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                                         Part V: Statistical Studies and the Hunt for a Meaningful Relationship
                                                    Many folks make the mistake of thinking that a correlation of –1 is a bad
                                                    thing, indicating no relationship. Just the opposite is true! A correlation of
                                                    –1 means the data are lined up in a perfect straight line, the strongest linear
                                                    relationship you can get. The “–” (minus) sign just happens to indicate a
                                                    negative relationship, a downhill line.
                                                    How close is close enough to –1 or +1 to indicate a strong enough linear

                                                    relationship? Most statisticians like to see correlations beyond at least +0.5
                                                    or –0.5 before getting too excited about them. Don’t expect a correlation to
                                                    always be 0.99 however; remember, this is real data, and real data aren’t perfect.
                                                    For my subset of the cricket chirps versus temperature data from the earlier
                                                    section “Picturing a Relationship with a Scatterplot,” I calculated a correlation
                                                    of 0.98, which is almost unheard of in the real world (these crickets are good!).
                                                    Examining properties of the correlation
                                                   Here are several important properties of the correlation coefficient:
                                                     ✓ The correlation is always between –1 and +1, as I explain in the preceding
                                                        section.
                                                     ✓ The correlation is a unitless measure, which means that if you
                                                        change the units of X or Y, the correlation won’t change. For example,
                                                        changing the temperature from Fahrenheit to Celsius won’t affect
                                                        the correlation between the frequency of chirps (X) and the outside
                                                          temperature (Y).
                                                     ✓ The variables X and Y can be switched in the data set without changing
                                                        the correlation. For example, if height and weight have a correlation of
                                                        0.53, weight and height have the same correlation.
                                         Working with Linear Regression
                                                    In the case of two numerical variables X and Y, when at least a moderate
                                                    correlation has been established through both the correlation and the scat-
                                                    terplot, you know they have some type of linear relationship. Researchers
                                                    often use that relationship to predict the (average) value of Y for a given
                                                    value of X using a straight line. Statisticians call this line the regression line.
                                                    If you know the slope and the y-intercept of that regression line, then you can
                                                    plug in a value for X and predict the average value for Y. In other words, you
                                                    predict (the average) Y from X. In the following sections, I provide the basics
                                                    of understanding and using the linear regression equation (I explain how to
                                                    make predictions with linear regression later in this chapter).









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