12.2 Scatter Plots

Scatter Plots

Scatter plots let you visualize the relationship between two quantitative variables. By plotting data as individual points on a graph, you can quickly spot patterns, trends, and outliers that would be hard to see in a raw data table. This skill is the foundation for regression analysis, which you'll use throughout statistics.

Scatter Plot Construction

A scatter plot places two variables on a coordinate plane so you can see how they relate to each other.

• The independent variable (also called the explanatory variable) goes on the x-axis. This is the variable you think might influence the other (e.g., age, study time).
• The dependent variable (also called the response variable) goes on the y-axis. This is the variable you think might be affected (e.g., height, test score).
• Each point on the plot represents one observation. Its position is determined by that observation's x and y values.

To create a scatter plot:

1. Choose your scales for each axis based on the range of your data. Make sure the scale covers all values without too much empty space.
2. Plot each data point at its corresponding (x, y) location.
3. Label both axes with the variable name and units (e.g., x-axis: Age (years), y-axis: Height (cm)).
4. Give the plot a descriptive title (e.g., "Relationship between Age and Height").

Interpreting Scatter Plot Relationships

When you look at a scatter plot, you're evaluating three things: direction, strength, and pattern.

Direction tells you whether the variables move together or in opposite ways:

• Positive relationship: As x increases, y also increases. Points trend from the bottom-left to the top-right. Example: height tends to increase with age in children.
• Negative relationship: As x increases, y decreases. Points trend from the top-left to the bottom-right. Example: test scores tend to decrease as the number of distractions increases.
• No relationship: Points are scattered randomly with no clear trend. Example: shoe size and IQ have no meaningful connection.

Strength describes how tightly the points cluster around a pattern:

• Strong: Points closely follow a clear pattern, with little scatter. Example: weight and daily calorie intake.
• Moderate: A trend is visible, but points spread out more around it.
• Weak: Points are loosely scattered, and the trend is hard to see. Example: hours of sleep and exam grades.

Strength can be quantified using the correlation coefficient (covered in detail in later sections), which gives a numerical value for how strong and in what direction the linear relationship is.

Pattern describes the shape the points form:

• Linear: Points roughly follow a straight line. Example: income tends to rise with years of education.
• Nonlinear: Points follow a curve. Example: running speed improves with age in childhood, peaks, then declines.
• Outliers are data points that fall far from the overall pattern. Even a single outlier can shift your interpretation of the relationship, so always note them. For instance, a few individuals with extremely high incomes could distort the apparent trend in a dataset about income and education.

Appropriateness of Regression Lines

A regression line (or line of best fit) summarizes the linear trend in a scatter plot. But it's not always the right tool. Before fitting one, check whether it actually makes sense for your data.

A regression line is appropriate when:

• The scatter plot shows a roughly linear relationship (not curved).
• There's a logical reason to think the independent variable influences the dependent variable (e.g., study time affecting test scores).
• Your goal is to predict values of the dependent variable from the independent variable (e.g., predicting sales from advertising spending).

A regression line is NOT appropriate when:

• The relationship is nonlinear. Fitting a straight line to curved data gives misleading results. Example: age vs. physical strength follows a curve, not a line.
• There's no clear relationship at all. If the points are randomly scattered, a line through them is meaningless.
• The relationship has no logical basis. Even if two variables happen to correlate, a regression line is misleading if there's no reason one would predict the other (e.g., number of cars owned and GPA).
• Significant outliers are pulling the line away from where most of the data sits.

Before calculating a regression line, also verify these assumptions:

1. Linearity: The relationship between the variables is actually linear.
2. Independence: The data points are independent of each other (one observation doesn't influence another).
3. Constant variance of residuals: The spread of points around the line stays roughly the same across the entire range of x-values. If the spread fans out or narrows, the regression line may not be reliable.

Advanced Scatter Plot Analysis

A few additional features are worth noticing when you examine scatter plots:

• Clustering occurs when groups of data points bunch together in distinct areas of the plot. This can signal subgroups in your data. For example, a scatter plot of height vs. weight might show two clusters if the data includes both children and adults.
• Strength of association can be assessed visually by how tightly points hug a trend line. The tighter the clustering around the line, the stronger the association.
• A scatterplot matrix is a grid that displays scatter plots for every pair of variables in a dataset. It's useful when you have more than two variables and want to quickly scan for relationships across all of them.