Scatterplots and Correlation Textbook Section 3.1

Explanatory & Response Variables Sometimes, we study the relationship of two quantitative variables.Sometimes, we study the relationship of two quantitative variables. The Response Variable measures the outcome of a study.The Response Variable measures the outcome of a study. The Explanatory Variable may help explain the changes in the response variable.The Explanatory Variable may help explain the changes in the response variable. We think that car weight helps explain accidental deathsWe think that car weight helps explain accidental deaths Car weight = explanatoryCar weight = explanatory Death rate = responseDeath rate = response We also think that smoking influences life expectancyWe also think that smoking influences life expectancy Number of cigarettes smoked = explanatoryNumber of cigarettes smoked = explanatory Life expectancy = responseLife expectancy = response

Check your understanding Identify the explanatory and response variables in each setting…Identify the explanatory and response variables in each setting… 1.How does drinking beer affect the level of alcohol in people’s blood? The legal limit for driving in all states 0.08%. In a study, adult volunteers drank different numbers of cans of beer. Thirty minutes later, a police officer measured their blood alcohol levels. 2.The National Student Loan Survey provides data on the amount of debt for recent college graduates, their current income, and how stressed they feel about college debt. A sociologist looks at the data with the goal of using amount of debt and income to explain the stress caused by college debt.

Displaying Relationships: Scatterplots The most useful graph for displaying the relationship between two quantitative variables is the scatterplot.The most useful graph for displaying the relationship between two quantitative variables is the scatterplot. The x-axis is always the explanatory variableThe x-axis is always the explanatory variable The y-axis is always the response variableThe y-axis is always the response variable In the rare cases in which explanatory and response cannot be distinguished, the axes can be interchangeable.In the rare cases in which explanatory and response cannot be distinguished, the axes can be interchangeable.

Describing Scatterplots Direction : Our graph moves up from left to right, so we say that there is a positive correlation. Form: There is a linear pattern to the graph. The overall pattern forms a straight line – the only options are linear or curved. Strength : Because the points do not vary much from the linear pattern, the relationship is fairly strong. There do not appear to be an outliers. Remember: F.O.D.S.

Measuring Linear Association: Correlation When discussing linear scatterplots, it is difficult to determine the strength of the relationship on our own – we need a numerical description for how close the points lie to an imaginary “line of best fit” through the data.When discussing linear scatterplots, it is difficult to determine the strength of the relationship on our own – we need a numerical description for how close the points lie to an imaginary “line of best fit” through the data. The correlation, r, measures the direction and strength of the linear relationship.The correlation, r, measures the direction and strength of the linear relationship. Correlation is ALWAYS between -1 and 1.Correlation is ALWAYS between -1 and 1. When r is positive, the direction is positive.When r is positive, the direction is positive. When r is negative, the direction is negative.When r is negative, the direction is negative. The closer the value of r is to 1 or -1, the stronger the relationship.The closer the value of r is to 1 or -1, the stronger the relationship. The value is ONLY 1 or -1 in the instance of a perfectly straight line.The value is ONLY 1 or -1 in the instance of a perfectly straight line.

Facts about Correlation - r 1.Correlation makes no distinction between explanatory and response variables. 2.Correlation does not change when we change the units of measurement. 3.Correlation itself has no unit. 4.Correlation does not imply causation – just a relationship. 5.Correlation only describes the strength of linear relationships. 6.Even strong correlation (close to 1 or -1), does not guarantee a linear relationship – ALWAYS PLOT YOUR DATA! 7.Correlation is not resistant. 8.Correlation is not a complete summary of two-variable data. 1.You should also give the mean and standard deviation of both x- and y- variables along with the correlation.