Unit 2 - Exploring Two-Variable Data

After covering single-variable statistics, it’s time to increase the complexity a little bit with two variable statistics! We can deal with two variable statistics in two ways. With categorical variables, we can use two way tables to represent the relationship between two different categories of categorical variables. With quantitative variables, we can show the relationship between these using scatterplots. We will also see whether there is a relationship between two variables in both situations. This will link to later units as well. Our study of two-way tables will link to probability (Unit 4) and Chi-squared tests for homogeneity or independence (Unit 8). Our study of scatterplots will link to regression inference as well (Unit 9).

We looked at new topics for this year's AP Statistics exams that were contained in Unit 2: Exploring Two-Variable Data. This includes making probability and analytical conclusions from mosaic plots and exploring departures from linearity in regression models.

Browse slides from the stream 'Exploring Two Variable Data.'

We review some facts about scatterplots and regression lines, then learn how to interpret "s", "r-squared," and read computer outputs, before doing some cumulative Q&A about linear regression.

Browse slides from the stream 'Advanced Linear Regression,' where we cover topics including Exploring Data.

Browse slides from the stream 'Least-Squares Regression Lines.'

In part two of a three-part series, we meet Least-Squares Regression Lines. We learn how to use them to make predictions, how to interpret the slope/intercept of the equation, and the meaning and dangers of "extrapolation." Then, we define "residuals" and discuss how to calculate and interpret them in context.

Browse slides from the stream 'Scatterplots & Association,' where we cover topics including Exploring Data.

In part 1 of a 3-part series on scatterplots and linear regression, we learn how to describe associations seen on scatterplots (including "direction," "form," and "strength"), define the specific terms used when describing association, and learn about the meaning of "r", a numerical values that describes the strength & direction of linear association.

We start off things with a review of categorical and quantitative data. Categorical data is data that is an attribute that is usually represented with a percentage or proportion, while quantitative data is data that can be represented by a number. If data can be averaged, it's quantitative.

There are many ways in which we can represent data from two categorical variables. Some of these are more graphical, like side-by-side bar graphs, segmented bar graphs, and mosaic plots, while others are numerical, like two-way tables (also called contingency tables).

Lets look back at the two-way table from Unit 2.2.

We can also have two sets of quantitative data to be compared as well, called a bivariate quantitative data set. Usually we have an independent, or explanatory variable, and a dependent, or response variable. That is, an explanatory variable explains a response to the response variable.

Correlation is when two variables are related to each other, and this is numerically represented with the correlation coefficient, which in stats we denote as r. The correlation coefficient shows the degree to which there is a linear correlation between the two variables, that is, how close the points are to forming a line. It can be positive or negative and this is the same as the direction of the scatterplot. The coefficient takes a value between -1 and 1, where r=-1 means that the points fall exactly on an decreasing line while r=1 means that the points fall exactly on a increasing line. A correlation coefficient of 0 means that there is no correlation between the data points.

The simplest form of regression is linear regression where we find a linear equation of the form ŷ=a+bx, where a is the y-intercept and b is the slope. Given a scatterplot, there can be infinitely many linear regression approximations, but there is only one best linear regression model, and this is called the least squares regression line (LSRL). We will discuss this and how to find the formula in a couple of sections.

When evaluating the effectiveness of a linear regression model, we use residuals to do this. But what is a residual? Well, they’re just the difference between the actual data and the value predicted by a linear regression model, or y-ŷ. A point closer to the best fit line has a smaller residual while a point farther from the best fit line has a larger residual. But what does this all mean? Well, if we have a positive residual, then the actual value is greater than the predicted value and we say that the model underestimates the true value by a certain amount. Likewise, if we have a negative residual, then the actual value is less than the predicted value and we say that the model overestimates the true value by a certain amount.

The least squares regression line is the best linear regression line that exists. It’s made by minimizing the sum of the squares of the residuals. Why square the residuals? This is because if we didn’t, negative and positive residuals would cancel out, reducing the impact of the residuals. Like regular regression models, the LSRL has a formula of ŷ=a+bx, with a being y-intercept and b being slope with each having their own formula using one-variable statistics of x and y.

Sometimes, the least squares regression model may not be the best for representing a data set. We’re going to list some reasons why.