The formulae for a Pearson sample product moment correlation coefficient (also called a Pearson correlation coefficient).

**PEARSON’S PRODUCT MOMENT CORRELATION COEFFICIENT AND ITS SAMPLE ESTIMATE**

The formulae for a Pearson sample product moment
correlation coefficient (also called a Pearson correlation coefficient) are
shown in Equations 12.1 and 12.2. The deviation score formula for *r* is

We will apply these formulae to the small sample of
weight and height measure-ments shown in Table 12.2. The first calculation uses
the deviation score formula (i.e., the difference between each observation for
a variable and the mean of the variable).

The data needed for the formulae are shown in Table
12.3. When using the cal-culation formula, we do not need to create difference
scores, making the calcula-tions a bit easier to perform with a hand-held
calculator.

We would like to emphasize that the Pearson product
moment correlation mea-sures the strength of the linear relationship between
the variables *X* and *Y*. Two variables *X* and *Y* can have an exact
non-linear functional relationship, implying a form of dependence, and yet have
zero correlation. An example would be the func-tion *y* = *x*^{2} for *x* between –1 and +1. Suppose that *X* is uniformly distributed on [0, 1] and
*Y* = *X*_{2} without any error term. For a bivariate distribution,
*r* is an estimate of the correlation (*ρ*) between *X* and *Y*, where

*ρ** *=* *Cov(*X*, *Y*) / √[Var(*X*)Var(*Y*)]

The covariance between *X* and *Y* defined by Cov(*X*, *Y*)
is *E*[(*X* – *μ** _{x}*)(

**TABLE 12.2. Deviation Score Method for Calculating r (Pearson Correlation Coefficient)**

**TABLE 12.3. Calculation Formula Method for Calculating r (Pearson Correlation Coefficient)**

**Display 12.1: Proof of Cov( X, Y) = 0 and **

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