Chapter 7 Scatterplots, Association, and Correlation

Examining Relationships Relationship between two variables Examples: Height and Weight Alcohol and Body Temperature SAT Verbal Score and SAT Math Score High School GPA and College GPA

Two Types of Variables Response Variable (Dependent) Measures an outcome of the study Explanatory Variable (Independent) Used to explain the response variable. Example: Alcohol and Body Temp Explanatory Variable: Alcohol Response Variable: Body Temperature

Two Types of Variables Does not mean that explanatory variable causes response variable It helps explain the response Sometimes there are no true response or explanatory variables Ex. Height and Weight SAT Verbal and SAT Math Scores

Graphing Two Variables Plot of explanatory variable vs. response variable Explanatory variable goes on horizontal axis (x) Response variable goes on vertical axis (y) If response and explanatory variables do not exist, you can plot the variables on either axis. This plot is called a scatterplot This plot can only be used if explanatory and response variables are both quantitative.

Scatterplots Scatterplots show patterns, trends, and relationships. When interpreting a scatterplot (i.e., describing the relationship between two variables) always look at the following: Overall Pattern Form Direction Strength Deviations from the Pattern Outliers

Interpreting Scatterplots Form Is the plot linear or is it curved? Strength Does the plot follow the form very closely or is there a lot of scatter (variation)?

Interpreting Scatterplots Direction Is the plot increasing or is it decreasing? Positively Associated Above (below) average in one variable tends to be associated with above (below) average in another variable. Negative Associated Above (below) average in one variable tends to be associated with below (above) average in another variable.

Example – Scatterplot The following survey was conducted in the U.S. and in 10 countries of Western Europe to determine the percentage of teenagers who had used marijuana and other drugs.

Example – Scatterplot 2434United States 3153Scotland 37Portugal 36Norway 1423North Ireland 819Italy 1637Ireland 15Finland 2140England 317Denmark 422Czech Republic Other DrugsMarijuanaCountry Percent who have used

Example – Scatterplot

The variables are interchangeable in this example. In this example, Percent of Marijuana is being used as the explanatory variable (since it is on the x-axis). Percent of Other Drugs is being used as the response since it is on the y-axis.

Example - Scatterplot The form is linear The strength is fairly strong The direction is positive since larger values on the x-axis yield larger values on the y-axis

Example - Scatterplot Negative association Outside temperature and amount of natural gas used

Correlation The strength of the linear relationship between two quantitative variables can be described numerically This numerical method is called correlation Correlation is denoted by r

Correlation A way to measure the strength of the linear relationship between two quantitative variables.

Correlation Steps to calculate correlation: Calculate the mean of x and y Calculate the standard deviation for x and y Calculate Plug all numbers into formula

Correlation

Calculating r. Femur (x) Humerus (y) Set up a table with columns for x, y,,,,, and

Calculating r yx

Calculating r Recall: So,

Calculating r Recall: So,

Calculating r. Put everything into the formula:

Properties of r r has no units (i.e., just a number) Measures the strength of a LINEAR association between two quantitative variables If the data have a curvilinear relationship, the correlation may not be strong even if the data follow the curve very closely.

Properties of r r always ranges in values from –1 to 1 r = 1 indicates a straight increasing line r = -1 indicates a straight decreasing line r = 0 indicates no LINEAR relationship As r moves away from 0, the linear relationship between variables is stronger

Properties of r Changing the scale of x or y will not change the value of r Not resistant to outliers Strong correlation ≠ Causation Strong linear relationship between two variables is NOT proof of a causal relationship!

Reading JMP Output The following is some output from JMP where I considered Blood Alcohol Content and Number of Beers. The explanatory variable is the number of beers. Blood alcohol content is the response variable.

Reading JMP Output

Summary of Fit

Reading JMP Output RSquare = r 2 This means I know this is positive because the scatterplot has a positive direction. The Mean of the Response is the mean of the y’s or