9.4: Significance of Correlation

We now know how to calculate \( r \) and how to interpret its sign and magnitude. But before we use a correlation to draw conclusions — or build a regression model — we need to ask a more fundamental question:

Could this correlation have occurred by random chance, even if the true relationship in the population is zero?

This is a question about statistical significance, and it calls for a hypothesis test.

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Why This Matters

Suppose we sample 8 students and calculate a correlation of \( r = 0.61 \) between study hours and exam scores. That sounds meaningful — but with only 8 data points, could a value that large have appeared just by chance, even in a population where study hours and exam scores are completely unrelated?

The answer is: possibly yes. Small samples are noisy. A moderate \( r \) from a small sample may not be strong evidence of a real relationship in the population. We need a formal test to decide.

This connects directly to the five-step hypothesis testing process from Chapter 8. The only thing that changes is the test statistic we use.

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Setting Up the Hypotheses

We use \( \rho \) (the Greek letter rho) to denote the true population correlation coefficient — the correlation we would find if we could measure every individual in the population. Our sample value \( r \) is an estimate of \( \rho \).

• \( H_0: \rho = 0 \)   (no linear relationship exists in the population)
• \( H_A: \rho \neq 0 \)   (a linear relationship does exist)

This is almost always a two-tailed test — we are asking whether the relationship is real, not specifically whether it is positive or negative. (One-tailed versions exist but are less common.)

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The Test Statistic

When \( H_0: \rho = 0 \) is true, the sample correlation \( r \) follows a predictable pattern. We can convert it into a t-test statistic using the formula below — and that t-statistic follows the Student t-distribution with \( n - 2 \) degrees of freedom.

Test statistic for the significance of a correlation:

\[ t = r \sqrt{\frac{n - 2}{1 - r^2}} \]

• \( r \) = the sample correlation coefficient
• \( n \) = the number of data pairs
• Degrees of freedom: \( df = n - 2 \)

Notice that \( r^2 \) appears in the denominator — the coefficient of determination from Section 9.3.1 is built right into this formula. The larger \( |r| \) is, and the larger \( n \) is, the larger \( |t| \) will be, and the stronger the evidence against \( H_0 \).

Technical note: We lose two degrees of freedom (rather than one) because estimating a linear relationship requires fixing both the slope and the intercept.

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Two Ways to Perform a Hypothesis Test for Correlation

Using the test statistic \( t \) above, we can reach a conclusion using either of two approaches in the next table. Since Desmos does not perform hypothesis testing calculations for correlation in its inference tool, we can choose one of these two methods:

Two approaches to testing the significance of a correlation.

p-value method	Critical value method
Calculate \( t \) using the formula above. Look up or compute \( P(|t_{df}| > |t_{obs}|) \) — the two-tailed p-value from the t-distribution with \( df = n-2 \).  You can use Excel's =T.DIST.2T function or tdist( ) in Desmos in addition to other calculators. If \( p < \alpha \), reject \( H_0 \). Best when you have access to a t-distribution calculator that gives exact probabilities.	Calculate \( t \) using the formula above. Find the critical value \( t^* \) from a t-table at your chosen \( \alpha \) with \( df = n-2 \). If \( |t_{obs}| > t^* \), reject \( H_0 \). Best when technology is limited. The \(r \) critical value table on the following page can also be used directly with \( r \) — see below.

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Shortcut: Critical Values for \( r \) Directly

Because the test statistic formula always uses the same inputs (\( r \) and \( n \)), statisticians often use pre-computed minimum values of \( |r| \) needed to reject \( H_0 \) at common significance levels. This gives us a table of critical values for \( r \) — a convenient shortcut that skips the test statistic calculation entirely.

How to use it: Find the row for your sample size \( n \) and the column for your chosen \( \alpha \). If \( |r| \) exceeds the table value, the correlation is statistically significant.

