Which of the following are assumptions required for the Pearson correlation coefficient to be valid?

The Pearson correlation coefficient, a statistic used to measure the strength and direction of a linear relationship between two variables, has several key assumptions that must be met for its results to be valid.

First, the assumption of a linear relationship between the two variables is critical. The Pearson correlation is designed specifically to assess linear relationships, which means that if the relationship is non-linear, the coefficient may underrepresent the true strength or direction of the association.

Second, the measurement scale of the two variables is significant. The Pearson correlation requires that both variables are measured at an interval or ratio scale, as these measurement types provide the necessary properties for meaningful arithmetic operations that underpin the calculation of the correlation coefficient.

Third, the presence of outliers can drastically affect the Pearson correlation coefficient. Outliers can skew the results, leading to a misinterpretation of the strength and nature of the relationship between the two variables.

Since all these conditions are integral to the proper use of the Pearson correlation coefficient, the correct response is that all of these assumptions must be satisfied for the calculation to yield valid insights regarding the relationship between the variables.