Relationships between quantitative variables Unit 8 relationships between quantitative variables This will build on what you know already Generally the one on the x axis is the one that we think of as explaining the other variable on a scatter plot Positive Association So we ve seen how a scatterplot tells you the direction of the association between two variables and can give you an idea of the strength of the association But it can also tell you about the form of the association Sometimes the association between two variables is linear the points fall roughly along a straight line the change in the y variable is about the same for any given change in the x variable We call any association of this kind non linear or curvilinear Some important things to note about the correlation coefficient r its value ranges only between 1 and 1 negative values go with a negative association positive values with a positive association there s a weak association values near 1 or 1 mean there s a strong association values near 0 mean There are several ways two variables might be associated First there might actually be a direct causal relationship think of hours spent studying and test scores Second the association between x and y might be due to an indirect causal relationship x influences some other variable which in turn influences y Think of hours spent studying and amount of approval from parents Third x and y could be associated because their values have a common cause there is some third variable that influences both Think of parents approval and your grad school prospects We ll now learn how to draw the line that best captures this relationship This line is called the least squares regression line or simply the regression line The regression line gives us a predicted y value for every value of x We call this predicted value y We call the unexplained parts of each y value residuals We say here that price displays heteroscedasticity when predicted by weight this simply means that there is more variability in y for some parts of the range of the x variable And here s another warning about interpreting linear relationships Don t extrapolate outside of the range of observed x values Df n 2