MATH125

Applying Chapter 2

Recall: two types of data: Qualitative (“quality”): That describe categories, or attributes, such as:

Number of males and females in a class. Number of Democrats and Republicans in a district. Religion (how many Catholics, Lutherans, Baptists, Jews,

Muslims, Hindus, etc… in a county). Job occupation (How many doctors, teachers, policemen..) Etc…

Quantitative (“quantity”)That can be measured and can have any number value, such as: Income Houses square footage (or any measurement) Gas mileage Height Weight Etc…

How to make…Graphs for Qualitative DataGraphs for Qualitative Data

Properties of Qualitative Graphs:Applies to all graphs:

1. The graph has a definitive title describing the data being graphed.

2. All data is accounted:a) The sum of the frequencies accounts for all data

values. b) The sum of the relative frequencies adds up to 1.00

(Percents sum to 100).Note: Due to rounding, the sum may be close to 1.00, usually from 0.98 to 1.02. (Percents range from 98% to 102%.)

Properties of Qualitative Graphs (continued):Specific to Bar Graphs:

3. Each axis has a general descriptive label.

4. Each axis has a specific label. On one axis, the bars are identified with a defining label. On the other axis, the bar’s lengths are quantified with a number. An appropriate scale is chosen to display the bar’s lengths.

Specific to Pie Charts:

3. Each slice is identified with a defining label.

4. The sum of the degrees in a pie chart adds to 360 degrees, but degree measures are NOT labeled on the pie chart. Either frequency or relative frequency label the slices.Note: Due to rounding, the sum may be close to 360 degrees, usually from 358 degrees to 362 degrees.

Example 1:Does the graph describing the distribution of students attending different grades (below) satisfy the rules for a graph? (source: USA TODAY)

Answers:

1. “Where Our Students Are” is the title which defines the graph as placing students in different grades.

2. The sum of the percentages (relative frequencies) is 93.6%. This sum is too far from 100% to be considered a rounding error.

3. The graph does not have descriptive labels on the horizontal (x) and vertical (y) axes.

4. The graph does have specific labels identifying the bars (pencils) and their lengths are quantified. However, the scale is absent from the horizontal axis.

The graph is not properly constructed.

Example 2:Does the graph below describing the driving habits of Americans in

response to increasing gasoline prices satisfy the rules for a graph? (source: USA TODAY)

Answers:

1. “Altering Driving Habits” defines the graph as drivers reactions to increased gasoline prices.

2. The numeric values are labeled as percents. The sum of the percentages is 100%. This sum is appropriate.

3. Each slice has a definitive label.

4. The chart’s slices complete 360 degrees. (It’s a complete circle.)

The graph is properly constructed.

Practice Exercise 1:Does the graph below, “Adjustment from Vacation” satisfy the properties of a graph?

Go over the checklist:

1. Is there a title?

Ans:

2. Is the sum of %’s acceptable?

Ans:

3. Is there a general label on the axis?

Ans:.

4. Are there specific labels with each bar and is a scaling present?

Ans:

Conclude:

Practice Exercise 2:Does the graph below, “How often we summer BBQ” satisfy the properties of a graph?

Go over the checklist:

1. Is there a title?

Ans:

2. Is the sum of %’s acceptable?

Ans:

3. Is there a defining label on each slice?

Ans:.

4. Does the chart complete a full circle?

Ans:

Conclude:

Constructing Graphs for Qualitative Data

The Data:Example: Donations made to the American Heart Association

RECIPIENT CENTS/DOLLAR

Administrative $0.086

Fund Raising $0.136

Community Services $0.117

Research $0.314

Education $0.244

Training $0.103

TOTALS $1.000

Bar Graph

RECIPIENT CENTS/DOLLAR RELATIVE FREQUENCY (%)

Administrative $0.086

Fund Raising $0.136

Community Services

$0.117

Research $0.314

Education $0.244

Training $0.103

TOTALS $1.000 Sum of all %’s = 100%

Constructing Bar GraphsCalculate the Relative Frequency Percentages:Add column: relative frequency = (each value)/(total) (express as %)

%6.8086.0000.1$

086.0$

%6.13136.0000.1$

136.0$

%7.110.117000.1$

$0.117

%4.310.314000.1$

0.314$

%4.240.244000.1$

0.244$

%3.10.1030000.1$

0.103$

Rules for Constructing a Bar Graph1) Must have a titletitle describing the data being graphed.

2) Must have a descriptive general label on each axisgeneral label on each axis.

