#set document(title: "4.4 Properties of r", author: "OpenStax")
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== 4.4#h(0.6em)Properties of r


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Prerequisites]


      Linear 
                Transformations, Introduction to Bivariate 
                Data  #linebreak()  #linebreak()
]


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Learning Objectives]


      + State the range of values for Pearson's correlation
+ State the values that represent perfect linear relationships
+ State the relationship between the correlation of Y with X and the
        correlation of X with Y
+ State the effect of linear transformations on Pearson's correlation
]


    A basic property of Pearson's r is that its possible 
                range is from -1 to 1. A correlation of -1 means a perfect negative 
                linear relationship, a correlation of 0 means no linear relationship, 
                and a correlation of 1 means a perfect positive linear relationship.


    Pearson's correlation is symmetric in the sense 
                that the correlation of X with Y is the same as the correlation 
                of Y with X. For example, the correlation of Weight with Height 
                is the same as the correlation of Height with Weight.


    A critical property of Pearson's r is that it is 
                unaffected by linear 
                transformations. This means that multiplying a variable by 
                a constant and/or adding a constant does not change the correlation 
                of that variable with other variables. For instance, the correlation 
                of Weight and Height does not depend on whether Height is measured 
                in inches, feet, or even miles. Similarly, adding five points 
                to every student's test score would not change the correlation 
                of the test score with other variables such as GPA.