Section2.1

Chapter Title

• Exploring Data with Tables and Graphs

Overview

• Focus on organizing and summarizing data through various methods including graphs and frequency distributions.

2-1 Frequency Distributions for Organizing and Summarizing Data

• A frequency distribution (or frequency table) displays how data values are distributed among categories.

• It aids in understanding the distribution's nature in large data sets.

2-2 Histograms

• Graphical representation of frequency distributions using bars to illustrate the frequency of data in specific intervals.

2-3 Graphs that Enlighten and Graphs that Deceive

• Importance of visual representation and potential for misleading interpretations.

• Understanding how to discern accurate data representations.

2-4 Scatterplots, Correlation, and Regression

• Visual methods for understanding relationships between two quantitative variables.

• Correlation indicates the strength and direction of a relationship; regression provides a way to model the relationship.

Frequency Distribution Definition

• Frequency Distribution (Frequency Table):

  • Organizes data into classes, indicating the frequency of data points in each category.

  • Useful for summarizing large data sets to observe patterns and trends.

Key Definitions

• Lower Class Limits: Smallest numbers in each class.

• Upper Class Limits: Largest numbers in each class.

• Class Boundaries: Separates classes without gaps, used for accurate representation.

• Class Midpoints: Calculated as (Lower Limit + Upper Limit) / 2.

• Class Width: Difference between two consecutive lower class limits.

Constructing a Frequency Distribution

• Procedure (Part 1):

  • Select the number of classes (typically between 5 and 20).

  • Calculate class width: (Maximum Data Value - Minimum Data Value) / Number of Classes.

  • Round up for convenience.

• Procedure (Part 2):

  • Choose the first lower class limit (minimum value or convenient value).

  • List subsequent lower limits using the class width.

  • Tally data values into respective classes, summing to find frequencies.

Example: Commute Time in Los Angeles (1 of 5)

• Based on data from Los Angeles' daily commute times.

• Data points to consider for frequency distribution.

Example: Commute Time in Los Angeles (2 of 5)

• Select 7 classes and calculate class width rounded to 15 for convenience.

Example: Commute Time in Los Angeles (3 of 5)

• Establish lower class limits: 0, 15, 30, 45, 60, 75, 90.

Example: Commute Time in Los Angeles (4 of 5)

• Corresponding upper class limits identified: 14, 29, 44, 59, 74, 89, 104.

Example: Commute Time in Los Angeles (5 of 5)

• Document tally marks for each class to find frequencies.

• Frequencies recorded: 0-14 (6), 15-29 (18), 30-44 (14), 45-59 (5), 60-74 (5), 75-89 (1), 90-104 (1).

Relative Frequency Distribution (1 of 2)

• Relative Frequency Distribution:

  • Involves expressing class frequencies as a total proportion of sum of all frequencies.

  • Important for understanding data in percentage terms.

Relative Frequency Distribution (2 of 2)

• The total must approximate to 100% for accuracy, accommodating rounding errors.

• Example frequencies for commute time in Los Angeles: 0-14 (12%), 15-29 (36%), etc.

Comparisons

• Combining relative frequency distributions for different data sets facilitates comparisons of trends.

Example: Comparing Daily Commute Time in NY, NY and Boise, ID (1 of 2)

• Displays relative frequencies from both locations.

• Helps highlight differences in commute times influenced by city size and density.

Example: Comparing Daily Commute Time in NY, NY and Boise, ID (2 of 2)

• Notable differences in commute times between cities; Boise shows lower times.

Cumulative Frequency Distribution

• Cumulative frequency sums frequencies of each class with all previous classes.

• Example data for commute times outlined, showing cumulative sums.

Critical Thinking: Using Frequency Distributions to Understand Data (1 of 2)

• Normal distributions typically show a pattern of increasing frequencies to a peak followed by a decrease.

Critical Thinking: Using Frequency Distributions to Understand Data (2 of 2)

• Normal distribution representation example provided, illustrating expected frequency trends.

Gaps

• Gaps can suggest data derived from different populations, although this is not universally true.

Example: Exploring Data: What Does a Gap Tell Us? (1 of 2)

• Frequency distribution of penny weights highlights gaps, indicating possibly distinct populations based on composition.

Example: Exploring Data: What Does a Gap Tell Us? (2 of 2)

• Analyzes the significant weight gap related to the different compositions of pennies pre- and post-1983.