Introduction to Graphs, Data Visualization, and Interpretation

The Utility and Function of Graphs

• Humans are generally poor at processing large datasets in their raw form. The brain lacks the capacity to easily grasp patterns and structures when data is presented in tables or raw sequences.

• Graphs simplify information while highlighting critical messages and ideas within the data.

• While graphs enhance the speed and ease of communication, they come with the cost of losing some precision compared to raw datasets.

• In a raw data table showing median income in Florida counties, identifying the minimum and maximum values is time-consuming.

  • Union County: Lowest median income ($13,590$).

  • St. Johns County: Highest median income ($36,836$).

• Visual representation allows for immediate observation of patterns:

  • Union County is not just the lowest; it sits an unusual distance away from other data clusters (an unusually low income).

  • St. Johns County and Collier County have unusually high incomes compared to the rest of the distribution.

  • Skewness can be seen visually: there are more counties with low median incomes than high median incomes in Florida.

  • Comparison to benchmarks: Most Florida counties fall below the U.S. median income ($31,128$) and even most fall below the Florida median ($27,598$).

Data Terminology and Classification

• In this context, the terms graphs, charts, figures, and data visualizations are used interchangeably to refer to any graphical representation of data.

• Dataset: The entire collection of data values, often represented as a table.

• Variable: A characteristic that can be measured or labeled (e.g., Name, Number of Children, Height, Age).

• Value: The specific measurement or label for a variable (e.g., "Daphne" for the Name variable, or "0" for the children variable).

• Data Point: A single row of data across the dataset, representing a specific entity (e.g., a person who is $21$ years old and $150\,cm$ tall).

Types of Numerical Data

Numerical data represents an amount of something and is divided into two categories:

• Discrete Data: Data that is counted rather than measured. These come in distinct chunks with no midpoints. Examples include the number of pets, shoe sizes, or the roll of a die (one cannot roll a $2.5$).

• Continuous Data: Data that is measured. There are meaningful midpoints between numbers, allowing for infinite precision depending on the measurement tool. Examples include height ($157.49\,cm$), speed ($101.11\,km/h$), and length.

Types of Categorical Data

Categorical data assigns labels to items rather than measuring an amount:

• Nominal Data: Categories that have no inherent order and can be rearranged without changing the meaning.

  • Examples: Countries (Australia, Japan), Tutors (Ryan, David), or Postcodes (the numbers are just labels for regions; $4000$ is not "more" of a suburb than $4100$).

• Ordinal Data: Categories that possess a natural, meaningful order.

  • Examples: Grades (Pass, Credit, Distinction, High Distinction), Medals (Gold, Silver, Bronze), or Sizes (Small, Medium, Large).

  • Note: While word labels for grades are ordinal, the raw score (e.g., $78\%$) is numerical.

Anatomy of a Graph

• Content: The actual data being shown, such as the shaded areas, lines, or bars representing the variables.

• Scaffolding: The container and context for the data. This includes axes, labels, grid lines, titles, captions, and legends.

Visual Encodings

Visual encodings are properties of a graph that vary according to the underlying data. Identifying these helps in interpreting complex or unfamiliar visualizations.

• Length / Height: The property seen in bar charts where the length or height of the bar is proportional to the value (e.g., comparing country populations).

• Position: The placement of a dot or marker on an axis (e.g., a dot's position on the x-axis representing median income).

• Area: The size or volume of a shape is proportional to the data (e.g., bubble maps where larger bubbles indicate higher populations in specific counties).

• Angle: Proportion represented by the degree of a slice in a circle (e.g., Asia representing $60\%$ of the world population corresponds to a $216^{\circ}$ angle, as $0.60 \times 360^{\circ} = 216^{\circ}$).

• Slope: The angle of a line segment in a line graph, representing the rate of change between two data points.

• Color Hue: Different colors used to distinguish categories (e.g., red for Spain, gray for Sweden).

• Color Shade: The intensity or lightness/darkness of a color representing a value (e.g., shades of red representing temperature variations relative to a baseline).

• Shape: Different symbols used to represent categories (e.g., a circle for lemonade sales and a cross for tea sales).

• Width / Thickness: The width of a line representing magnitude (e.g., a "tentacle" diagram where the branch width is proportional to the number of individual targets Trump attacked on Twitter).

Principles of Good Graph Design

These guidelines are not absolute rules but should be followed unless a fairer representation of data requires otherwise.

• Sensible Axes for Bar Graphs: Bar graphs emphasize the magnitude of differences. The dependent variable (y-axis) should usually start at zero to maintain proportionality.

  • Exception: If chance-level performance is high (e.g., $25\%$ for a 4-option multiple-choice quiz or $50\%$ for a face-matching task), starting the axis at the chance level may be more meaningful.

• Sensible Axes for Line Graphs: Line graphs emphasize change. It is often detrimental to start the y-axis at zero because it can flatten the data, making small but significant changes impossible to see.

• Consistent Axes: When comparing multiple graphs side-by-side, they must use identical axes (same scale, order, and positioning) so that differences in the data are immediately apparent.