3) Both axes must also have specific labelsspecific labels. On one axis, the bars are identified with a defining label. On the other axis, their lengths are quantified with a number. An appropriate scalescale is chosen to display the bar’s lengths.

4) All data is accounted: The sum of the relative frequencies sum of the relative frequencies adds up to 1.00adds up to 1.00 (Percents add up to 100%Percents add up to 100%).Note: Due to rounding, the sum may be close to 1.00, usually from 0.98 to 1.02. (Percents range from 98% to 102%.)

Types of bar graphs:

What QUANTITY to chart:

Frequency (count)

Relative Frequency (percentage, or fraction).

Relative Frequency Bar Chart for the Donations made to the American Heart Association:

• The title is definitive.

• Each axis has a general and specific label.

• The specific labels identifies and quantifies each bar.

• The scale is 5%. The sum of the bar’s percents is 100% .

Pie Chart

Constructing a Pie ChartCalculate the angles for the circle slices:Add column: angle = relative frequency x 360o

Rules for Constructing a Pie Chart1) The pie chart has a definitive titletitle describing the data being

graphed.

2)2) Descriptive labelDescriptive label on each slice.

3)3) Each Each slice is labeled with either frequencyfrequency or relative frequencyrelative frequency (percent).DO NOT use degrees as a label.

4) The sum of the frequencies accounts for all data valuesall data values.

5) The sumsum of the relative frequencies adds to 1.001.00 (Percents sum to Percents sum to 100100%). Note: Due to rounding, the sum may be close to 1.00, usually from 0.98 to 1.02. (Percents range from 98% to 102%.)

6) The sum of the degrees in a pie chart adds up to 360 degrees360 degrees.Note: Due to rounding, the sum may be close to 360 degrees, usually from358 degrees to 362 degrees.

Pie Chart of Relative Frequencies for the Donations made to the American Heart Association:

• The title is definitive.

• Each slice has a descriptive label that identifies it.

• Each slice has a specific label of its relative frequency.

• The sum of the percent frequencies is 100% .

Practice!

Problem 1Construct a frequency bar graph displaying the sources of the US federal government’s income for 2002. The federal government’s income for 2002 was $1.9 trillion ($1,900,000,000,000 = $1,900 billion). Social Security contributed $735 billion, Personal Income Taxes contributed $903 billion, Excise Taxes contributed $147 billion, Corporate Income Taxes contributed $147 billion and Borrowing contributed $168 billion.

2002 US Federal Government Income

Sources of Income $ Billions

Social Security

Personal Income Taxes

Excise Taxes

Corporate Income Taxes

Borrowing

Answer:

1) Fill the table

Problem 1 (answer continued)

2) Construct the chart:

a) Choose a scale

b) Put in the values of the bar heights

c) Follow all rules: Title, general and specific labels

Pb1-Ans

Problem 2The following is the distribution of funds for every dollar donated to the United Ways. Construct a relative frequency pie chart.

Distribution Cents / Dollar

Administrative Costs $0.128

Nurturing Children $0.253

Develop Self-Sufficiency $0.174

Health & Wellness $0.140

Strengthen Families $0.192

Strong Communities $0.113

Problem 2 (answer)1) Add a column for relative frequencies and a column for angles of the pie slices:

DistributionCents /

DollarRelative

Frequency (%)Degrees

Administrative Costs $0.128

                                                         

Nurturing Children $0.253

                                                        

Develop Self-Sufficiency $0.174

                                                        

Health & Wellness $0.140

                                                    

Strengthen Families $0.192

                                                        

Strong Communities $0.113

                                                        

Problem 2 (answer continued)

2) Construct the chart:

a) Draw a circle

b) Divide into the appropriate slices

c) Follow all rules: Title, general and specific labels on each slice, sum of all percent frequencies must add up to 100%

Pb2-Ans

How to make…Graphs for Quantitative DataGraphs for Quantitative Data

Graphs for Quantitative Data Histograms and Polygons:

Example: Distribution of SUV prices. Example: Distribution of SUV prices.

Histogram Polygon

Data for SUV prices:Data for SUV prices:

Prices of 40 four-wheel drive (4WD) Sport Utility Vehicles (SUV). These vehicles range from the Geo Tracker to the Range Rover.

