Scatterplots and Correlation: Analyzing Bivariate Data, Summaries of Statistics

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AP Statistics

Chapter 6

Scatterplots, Association, and

Correlation

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Objectives:

• Scatterplots
• Association
• Outliers
• Response Variable
• Explanatory Variable
• Correlation
• Correlation Coefficient
• Lurking Variables

Scatterplots

4 Scatterplots usually don’t show the origin (or have breaks), because often neither variable has values near 0. The plot should focus on the part of the coordinate plane that actually contains the data.

Analyzing Scatterplots

Analysis of bivariate data builds on the tools used for examining univariate data.

1. First plot the data, then compute numerical summaries.
2. Look for overall patterns and deviations from those patterns.
3. When the overall pattern is quite regular, use a compact mathematical model to describe it. 5

Scatterplot Variables

• Response Variable - measures the outcome of a study (the variable of interest). The variable you hope to predict or explain. (The dependent variable, plotted on the y-axis).
• Explanatory or Predictor Variable – helps explain or predict changes in a response variable. Attempts to explain the observed response. (The independent variable, plotted on the x-axis.) 7

Roles for Variables

• It is important to determine which of the two quantitative variables goes on the x - axis and which on the y - axis.
• This determination is made based on the roles played by the variables.
• When the roles are clear, the explanatory or predictor variable goes on the x - axis, and the response variable (variable of interest) goes on the y - axis.

Example

• Here we plotted prediction error on the y-axis against the year on the x-axis because we are interested in how the predictions have changed over time.
• Could we have plotted them the other way?
• In this case, it is hard to imagine reversing the variables - knowing the prediction error and wanting to guess in what year it happened. 10 Average Error in Predicted Position of Atlantic Hurricanes

TI-83/84: Graph on a Scatterplot

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Association

• The scatterplot displays the association (relationship) between the explanatory variable and the response variable.
• Calling one variable explanatory and the other response doesn’t necessarily mean that changes in one causes changes in the other.
• We will discuss causation in more detail later in the chapter. 13

Looking at Scatterplots

• When looking at scatterplots, we will look for direction, form, strength, and unusual features.
• Direction:
  • A pattern that runs from the upper left to the lower right is said to have a negative direction.
  • A trend running the other way has a positive direction.

Looking at Scatterplots

• This example shows a negative association between central pressure and maximum wind speed.
• As the central pressure increases, the maximum wind speed decreases.

Looking at Scatterplots

• The figure shows a negative direction between the years since 1970 and the prediction errors made by NOAA.
• As the years have passed, the predictions have improved (errors have decreased). Can the NOAA predict where a hurricane will go?

Looking at Scatterplots

• Form:
  • If there is a straight line (linear) relationship, it will appear as a cloud or swarm of points stretched out in a generally consistent, straight form.

Looking at Scatterplots

• Form:
  • If the relationship isn’t straight, but curves gently, while still increasing or decreasing steadily, we can often find ways to make it more nearly straight.