• Complete Data: All available scale points should be included. Omitting years or data points reduces the resolution of the data and can be misleading.

• Consistent Scale Intervals: The distance between points on an axis must represent the same unit (e.g., every step on the x-axis should represent exactly one year). Gaps should be indicated if data is missing for a specific interval.

• Consistent Bin Sizes: When grouping data into bins (e.g., counting flu instances in age groups $0-9$, $10-19$, etc.), every bin must be the same size to allow for accurate visual comparison of frequencies.

The Principle of Proportional Ink

• When a shaded region represents data, the area of that ink must be directly proportional to the value it represents.

• In a pie chart, $50\%$ of the data must use $50\%$ of the ink.

• In a bar chart, a bar representing $70$ should take up exactly $70\%$ of the available vertical space from zero.

• In a shaded line graph (where the area under the curve is filled), the y-axis must start at zero. If it does not, the shaded area is no longer proportional to the values, misleading the viewer.

Interpreting Data and Avoiding Assumptions

Graphs only show the data; they do not dictate why the data exists. Interpretation often involves bringing outside assumptions that may lead to errors.

• The Sensodyne Example: An ad stated "9 out of 10 dentists recommend Sensodyne."

  • Investigation revealed dentists could recommend multiple brands.

  • Sensodyne was recommended as often as its competitors, but the ad implied a preference that the data did not actually support.

• Homicide Normalization: Raw counts of murders in large cities look high compared to rural areas, leading to the assumption that cities are more dangerous. However, if the data isn't normalized by population (murders per $100,000$), the density of people is the cause of the high number, not the inherent danger.

• Google Search and the Solar Eclipse: Searches for "eyes hurt" showed a geographical band across the U.S. matching the path of the solar eclipse on April 8, 2024.

  • Scientists use the data to generate hypotheses (the eclipse caused eye pain), then test via temporal comparisons (searching for the pattern before and after April 8).

• Taylor Swift Swear Word Frequency: A graph showed an increase in swear words over time. While social media interpreted this as her "going bad," other scientific explanations include:

  • Increased creative freedom from new contracts.

  • Changing social norms regarding swearing in music.

  • Maturing life experiences and a maturing fan base.

  • Success allowing for more authenticity and risk-taking.

  • Transitioning from Country to Pop music genres.

  • More songs per album providing more opportunities to swear.

  • Shifting from radio (which has censorship) to streaming platforms like Spotify.

  • Repetition of specific lyrics within choruses.

  • Market demands/audience preferences.

Case Study: Dolphin Signature Whistles

Research on 19 adult female dolphins in Sarasota Bay, Florida, suggests they use "motherese" (baby talk).

• Recording Method: Suction-cup hydrophones were placed on the dolphins' "melon" (forehead) during health assessments.

• Data Analysis: The y-axis showed the Max Frequency Z-score (the pitch of the dolphin's voice).

• Key Finding: Whistles recorded when dolphins were with their calves (orange markers) consistently showed a higher pitch than whistles recorded without calves (blue markers).

• Graph Oddity: Some data points appeared "horned."

  • The width of the bars reflected confidence intervals around the mean.

  • "Horns" occurred simply when the numerical values for the first quartile and the mean's confidence interval were flipped relative to the standard orientation.

Case Study: Wine Competitions

Data from wine competitions suggests that awards may not always reflect stable quality.

• Correlation Matrix: Correlations between various competitions (e.g., Riverside International, San Francisco Chronicle, Orange County Fair) are remarkably low, with many values near $0.02$ or $0.10$.

  • A correlation of $1.00$ represents a perfect match; low values suggest a wine winning Gold in one competition has little bearing on its result in another.

• Distribution Performance: A study of $375$ wines entered into five competitions showed that the number of Gold medals awarded was almost identical to what would be predicted by chance alone.

• Award Consistency: Among wines entered in more than three competitions, $53\%$ received varied awards (e.g., G:NA/Other), whereas very few showed consistent high marks across all entries.

Assignment: Data in the Wild

• Task: Find a problematic graph published in the real world (newspaper, social media, billboard, etc.).

• Format: Write a professional letter to the editor or publisher identifying the issues.

• Requirements:

  • The graph must be from February 2024 or later.

  • It must contain at least two distinct issues based on lecture content.

  • Include the original graph and a "new and improved" version (hand-drawn or computer-generated).

  • The letter must explain the intended message, why the original is problematic, how the new version fixes it, and what good elements were retained.

  • Length: Maximum $1,000$ words.

  • Weight: $20\%$ of the total grade.

  • Due Date: Thursday of Week 10.

Questions & Discussion

• Question: In the dolphin study, how do we know the blue data (without calf) wasn't just the mother dolphin calling for her calf in a lower voice?

• Response: The blue data was likely recorded before the dolphin even had a calf, which would invalidate the "calling for the baby" explanation. In science, the data shows the pattern; if an alternative explanation is possible, follow-up studies are conducted to test those hypotheses.

• Question: What counts as a graph being "republished" for the assignment?

• Response: If a graph originally from the 1990s is picked up and retweeted or posted by a news outlet in 2024 or 2025, it is considered a recent publication and is eligible for the assignment.