$14,655 $14,799 $63,500 $15,605 $31,985 $32,250 $26,268 $17,990 $19,300 $32,950 $33,595 $33,790 $22,708 $23,240 $23,920 $27,815 $23,405 $29,099 $29,249 $30,585 $30,645 $16,395 $16,798 $34,590 $35,550 $36,300 $27,910 $28,680 $28,950 $38,175 $41,188 $25,999 $26,185 $20,000 $25,176 $42,660 $54,950 $56,000 $21,995 $22,195

Steps to construct a Graph:Steps to construct a Graph:

1. Organize the data in a table in ascending or descending order.

2. Choose the number of classes.3. Choose the class width [must be about equal or greater than

(maximum data value – minimum data value)/number of classes].4. Choose the starting point of the first class. 

Conditions that must be satisfied:

1. The class width, the size of each interval, should be the same for each class.

2. The minimum data value must be contained in the first class. 3. The last class must contain the maximum data value.

(The first and last class cannot be empty.)

1. SUV prices sorted from smallest to largest:

$14,655 $14,799 $15,605 $16,395 $16,798 $17,990 $19,300 $20,000$21,995 $22,195 $22,708 $23,240 $23,405 $23,920 $25,176 $25,999$26,185 $26,268 $27,815 $27,910 $28,680 $28,950 $29,099 $29,249$30,585 $30,645 $31,985 $32,250 $32,950 $33,595 $33,790 $34,590$35,550 $36,300 $38,175 $41,188 $42,660 $54,950 $56,000 $63,500

2. Let’s choose 6 classes (you could choose more or less).

3. Class width:The maximum data value is $63,500 and the minimum data value is $14,655. Therefore: ($63,500 - $14,655)/6 $8,140.83333

→ Let’s choose a class width of $10,000.

4. Choose $10,000 as the starting point. Add the class   width to your starting point.

Apply these steps to construct a Graph for the Apply these steps to construct a Graph for the SUV prices:SUV prices:

Classes:Classes:

Class Dollars ($1,000’s)

1st $10 $20

2nd $20 $30

3rd $30 $40

4th $40 $50

5th $50 $60

6th $60 $70

Note: The classes are scaled and satisfy the three rules for quantitative data: a. The six classes have the same class width, $10,000.   b. The minimum value of $14,655 falls in the first class.   c. The maximum value of $63,500 falls in the last class.

Table of frequencies and relative frequencies for SUV Table of frequencies and relative frequencies for SUV example:example:

Note: the lower limit of the class belongs to the class, the upper limit belongs to next class (for example: The data item $20,000 is counted in the 2nd class, $20,000 to $30,000, but the data item $30,000 is counted in the 3rd class.)

Class Dollars ($1,000’s) FrequencyRelative

Frequency.

1st $10 $20 7                         2nd $20 $30 17                         3rd $30 $40 11                         4th $40 $50 2                         5th $50 $60 2                         6th $60 $70 1                         

Totals   40                         

HistogramHistogram

Properties of Histograms (Bar Graphs for Quantitative Data)

1. The histogram has a definitive title describing the data being graphed.

2. Each axis has a descriptive label.

3. Each axis has a scale. On one axis, the bars are quantified using the classes from the data table. On the other axis, the bar’s lengths are quantified with an appropriate scale chosen to display the bar’s lengths.

4. The sum of the:a) Frequencies accounts for all data values. b) Relative frequencies adds to 1.00 (Percents sum to 100).

Note: Due to rounding, the sum may be close to 1.00 (100%.)

Histograms for the SUV prices:Histograms for the SUV prices:

The properties of histograms are satisfied:1. The graph has a definitive title, “SUV Prices.”2. The sum of frequencies accounts for all the data values.3. Each axis has a descriptive title “Dollars ($1,000’s)” and “Frequency”.4. Each axis has a quantifying scale. One scale uses the classes to define the bars. The other scale quantifies the height of the bars.

Frequency Histogram Relative Frequency Histogram

The properties of histograms are satisfied:1. The graph has a definitive title, “SUV Prices.”2. The sum of the percentages is 100%.3. Each axis has a descriptive title “Dollars ($1,000’s)” and “Percents”.4. Each axis has a quantifying scale. One scale uses the classes to define the bars. The other scale quantifies the height of the bars.

PracticeHome prices vary in different neighborhoods of Chicago. Make a table with seven classes and use it to construct a frequency histogram.

$ 55,000 $ 56,000 $ 64,900 $ 77,900 $ 80,000 $ 87,500 $ 90,000$ 95,000 $ 98,000 $109,700 $110,000 $118,000 $120,000 $128,000$133,900 $135,000 $140,000 $142,000 $147,000 $150,000 $153,500$159,900 $160,000 $171,000 $182,000 $187,900 $188,000 $195,000$196,000 $220,000 $220,900 $230,000 $230,000 $245,075 $285,000$294,000 $297,000 $349,925 $367,000