Embed Size (px)
Citation preview
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 69
54 56 69 68 78 65 85 49 69 61
48 81 68 37 43 78 82 43 64 67
52 56 81 77 79 85 40 85 59 80
60 71 57 61 69 61 83 90 87 74
Class Limits
Tally Frequency
35 – 41 /// 3
42 – 48 /// 3
49 – 55 //// 4
56 – 62 //// //// 10
63 – 69 //// //// 10
70 – 76 //// 5
77 – 83 //// //// 10
84 - 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 O
O O B AB B
B B O A O
A 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 /// 3
31 – 37 30.5 – 37.5 / 1
38 – 44 37.5 – 44.5 //// 5
45 – 51 44.5 – 51.5 //// //// 9
52 – 58 51.5 – 58.5 //// / 6
59 – 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.
Miles0
1
2
3
4
5
6
.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 - Histograms
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
99.5
-104
.5
109.
5-11
4.5
119.
5-12
4.5
129.
5-13
4.5
0
4
8
12
16
20
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 38 430
1
2
3
4
5
6
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
20
Record High Temperatures
Frequency
Class Boundaries
Midpoints
Frequency
99.5 – 104.5 102 2
104.5 – 109.5 107 8
109.5 – 114.5 112 18
114.5 – 119.5 117 13
119.5 – 124.5 122 7
124.5 – 129.5 127 1
129.5 – 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
25
Cumulative 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
cy
Less than 99.5 0
Less 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
50
99.5
109.
5
119.
5
129.
50
10
20
30
40
50
60
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 axis
2. 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
100
200
300
400
500
600
700
800
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
Flor
ida
Penn
sylv...
Oklah
oma
Mai
ne
Indi
ana
0
1
2
3
4
5
6
7Average Cost per mile
FloridaPennsylvaniaOklahomaMaineIndiana
Time Series Graph
Represents data that occur over a specific period of time
How to create a Time Series Graph
1. 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 20050
1
2
3
4
5
6
7
8
9
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 Plot
1. 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 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
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 1
3 0 0 0 0 2 2 3 4 6 6 6 8 8 9 9
7 4 4 0 0 4 0 0 0 0
5 3 2 2 0 0 5 0 3 4 8
3 0 6 1
0 7
APPLYING THE CONCEPTS 2-3
Answers – Applying the Concepts 2-3
1. year, cause of death, and rate of death per 100,000 men
2. Cause of death is qualitative, year and death rates are quantitative
3. 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.
