Correlation
Correlation is a numerical measure of the direction and strength of the relationship between two quantitative variables. If you are describing the relationship between two categorical variables or the relationship between a categorical and quantitative variable, use the word association.
The correlation coefficient r measures the direction and strength of the linear relationship between the two quantitative variables. Remember: the correlation coefficient only has a viable value if the relationship is linear. If a scatter plot is not already given, make sure to plot the data to make sure it is a linear relationship.
Rules to keep in mind:
- Correlation does not imply causation
- Correlation requires that both variables be quantitative
- Correlation only measures the strength of a linear relationship between two variables
- A value of r close to 1 or -1 doesn’t guarantee a linear relationship
- Like a mean and standard deviation, the correlation is not resistant
- Correlation is not a complete summary of two-var data
Regression
A regression line describes how a response variable, y, changes as an explanatory variable, x, changes. We often use a regression line to predict the value of y for a given value of x. Remember that data values won’t lie perfectly on the regression line.
Basic structure of a regression equation: ŷ = a + bx
ŷ: the predicted value of the response variable y for a given value of the explanatory variable x. Y-hat stands for predicted value. B: the slope, the amount by which y is predicted to change when x increases by one unit A: the y-intercept, the predicted value of y when x = 0
Extrapolation is the use of a regression line for prediction far outside the interval of values of the explanatory variable x used to obtain the line. Such predictions are often not accurate.
Few relationships are linear for all values of the explanatory variable. Don’t make predictions using values of x that are much larger or much smaller than those that actually appear in your data.
In most cases, no line will pass exactly through all the points in a scatterplot. Because we use the line to predict y from x, the prediction errors we make are errors in y, the vertical direction in the scatterplot. A good regression line makes the vertical deviations of the points from the line as small as possible.