Graphs for Quantitative Data and Misleading Graphs

Chapter 2.3 & 2.4: Graphs for Quantitative Data and Misleading Graphs

Introduction

Instructor & Textbook

  • Instructor: Melony Parkhurst

  • Textbook: Essential Statistics, 2nd Ed., Navidi & Monk

    • Images used were created or obtained from personal files, course textbook files, credited websites, subscription to shutterstock.com, or free files (e.g., unsplash.com).

Place Value

  1. Understanding rounding based on place value:

    • Rounding rules depend on the specific 'place' requested:

      • Thousandths place

      • Hundredths place

      • Tenths place

  2. Observation on labels for values after the decimal point:

    • Importance of identifying the specific place being referenced for rounding.

  3. Connections to Math homework - understanding rounding is frequently applied in mathematical assignments.

Section 2.3: Stem-and-Leaf Plots

Definition and Construction

  • Stem-and-leaf plots are a tool used to display quantitative data visually.

  • Structure of Stem-and-Leaf Plots:

    1. Stem - non-decimal part of the number.

    2. Leaf - rightmost digit.

Example Data:

  • U.S. Census Bureau projection for the percentage of the population aged 65 and over for each state and the District of Columbia:

    • Alabama: 14.1

    • Rhode Island: 14.1

    • Nevada: 12.3

    • Kentucky: 13.1

    • Arkansas: 14.3

    • Tennessee: 13.3

    • New Mexico: 14.1

    • Maryland: 12.2

    • Connecticut: 14.4

    • Vermont: 14.3

    • North Dakota: 15.3

    • Minnesota: 12.4

    • Florida: 17.8

    • West Virginia: 16.0

    • Oregon: 13.0

    • Montana: 15.0

    • Idaho: 12.0

    • Alaska: 8.1

    • South Carolina: 13.6

    • New Hampshire: 12.6

    • Iowa: 14.9

    • California: 11.5

    • Texas: 10.5

    • New York: 13.6

    • Louisiana: 12.6

    • Delaware: 14.1

    • Virginia: 12.4

    • Ohio: 13.7

    • Massachusetts: 13.7

    • Georgia: 10.2

    • Wisconsin: 13.5

    • Pennsylvania: 15.5

    • Mississippi: 12.8

    • Illinois: 12.4

    • Arizona: 13.9

    • South Dakota: 14.6

    • Nebraska: 13.8

    • Kansas: 13.4

    • Colorado: 10.7

    • Utah: 9.0

    • New Jersey: 13.7

    • Maine: 15.6

    • D.C.: 11.5

    • Washington: 12.2

    • North Carolina: 12.4

    • Michigan: 12.8

    • Hawaii: 14.3

    • Wyoming: 14.0

    • Oklahoma: 13.8

    • Missouri: 13.9

    • Indiana: 12.7

Steps to Create a Stem-and-Leaf Plot:

  1. List all values in ascending order.

  2. Draw a vertical line to the right of this list.

  3. For each value, write the leaf next to its respective stem.

  4. Arrange the leaves in ascending order next to the appropriate stem.

    • The leaf represents the tenths place in the examples.

Special Notes:

  • Split Stem-and-Leaf Plots:

    • These are used when one or two stems contain most of the data and each stem is duplicated to allow for clearer representation.

    • Caution against creating split plots for quizzes, exams, or homework.

Section 2.3: Back-to-Back Stem-and-Leaf Plots and Dotplots

Back-to-Back Stem-and-Leaf Plots

  • Useful for comparing two different datasets on one plot.

  • The middle section represents the stem for both datasets.

Dotplots

  • Definition:

    • A dotplot provides a rough impression of the distribution of a dataset.

  • Structure:

    • A vertical column of dots for each value in the dataset, where:

    • Number of dots per column indicates how many times each value appears.

  • Applications:

    • Best utilized when the dataset is relatively small and has distinct values.

Example of Dotplots

  • Data: U.S. Presidents and their wives with corresponding values:

    • 0, 2, 10, 2, 5, 3, 1, 2, 2, 4, 1, 5, 4, 15, 3, 4, 5, 3, 2, 3, 4, 2,
      6, 0, 0, 0, 8, 3, 3, 6, 2, 4, 2, 0, 4, 6, 4, 7, 2, 0, 1, 2, 6

  • Analysis Questions:

    • What is the sample size?

    • What was the most frequent number of children?

    • How many presidents had more than 6 children?

    • Identify where the gaps are located.

    • What was the highest number of children (notably observed: Tyler in the 1800s)?

Example: Time-Series Plots

Dow Jones Industrial Average

  • The time series plot displays year-end closing values from 2000 to 2012:

    • 2000: 10,786.85

    • 2001: 10,021.50

    • 2002: 8,341.63

    • 2003: 10,453.92

    • 2004: 10,783.01

    • 2005: 10,717.50

    • 2006: 12,463.15

    • 2007: 13,264.82

    • 2008: 8,776.39

    • 2009: 10,428.05

    • 2010: 11,557.51

    • 2011: 12,217.56

    • 2012: 13,104.14

  • Key analysis questions:

    • What year had the highest closing value?

    • In what years did the values decline?

Section 2.4: Misleading Graphs

Definition: Misrepresentation in Graphs

  • Improper use of statistical graphs can distort the data and mislead interpretations.

  • Common forms of misrepresentation include:

    1. Incorrect sizing of graphical images (related to the Area Principle).

    2. Misleading perspective in three-dimensional diagrams.

Positioning the Vertical Scale

  • Critical understanding of where the vertical axis intersects with the horizontal axis is essential, referred to as the baseline.

  • Misleading representations occur when the baseline does not start at zero. For instance:

    • Two bar graphs presenting the number of passengers at Denver International Airport could be misleading if one uses a baseline at a point other than zero, exaggerating the differences between data points.