2.1 Organizing Qualitative Data

Creating Bar Charts and Pie Charts

Frequency Vs. Relative Frequency

• A frequency distribution lists each category of data and the number of occurrences for each category of data.

• The relative frequency is the proportion (or percent) of observations within a category and is found using the formula:

• A relative frequency distribution lists the relative frequency of each category of data.

frequencyrelative frequency

sum of all frequencies

Bar Graphs• A bar graph is constructed by labeling each

category of data on either the horizontal or vertical axis and the frequency or relative frequency of the category on the other axis.

• A Pareto chart is a bar graph where the bars are drawn in decreasing order of frequency or relative frequency.

• Use the M&M data to construct – a frequency bar graph and – a relative frequency bar graph.

Frequency Bar Graph

Relative Frequency Bar Graph

Pareto Chart of M&M’s

Comparing Two Data Sets

• The following data represent the marital status (in millions) of U.S. residents 18 years of age or older in 1990 and 2006.– Draw a side-by-side

relative frequency bar graph of the data.

Marital Status

1990 2006

Never married

40.4 55.3

Married 112.6 127.7

Widowed 13.8 13.9

Divorced 15.1 22.8

Side-by-Side Bar Graph

What percentage of college students get their information from television?

A. 181%

B. 32.9%

C. 45.3%

D. 60.3%

Slide 2- 10Copyright © 2010 Pearson Education, Inc.

Pie Charts• A pie chart is a circle

divided into sectors.

• Each sector represents a category of data.

• The area of each sector is proportional to the frequency of the category.

Using a Calculator• You can use a calculator to create graphs

ONLY if you have raw data. You cannot use a frequency distribution in the calculator.

Favorite Colors of 15 Friends

Blue Green Green Red Pink

Blue Pink Red Red Red

Green Blue Red Yellow Red

TI-nspire Qualitative Data1. Create a new lists &

spreadsheets

2. Title column

3. Enter data into column

4. Create a new Data & Statistics Page

5. Click on x-axis box to add variable – choose, the name you just created

6. Click Menu and change graph type to desired graph.

• Create a dot plot, bar chart, and pie chart for the following data.

Favorite Colors of 15 Friends

Blue Green Green Red Pink

Blue Pink Red Red Red

Green Blue Red Yellow Red

2.2 Organizing Quantitative Data• The first step in summarizing quantitative data is

to determine whether the data is discrete or continuous.

• If the data is discrete and there are relatively few different values of the variable, the categories of data will be the observations (as in qualitative data).

• If the data is discrete, but there are many different values of the variable, or if the data is continuous, the categories of data (called classes) must be created using intervals of numbers.

Constructing Frequency and Relative Frequency Distribution from Discrete Data

• The following data represent the number of available cars in a household based on a random sample of 50 households.

– Construct a frequency and relative frequency distribution.

Histograms

• A histogram is constructed by drawing rectangles for each class of data whose height is the frequency or relative frequency of the class.

• The width of each rectangle should be the same and they should touch each other.

Histograms

• Draw a frequency and relative frequency histogram for the “number of cars per household” data.

Number of cars per household

Continuous Data• Categories of data are created for continuous data using

intervals of numbers called classes.

• The following data represents the number of persons aged 25 - 64 who are currently work disabled.

• The class width is the difference between consecutive lower class limits. The class width of the data given above is 35 - 25 = 10.

Age Number (in thousands)

25 – 34 2,132 35 – 44 3,928 45 – 54 4,532 55 – 64 5,108

Continuous Data• The lower class limit of a class is the smallest value

within the class

– The lower class limit of first class is 25.

– The lower class limit of the second class is 35.

• The upper class limit of a class is the largest value within the class.

– The upper class limit of the first class is 34.

Age Number (in thousands)

25 – 34 2,132 35 – 44 3,928 45 – 54 4,532 55 – 64 5,108

Guidelines• Should be between 5 and 20 classes

– Choose a class width the you think will summarize the data well

• Try to avoid open ended classes– 60 and older

• Watch out for tables with class widths that overlap– Class of 20-30 and 30-40

Sample Problem - Organizing Continuous Data into a Frequency and Relative Frequency Distribution

• The following data represent the time between eruptions (in seconds) for a random sample of 45 eruptions at the Old Faithful Geyser in California. Construct a frequency and relative frequency distribution of the data.

Sample Problem - Organizing Continuous Data into a Frequency and Relative Frequency Distribution

• The smallest data value?– 672, so maybe we should start the class at 670?

• The largest data value?– 738, so maybe we should end the class at 740?

Sample Problem - Organizing Continuous Data into a Frequency and Relative Frequency Distribution

• Now select a width…maybe 5 or 10? See what looks good with the data.

670 - 679

680 - 689

690 - 699

700 - 709

710 - 719

720 - 729

730 - 739

670 – 674 705 - 709

675 – 679 710 - 714

680 – 684 715 - 719

685 – 689 720 - 724

690 – 694 725 - 729

695 – 699 730 - 734

700 – 704 735 - 739

Using class width of 10:

Using class width of 5:

Find the class width.

