to use is up to the researcher.

Ogive – (pronounced ojive) A line graph that represents the

cumulative frequencies for the classes in a frequency distribution.

Cumulative frequency – the sum of the frequencies accumulated up to the upper boundary of a class in the distribution.

Ogives look like

5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.50

5

10

15

20

25Cumulative Frequency

Cumulative Frequency

Steps to Create Ogives

Step 1: Find the cumulative frequency for each class. Add each frequency

one at a time so that the last class has the total.

Step 2: Draw the x and y axes. Label the x axis with the class boundaries. Use an appropriate scale for the y axis to represent the cumulative frequencies.

Steps (cont’)

Step 3: Plot the cumulative frequency at each upper class boundary.

Step 4: Starting with the first upper class boundary connect adjacent points with line segments. Extend the graph to the first lower class boundary on the x axis.

Example - OgiveCumulati

ve Frequen

cyLess than 99.5 0Less than 104.5

2

Less than 109.5

10

Less than 114.5

28

Less than 119.5

41

Less than 124.5

48

Less than 129.5

49

Less than 134.5

5099

.510

9.511

9.512

9.50

102030405060

Record High Temper-atures

Cumulative Frequency

Cumulative frequency graphs are used to visually show how many values are below a certain upper class boundary.

To find how many values are less than a specific boundary, draw a vertical line up to the graph and then a horizontal line to the y axis.

Relative Frequency Graphs

Histograms, frequency polygons and ogives that use proportions of the raw data.

To find the proportion, divide the frequency by the total items.

The cumulative relative frequency will always add to one.

Class Boundaries

Midpoint

Frequency

Relative Frequency

Cumulative Frequency

Cumulative Relative Frequency

5.5 – 10.5 8 1 0.05 Less than 5.5

0 0.00

10.5 – 15.5 13 2 0.10 Less than 10.5

1 0.05

15.5 – 20.5 18 3 0.15 Less than 15.5

3 0.15

20.5 – 25.5 23 5 0.25 Less than 20.5

6 0.30

25.5 – 30.5 28 4 0.20 Less than 25.5

11 0.55

30.5 – 35.5 33 3 0.15 Less than 30.5

15 0.75

35.5 – 40.5 38 2 0.10 Less than 35.5

18 0.90

Less than 40.5

20 1.00

5.5 – 10.5

10.5 –

15.5

15.5 –

20.5

20.5 –

25.5

25.5 –

30.5

30.5 –

35.5

35.5 –

40.5

0

0.05

0.1

0.15

0.2

0.25

0.3

Runners’ Times

Relative Frequency

8 13 18 23 28 33 380

0.05

0.1

0.15

0.2

0.25

0.3

Runners’ Times

Relative Frequency

5.5 10.5 15.5 20.5 25.5 30.5 35.5 40.50

0.2

0.4

0.6

0.8

1

1.2

Runners’ Times

Cumulative Relative Frequency

Applying Concepts 2-2

OTHER TYPES OF GRAPHS

Section 2-3

0 1 2 3 4 5 6

Bar Graphs

Represent the data by using vertical or horizontal bars, whose heights and lengths represent the frequencies of the data

How to create bar graph

1. Draw and label x and y axes. Horizontal – frequency goes on the x

axis, categories go on the y axis Vertical – frequency goes on the y axis,

categories go on the x axis2. Draw the bars corresponding to the

frequencies.

Example – Horizontal Bar Graph

Electronics

Dorm Décor

Clothing

Shoes

0 100200300400500600700800

1st Year College Spending

1st Year College Spending

Example – Vertical Bar Graph

Electron-ics

Dorm Décor

Clothing Shoes0

100200300400500600700800

1st Year College Spending

1st Year College Spending

Pareto Charts

Represent a frequency distribution for a categorical variable, and the frequencies are displayed by the heights of vertical bars, which are arranged in order from highest to lowest

How to create a Pareto Chart1. Arrange the data from highest to

lowest according to frequency. 2. Draw and label the x and y axes.

The x axis is the category and the y axis is the frequency.

3. Draw the bars corresponding to the frequencies. Make sure the bars are touching.

Example – Pareto Chart

Florid

a

Penn

sylv..

.

Oklaho

maMain

e

Indian

a01234567

Average Cost per mile

FloridaPennsylvaniaOklahomaMaineIndiana

Time Series Graph

Represents data that occur over a specific period of time

How to create a Time Series Graph1. Draw and label the x and y axes.

The x axis is the time and the y axis is the data.

2. Plot each point from the table. 3. Draw a line connecting the points.

The line does not have to be a smooth curve or straight line, just connect the dots.

Example – Time Series Graph

2001 2002 2003 2004 20050123456789

Damage (in millions)

Damage (in mil-lions)

Pie Graph

A circle that is divided into sections or wedges according to the percentage of frequencies in each category of the distribution.

How to create a Pie Graph

1. Since there are 360 degrees in a circle, the frequency for each class must be converted into a proportional part of the circle. This is done by using the formula where f is the frequency and n is the sum of the frequencies.

2. Convert each frequency to a percentage by using the formula

Steps continued

3. Use a protractor and a compass to draw the graph using the appropriate degree measures and label the sections with the name and percentages of each category

Example – Pie Chart

37.3%

27.3%

14.3%

12.7%

8.3%

Pounds

Potato ChipsTortilla ChipsPretzelsPopcornSnack Nuts

Stem and Leaf Plots

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.

How to create a Stem and Leaf Plot1. Arrange the data in order from

lowest to highest. 2. Separate the data according to the

first digit.3. In a table with two columns, the left

column being the stem and the right being the leaves, place the first digit of the groups in the stem column and the digit that follow in the leaf column.

Things to remember

Only one number can be a leaf Stems can be multiple digits Leaves are arranged in order from

lowest to highest, separated by a space or comma

Back to Back Stem and Leaf Plot If you are comparing two sets of data, you

can use a back to back stem and leaf plot A back to back stem and leaf plot has three

columns. The middle column is the stem, the left column is the leaves of the first set of data with the leaves being arranged from right to left, lowest to highest. The right column is the leaves of the second set of data with the leaves arranged from left to right, lowest to highest.

Example – Stem and Leaf Plot

Stem

Leaves

0 21 3 42 0 3 5 3 1 2 2 2 2 3 64 3 4 4 55 1 2 7

Example – Back to Back Stem and Leaf Plot

Atlanta Stem

Philadelphia

9 8 6 2 5 8 6 4 4 2 2 2 2

2 13 0 0 0 0 2 2 3 4 6 6 6

8 8 9 97 4 4 0 0 4 0 0 0 0

5 3 2 2 0 0 5 0 3 4 83 0 6 1

0 7

APPLYING THE CONCEPTS 2-3

Answers – Applying the Concepts 2-31. year, cause of death, and rate of

death per 100,000 men2. Cause of death is qualitative, year

and death rates are quantitative3. Year is discrete, death rate is

continuous, cause of death is neither

4. Line graph

Answers continued

5. No, pareto charts can only have one qualitative and one quantitative variable6. No, same reason as pareto chart7. Pareto chart typically shows categorical variable listed from highest frequency category to the lowest frequecy category8. Time series chart is used to see trends in the data, can also be used for forecasting and predicting.