### Chapter 2. Correlation: Correlation

### Strength of a Linear Relationship: Pearson Correlation Coefficient

To determine the direction and strength of the linear relationship between two variables, calculate the *Pearson correlation coefficient*.

#\phantom{0}#

**Definition**

The **Pearson** **correlation coefficient** is the standardized form of the covariance.

It is used to measure the direction and strength of the *linear *relationship between two *quantitative variables*.

The population and sample *Pearson correlation coefficient *are denoted by #\rho# and #r#, respectively.

**Formula**

\[\begin{array}{rcl}

\rho(X,Y)&=&\dfrac{\sigma_{X,Y}}{\sigma_X \sigma_Y}\\\\

r(X,Y)&=&\dfrac{s_{X,Y}}{s_Xs_Y}\\

\end{array}\]

Computation of the Sample CovarianceTo compute the *sample* *Pearson correlation coefficient *between two variables #X# and #Y# in **Excel**, make use of the following function:

CORREL(array1, array2)

array1: The range of cells containing the values of variable #X#.array2: The range of cells containing the values of variable #Y#.

To compute the *sample* *Pearson correlation coefficient *between two variables #X# and #Y# in **R**, make use of the following function:

cor(x, y)

x: The numeric vector that contains the values for variable #X#y: The numeric vector that contains the values for variable #Y#

The *Pearson correlation coefficient* always takes on a value between #-1# and #+1#:

- A value of #+1# indicates a
*perfect positive linear*relationship between two variables. - A value of #-1# indicates a
*perfect negative linear*relationship between two variables. - A value of #0# indicates the variables are
*linearly unrelated*.

It is important to remember that the *Pearson correlation coefficient *only measures the *linear* relationship between two variables. Consequently, finding a Pearson coefficient of #0# does not necessarily mean the two variables are completely unrelated, it simply indicates that there is no *linear *relationship.

The scatterplot below shows an example of two variables that have a *Pearson correlation coefficient *of #0#, but do have a perfect quadratic relationship.

Another thing to watch out for is that the *Pearson correlation coefficient *is very sensitive to outliers. A single outlier can have drastic effects on the magnitude of the coefficient.

**Pass Your Math**independent of your university. See pricing and more.

Or visit omptest.org if jou are taking an OMPT exam.