A. 3

B. 4

C. 5

D. 19

Class Frequency, f

1 – 5 21

6 – 10 16

11 – 15 28

16 – 20 13

Slide 2- 30Copyright © 2010 Pearson Education, Inc.

Identify the type of graph shown.

A. Bar Graph

B. Pie Chart

C. Pareto Chart

D. Histogram

Slide 2- 31Copyright © 2010 Pearson Education, Inc.

Which class has the highest frequency?

A. 53

B. 58

C. 18 – 22

D. 18 – 23 18 23 28 33 38 43 48 53 58

Slide 2- 32Copyright © 2010 Pearson Education, Inc.

Stem-and-Leaf Plot• A stem-and-leaf plot uses digits to the left

of the rightmost digit to form the stem.

• Each rightmost digit forms a leaf. – For example, a data value of 147 would have

14 as the stem and 7 as the leaf.

2 83 8883929224 7820301047065 01457832373352536 894839406257 58 5

State Unemployment Rate

State Unemployment Rate

State Unemployment Rate

Alabama 4.7 Kentucky 6.3 North Dakota 3.2

Alaska 6.8 Louisiana 3.8 Ohio 6.6

Arizona 4.8 Maine 5.3 Oklahoma 3.9

Arkansas 5.0 Maryland 4.0 Oregon 5.5

California 6.9 Mass 5.2 Penn 5.2

Colorado 5.1 Michigan 8.5 Rhode Island 7.5

Conn 5.4 Minnesota 5.3 South Carolina 6.2

Delaware 4.2 Mississippi 6.9 South Dakota 2.8

Dist Col 6.4 Missouri 5.7 Tenn 6.5

Florida 5.5 Montana 4.1 Texas 4.4

Georgia 5.7 Nebraska 3.3 Utah 3.2

Hawaii 3.8 Nevada 6.4 Vermont 4.7

Idaho 3.8 New Hamp 4.0 Virginia 4.0

Illinois 6.8 New Jersey 5.3 Washington 5.5

Indiana 5.8 New Mexico 3.9 W. Virginia 5.3

Iowa 4.0 New York 5.3 Wisconsin 4.6

Kansas 4.3 North Carolina

6.0 Wyoming 3.2

© 2010 Pearson Prentice Hall. All rights reserved

Sample Problem• An individual is considered to be unemployed if they do

not have a job, but are actively seeking employment. The following data represent the unemployment rate in each of the fifty United States plus the District of Columbia in June, 2008.

• We let the stem represent the integer portion of the number and the leaf will be the decimal portion. – For example, the stem of Alabama will be 4 and the leaf will be 7.

2 83 8883929224 7820301047065 01457832373352536 894839406257 58 5

2 83 2223888994 0000123467785 01223333345557786 023445688997 58 5

Stem-and-Leaf Plot

A split stem-and-leaf plot: Best used when data appears bunched up

2 83 22233 888994 000012344 67785 01223333345 5557786 023446 5688997 7 58 8 5

This stem represents 3.0 – 3.4

This stem represents 3.5 – 3.9

2-37

Advantage of Stem-and-Leaf Diagrams over Histograms

• Once a frequency distribution or histogram of continuous data is created, the raw data is lost (unless reported with the frequency distribution)

• However, the raw data can be retrieved from the stem-and-leaf plot.

• Stem-and-leaf plots are best used when the data set is SMALL

What is the maximum data entry?

A. 96

B. 38

C. 79

D. 56

3 8 94 0 2 7 5 1 1 4 86 3 3 3 8 9 9 7 0 0 1 1 2 4 7 8 8 8 8 98 2 2 3 4 7 7 8 9 99 1 1 4 5 6

Key: 3 | 8 = 38

Slide 2- 39Copyright © 2010 Pearson Education, Inc.

Dot Plot

• A dot plot is drawn by placing each observation horizontally in increasing order and placing a dot above the observation each time it is observed.

Dot Plot• The following data represent the number

of available cars in a household based on a random sample of 50 households.

• Draw a dot plot of the data.

3 0 1 2 1 1 1 2 0 24 2 2 2 1 2 2 0 2 41 1 3 2 4 1 2 1 2 23 3 2 1 2 2 0 3 2 22 3 2 1 2 2 1 1 3 5

Data based on results reported by the United States Bureau of the Census.

Dot Plot

Distribution Shape

True or false: The distribution is skewed right.

A. True

B. False

18 23 28 33 38 43 48 53 58

Slide 2- 44Copyright © 2010 Pearson Education, Inc.

TI-nspire Quantitative Data1. Create a new lists &

spreadsheets

2. Title column

3. Enter data into column

4. Create a new Data & Statistics Page

5. Click on x-axis box to add variable – choose, the name you just created

6. Click Menu and change graph type to histogram

• Create a Histogram for the following data

40 Randomly Selected 20-29 yr. olds.Serum HDL Cholesterol

70 54 70 2836 66 53 4538 45 58 4456 46 56 5346 63 51 4856 55 33 3549 48 52 5173 60 62 6032 39 51 5269 50 44 48