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      <text>43.2.04 Significance of Pearson’s r</text>
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      <text><![CDATA[<h3>The significance of Pearson’s \(r\)</h3>

In order to test the significance of a calculated value of \(r\) we assume that both \(x\) and \(y\) are normally distributed and set up the hypotheses:
\[ H_0: \rho = 0 \hspace{2cm} H_1: \rho \neq 0 \]
where \(\rho\) is the `true' value of the population correlation. If the assumption of normality is false the test must not be used. We know that the value of \(-1 \leq r \leq 1\) and we wish to know whether our correlation coefficient is significantly different to zero.

<p>A generalisation of the third law of indices states:</p>
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<h4>Key Point</h4>
<p><b>Significance of Pearson's \(r\)</b></p>

<p></p> It can be shown that the test statistic 
\[ r_{test} = \dfrac{\lvert r \rvert \sqrt{n - 2}}{\sqrt{1-r^2}} \]
calculated from a sample of \(n\) pairs of values, follows a \(t\)-distribution with \(n-2\) degrees of freedom.
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Note that many authors simply miss out the modulus sign and ignore the sign of \(r\) should it be
negative. The test statistic is then written
\[ r_{test} = \dfrac{r  \sqrt{n - 2}}{\sqrt{1-r^2}} \]
and critical values depending on the level of significance required are read off from \(t\)-tables in the usual way. A copy of \(t\)-distribution tables is included at the end of this Section (Table 2).]]></text>
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