AP Stats -- Chapter 3

Univariate Data

Data with a singular variable.

Bivariate Data

Data with multiple variables.

Steps for Data Analysis

• Plot the data, pointing out patterns and outliers.

• Use a numerical summary including r, r² for bivariate.

• Overall regression pattern to make a simplified model.

Models for Data

Univariate: Normal distribution; bivariate: linear, power, exponential model.

Explanatory Variable

A variable that predicts or explains changes in a response variable.

Response Variable

A variable that measures the outcome of a data set.

Scatterplots

A measure of the relationship between two quantitative variables for the same individuals.

Describing Scatterplots

• Direction: positive, negative, no association.

• Form: linear, non-linear.

• Strength: weak, moderate, very strong

• Context: mention variables, question asked.

• Unusual features: outliers, deviation, clusters.

Correlation r

Describes the correlation between two variables in a linear model; between -1 and 1. Does not describe causation or form and is not resistant.

Regression Line

A line that summarizes the relationship between two variables when one variable is explanatory; y hat = a + bx.

Residuals

Leftover variation in response variables from the predicted response variable; y - y hat.

a

The predicted y hat when x is equal to 0.

b

The predicted change in y hat per x increasing by one unit.

Least Squares Regression Line

The line that makes the sum of squared residuals as small as possible.

Residual Plot

A plot that displays residuals on the y-axis and the explanatory variable on the x-axis.

Linear Regression on a Residual Plot

Should have no obvious patterns, small in size, and no curved pattern.

Standard Deviation of Residuals

The typical size of a residual; measures the typical distance between y and y hat.

R²

The coefficient of determination, which measures the percent of variability in y accounted for by the regression line, y hat.

Computer Regression Output

• slope: b (in response variable)

• y-intercept: a

• s: standard deviation

• r and r²

Using Summary Statistics to Create a Regression Line

• Slope: b=r(s_y/s_x)

• Y=intercept: a=mean y minus b(mean x)

High leverage

Much larger or smaller x-value than other points.

Outliers

Has a large residual, usually in the y-value.

Influential Point

A point that if removed, substantially changes the slope of the regression line.