Section 3.1 Scatterplots

Two-Variable Quantitative Data  Most statistical studies involve more than one variable.  We may believe that some of the variables explain or even cause changes in the variables. Then we have explanatory and response variables.  Explanatory—like the independent variable, it attempts to explain the observed outcomes.  Response—like the dependent variable, it measures an outcome of a study.

Examples Identify the explanatory and response variables: Alcohol causes a drop in body temperature. To measure this, researchers give several different amounts of alcohol to mice, then measure the change in their body temperature after 15 minutes.Alcohol causes a drop in body temperature. To measure this, researchers give several different amounts of alcohol to mice, then measure the change in their body temperature after 15 minutes. If an object is dropped from a height, then its downward speed theoretically increases over time due to the pull of gravity. To test this, a ball is dropped and at certain intervals of time, the speed of the ball is measured.If an object is dropped from a height, then its downward speed theoretically increases over time due to the pull of gravity. To test this, a ball is dropped and at certain intervals of time, the speed of the ball is measured.

Scatterplots  Used for two-variable quantitative data!  Explanatory variable goes on the x-axis  Response variable goes on the y-axis  The explanatory variable does not necessarily “CAUSE” the change in the response variable.

Scatterplots and Correlation  Displaying Relationships: Scatterplots Make a scatterplot of the relationship between body weight and pack weight. Since Body weight is our eXplanatory variable, be sure to place it on the X-axis! Body weight (lb) Backpack weight (lb)

Interpreting Graphs One Variable Quantitative Data Two-Variable Quantitative Data CenterForm Linear? Clusters? Gaps? ShapeDirection Positive? Negative? SpreadStrength Strong? Weak? Moderate? OutliersOutliers

In sentence form…  There is a (strong/weak), (positive/negative), (linear/non-linear) relationship between (your two variables).

Scatterplots and Correlation  Interpreting Scatterplots Direction Form Strength Outlier There is one possible outlier, the hiker with the body weight of 187 pounds seems to be carrying relatively less weight than are the other group members. There is a moderately strong, positive, linear relationship between body weight and pack weight. It appears that lighter students are carrying lighter backpacks.

Adding Categorical Variables to Scatterplots  You can use different plotting symbols or different colors to designate a categorical variable.  You still have two quantitative variables, but you can add a “category” to these variables.

Some quick tips for drawing scatterplots  Choose an appropriate scale for the axes. Use a break if appropriate.  Label, Label, Label…  If you are given a grid, try to use a scale that will make the scatterplot use the whole grid.

Section 3.2 Correlation We are not good judges!  We shouldn’t just rely on our eyes to tell us how strong a linear relationship is.  We have a numerical indication for how strong that linear relationship is – it’s called CORRELATION.

Scatterplots and Correlation Definition: The correlation r measures the strength of the linear relationship between two quantitative variables. r is always a number between -1 and 1 r > 0 indicates a positive association. r < 0 indicates a negative association. Values of r near 0 indicate a very weak linear relationship. The strength of the linear relationship increases as r moves away from 0 towards -1 or 1. The extreme values r = -1 and r = 1 occur only in the case of a perfect linear relationship.

Facts About Correlation  It does not require a response and explanatory variable. Ex. How are SAT math and verbal scores related?  If you switch the x and the y variables, the correlation doesn’t change.  If you change the units of measurement for x and/or y, the correlation doesn’t change.  Positive r values indicate a positive relationship; negative values indicate a negative relationship. Remember… not cause.

More Facts  Correlation measures the strength of the LINEAR relationship. It doesn’t measure curved relationships.  Correlation is strongly affected by outliers.  r does not have a unit.

Homework Chapter 3 #11, 13, 14, 15, 17, 20, 22, 26