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.

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

Copyright © 2010 Pearson Education, Inc. What percentage of college students get their information from television? 181% 32.9% 45.3% 60.3% Copyright © 2010 Pearson Education, Inc. Slide 2- 10 10

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.

Favorite Colors of 15 Friends 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 Red Pink Yellow YES! NO!

TI-nspire Qualitative Data Create a new lists & spreadsheets Title column Enter data into column Create a new Data & Statistics Page Click on x-axis box to add variable – choose, the name you just created 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 Red Pink Yellow

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.

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.

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 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.

The smallest data value? The largest data value? 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?

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

Copyright © 2010 Pearson Education, Inc. Find the class width. 3 4 5 19 Class Frequency, f 1 – 5 21 6 – 10 16 11 – 15 28 16 – 20 13 Copyright © 2010 Pearson Education, Inc. Slide 2- 30 30

Copyright © 2010 Pearson Education, Inc. Identify the type of graph shown. Bar Graph Pie Chart Pareto Chart Histogram Copyright © 2010 Pearson Education, Inc. Slide 2- 31 31

Copyright © 2010 Pearson Education, Inc. Which class has the highest frequency? 53 58 18 – 22 18 – 23 18 23 28 33 38 43 48 53 58 Copyright © 2010 Pearson Education, Inc. Slide 2- 32 32

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 8 3 888392922 4 782030104706 5 0145783237335253 6 89483940625 7 5 8 5

© 2010 Pearson Prentice Hall. All rights reserved 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 Colorado 5.1 Michigan 8.5 Rhode Island 7.5 Conn 5.4 Minnesota South Carolina 6.2 Delaware 4.2 Mississippi South Dakota 2.8 Dist Col 6.4 Missouri 5.7 Tenn 6.5 Florida Montana 4.1 Texas 4.4 Georgia Nebraska 3.3 Utah Hawaii Nevada Vermont Idaho New Hamp Virginia Illinois New Jersey Washington Indiana 5.8 New Mexico W. Virginia Iowa New York Wisconsin 4.6 Kansas 4.3 North Carolina 6.0 Wyoming © 2010 Pearson Prentice Hall. All rights reserved 34

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 8 3 888392922 4 782030104706 5 0145783237335253 6 89483940625 7 5 8 5 2 8 3 222388899 4 000012346778 5 0122333334555778 6 02344568899 7 5 8 5

Stem-and-Leaf Plot

A split stem-and-leaf plot: Best used when data appears bunched up 2 8 3 2223 3 88899 4 00001234 4 6778 5 0122333334 5 555778 6 02344 6 568899 7 5 8 5 This stem represents 3.0 – 3.4 This stem represents 3.5 – 3.9 2-37 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

Copyright © 2010 Pearson Education, Inc. What is the maximum data entry? 96 38 79 56 Key: 3 | 8 = 38 3 8 9 4 0 2 7 5 1 1 4 8 6 3 3 3 8 9 9 7 0 0 1 1 2 4 7 8 8 8 8 9 8 2 2 3 4 7 7 8 9 9 9 1 1 4 5 6 Copyright © 2010 Pearson Education, Inc. Slide 2- 39 39

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 2 4 2 2 2 1 2 2 0 2 4 1 1 3 2 4 1 2 1 2 2 3 3 2 1 2 2 0 3 2 2 2 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

Copyright © 2010 Pearson Education, Inc. True or false: The distribution is skewed right. True False 18 23 28 33 38 43 48 53 58 Copyright © 2010 Pearson Education, Inc. Slide 2- 44 44

TI-nspire Quantitative Data Create a new lists & spreadsheets Title column Enter data into column Create a new Data & Statistics Page Click on x-axis box to add variable – choose, the name you just created 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 28 36 66 53 45 38 58 44 56 46 63 51 48 55 33 35 49 52 73 60 62 32 39 69 50