Representing the Relationship Between Two Quantitative Variables Chapter Notes | AP Statistics - Grade 9 PDF Download

When analyzing data with two quantitative variables, we often deal with a bivariate dataset where the variables are interconnected. One variable, known as the explanatory or independent variable (x), is considered to influence the other, called the response or dependent variable (y). The explanatory variable helps predict or explain changes in the response variable.
For instance, in a study exploring how age affects blood pressure, age would be the explanatory variable, while blood pressure serves as the response variable. By analyzing age, we can estimate its impact on blood pressure levels.

What is a Scatterplot?

Describing Scatterplots

Form

• Linear: Points align in a straight line, suggesting a consistent relationship.
• Curved: Points form a nonlinear pattern, such as a quadratic curve, indicating a more complex relationship.

For example, a linear scatterplot might show a steady increase in one variable as the other rises, indicating a strong positive relationship. A curved scatterplot could suggest a nonlinear connection, like a parabolic trend. In the referenced graphs, Graph 1 exhibits a curved form, while Graph 2 is distinctly linear.

Direction

• Positive correlation: As the explanatory variable increases, the response variable also increases, reflected by a positive slope in a linear model.
• Negative correlation: As the explanatory variable increases, the response variable decreases, indicated by a negative slope.

In Graph 1, the direction is decreasing, as the response variable values drop from left to right. In Graph 2, the direction is increasing, with response variable values rising. For example, in a study of age versus height, a positive slope suggests that height increases with age, while a negative slope indicates a decrease in height as age rises.

Strength

• Strong: Points tightly cluster around the model.
• Moderate: Points are somewhat scattered but still follow the model.
• Weak: Points are widely dispersed, showing little adherence to the model.

Graph 1 demonstrates a moderate strength correlation, while Graph 2 shows a strong correlation. The strength is quantitatively assessed using the correlation coefficient, discussed in later sections.

Unusual Features

• Clusters: Groups of points that are tightly packed, suggesting subgroups or distinct patterns within the data.
• Outliers: Points that deviate significantly from the main trend, possibly due to measurement errors or unique cases in the dataset.

These features are critical as they can affect the interpretation of the relationship between variables and influence statistical analyses.

Example Analysis

Outliers, Influential Points, and High Leverage Points

Understanding the distinctions between outliers, influential points, and high leverage points is crucial for accurate data analysis:

• Outlier: A data point that significantly deviates from the rest, potentially skewing statistical results.
• Influential Point: A point that significantly affects the regression line’s slope or shape but may not be an outlier. It can alter the model’s direction or curvature.
• High Leverage Point: A point with an extreme value in the explanatory variable, exerting significant influence on the regression model, whether or not it’s an outlier.

These points can dramatically impact scatterplot trends and regression analyses, requiring careful consideration.

Key Terms to Understand