Critical Values of \( r \) for a Two-Tailed Test (\( H_A: \rho \neq 0 \))

Sample size \( n \)	Critical Value for \(r\) at \( \alpha = 0.05 \) significance level
4	0.950
5	0.878
6	0.811
7	0.754
8	0.707
9	0.666
10	0.632
12	0.576
15	0.514
20	0.444
25	0.396
30	0.361
40	0.312
50	0.279
60	0.254
80	0.220
100	0.197

Reading the table: To be statistically significant at \( \alpha = 0.05 \) with \( n = 10 \) pairs, you need \( |r| > 0.632 \). Our dataset from Section 9.3 had \( n = 10 \) and \( r \approx 0.99 \) — well above this threshold.

Notice how the required \( |r| \) decreases as \( n \) increases. With a large enough sample, even a small correlation can be statistically significant. This is why we always consider both statistical significance and practical significance.

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Worked Example: Study Hours and Exam Scores

Let's apply the full five-step process to our familiar dataset from Section 9.3: \( n = 10 \), \( r = 0.99 \).

Step 1: State the Hypotheses

• \( H_0: \rho = 0 \)   (no linear relationship between study hours and exam scores)
• \( H_A: \rho \neq 0 \)   (a linear relationship exists)

Step 2: Calculate the Test Statistic

Plug in \( r = 0.99 \) and \( n = 10 \):

\[ t = r\sqrt{\frac{n-2}{1-r^2}} = 0.99 \times \sqrt{\frac{10-2}{1-(0.99)^2}} = 0.99 \times \sqrt{\frac{8}{1-0.9801}} = 0.99 \times \sqrt{\frac{8}{0.0199}} \]

\[ = 0.99 \times \sqrt{401.99} \approx 0.99 \times 20.05 \approx 19.85 \]

Degrees of freedom: \( df = 10 - 2 = 8 \).

Step 3: Choose a Significance Level

Use \( \alpha = 0.05 \).

Step 4: Make the Decision

Using the critical value method: From a t-table at \( df = 8 \) and \( \alpha = 0.05 \) (two-tailed), \( t^* = 2.306 \). Since \( |t_{obs}| = 19.85 \gg 2.306 \), we reject \( H_0 \).

Using the \( r \) critical value table: At \( n = 10 \), \( \alpha = 0.05 \), the critical value is \( 0.632 \). Since \( |r| = 0.99 > 0.632 \), we reject \( H_0 \).

Both methods agree — as they always will.

Step 5: State the Conclusion

There is strong statistical evidence of a linear relationship between study hours and exam scores (\( r = 0.99, t(8) = 19.85, p < 0.05 \)). We reject the hypothesis that no linear relationship exists in the population.

Reflect: A test statistic of 19.85 is enormous — the correlation is so strong that there is virtually no chance it arose by random sampling variation. This gives us confidence that the linear model we build in Section 9.6 is describing something real.

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A Caution: Significance Is Not the Whole Story

Statistical significance tells us the correlation is unlikely to be zero — but it does not tell us the correlation is large or meaningful. With a big enough sample, even a tiny correlation like \( r = 0.05 \) can be statistically significant. Always report \( r \) itself (and \( r^2 \)) alongside the significance test so readers can judge both the reality and the size of the relationship.

What significance of \( r \) does and does not tell us.

It tells us…	It does NOT tell us…
The linear relationship is unlikely to be zero in the population	The relationship is strong or practically important
We have enough evidence to proceed with a linear model	One variable causes the other
The direction of the relationship (positive or negative) is real	The relationship is linear rather than curved

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Think About It:

• In our study hours example, \( n = 10 \) and \( r = 0.99 \). What if we had the same \( r = 0.99 \) but only \( n = 4 \) data points? Use the critical value table to check whether the correlation would still be significant at \( \alpha = 0.05 \).
• A study of 500 people finds \( r = 0.09 \) between daily steps and resting heart rate. The p-value is 0.044. Is this significant? Is it important? What would you tell a friend who read the headline "Step count linked to heart rate"?
• Why do we use \( df = n - 2 \) rather than \( df = n - 1 \) for this test?

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Now that we have established that the linear relationship in our data is real and not simply a product of chance, we are ready to quantify how much of the variation in \( y \) that relationship explains. That is the focus of the next section: the coefficient of determination, \( r^2 \).