Understanding Distributions and Graphs

Chapter Overview

• Focus on Chapter Two material, which encompasses:

  • Distributions

  • Graphs

  • Analyzing and interpreting data through visualization

Data Collection and Population

• Introduction of Sample and Population concept:

  • Example used: Toasters

  • Population: All toasters

  • Sample: Specific toasters found online (e.g., through Google Shopping)

• Variable of interest: Price of toasters

  • Example toaster prices collected:

  • $69.99

  • $299.99 (not a joke, specifically a KitchenAid toaster)

  • $36.46

  • $39.99

  • $299.95

Data Organization

• Importance of organizing data:

  • Presentation of raw data can be confusing without organization.

  • Actual question: What is the average price for toasters in 2020?

Frequency Distribution

• Definition: A way to describe the structure of a dataset by breaking values into categories or classes and counting occurrences within each class.

  • Frequency: Number of occurrences within each class.

  • Examples of Classes for Toaster Prices:

  • $0 - $49.99

  • $50 - $99.99

• Counting frequency:

  • Example findings:

  • Nine toasters priced between $0 and $49.99

  • Five toasters priced between $50 and $99.99

  • Two toasters priced between $250 and $299.99

• Conclusion drawn: Majority of toasters priced below $50.

Vocabulary Related to Classes

• Lower Class Limit: The smallest number in a class.

  • Example: For the class $50 - $99.99, the lower class limit is $50.

• Upper Class Limit: The largest number in a class.

  • For the class $50 - $99.99, the upper limit is $99.99.

• Class Width: The range of values within a class.

  • Calculated as the difference between two consecutive lower class limits (e.g., $50 - $0 = $50).

• Class Boundary: The midpoint between two classes.

  • Example: Class boundary between $49.99 and $50 is $49.995.

• Midpoint: Average of the lower and upper class limits.

  • Example for class $0 - $49.99:

  • ext{Midpoint} = rac{0 + 49.99}{2} = 24.995

Graphical Representations

• Introduction to Graphs: Valuable for understanding data sets visually, making them easier to interpret than raw data or frequency distributions.

Histogram

• Definition: A graph displaying the frequency of classes using bar heights.

  • X-axis: Represents class boundaries (e.g., prices for toasters).

  • Y-axis: Represents frequency.

• Characteristics:

  • Bars in a histogram touch because they represent continuous data ranges.

  • Example interpretation of a histogram for toaster prices.

Other Types of Graphic Displays

• Pie Chart: Represents qualitative data.

  • Size of pie slices correlates with the percentage of the total.

  • Warning on misrepresentation and visual misleading:

  • Example with pumpkin (36%) vs. strawberry (2%).

• Bar Graph: Similar to histograms, but for categorical data (not necessarily continuous).

  • Height of bars indicates frequencies or amounts of different categories (e.g., candy bar chocolate content).

  • Important to ensure correct representation of values (e.g., weights, dimensions).

Stem-and-Leaf Plot

• Definition: Displays individual data points while maintaining the frequency structure.

  • Each stem represents the leading digit(s), while each leaf represents the trailing digit (e.g., stem 5, leaf 6 = 56).

• Advantage: Retains all individual data values unique to the dataset.

Distribution Shape

• Three types of distribution shapes:

  • Symmetric Distribution: Data is evenly distributed around the mean.

  • Skewed Left (Left-Skewed): Data is concentrated on the right; has a longer tail on the left.

  • Skewed Right (Right-Skewed): Data is concentrated on the left; has a longer tail on the right (e.g., toaster prices).

• Example: Annual household income in the U.S.

  • Heavy concentration of households earning less than $90,000, with a significant tail towards higher incomes.

• Importance of understanding skewness for interpretation of data sets.