Tests of a Population Correlation Coefficient

Updated for the 2026-2027 CFA® Level I curriculum.

A sample correlation rarely equals exactly zero, even when two variables are unrelated in the population. Sampling variation can produce a small positive or negative coefficient simply because the analyst is working with a limited set of observations.

A test for a correlation coefficient helps you decide whether the observed relationship is strong enough, given the sample size, to support a nonzero correlation in the population. For CFA Level I, you need to understand the parametric Pearson test and the nonparametric Spearman rank correlation test.

Key Takeaways About Tests of a Population Correlation Coefficient

• The sample correlation estimates the population Pearson correlation .

• The sample Spearman rank correlation estimates the population rank correlation .

• The usual Pearson null hypothesis is .

• Testing a Pearson correlation is a parametric procedure.

• Testing a Spearman rank correlation is a nonparametric procedure.

• The Pearson test statistic follows a t-distribution with degrees of freedom under the test assumptions.

• A larger absolute correlation produces a larger absolute test statistic when sample size remains constant.

• A larger sample can make a modest correlation statistically significant.

• Rejecting the null supports evidence of association. It does not establish causation.

What You Need to Know for CFA Level I

For CFA Level I, you should be able to:

• Distinguish the sample correlation from the population correlation.

• State the correct null and alternative hypotheses.

• Select Pearson or Spearman based on the data and relationship.

• Calculate the Pearson correlation t-statistic.

• Determine the correct degrees of freedom.

• Apply a critical-value or p-value decision rule.

• Interpret the conclusion in the context of the variables.

• Avoid treating statistical association as evidence of causation.

What Does a Test for a Correlation Coefficient Measure?

A correlation test evaluates whether the association observed in a sample provides enough evidence of a relationship in the wider population.

The sample Pearson correlation coefficient is calculated from the paired observations available to the analyst. It changes from one sample to another.

The population Pearson correlation coefficient is a fixed but unknown parameter. It describes the linear association between the two variables across the entire population.

A nonzero sample coefficient does not automatically imply that the population coefficient is nonzero. For example, a sample could produce even when , particularly when the number of observations is small.

The hypothesis test considers both the size of the sample correlation and the number of paired observations. A stronger correlation or a larger sample generally provides more evidence against a zero population correlation.

How Do You State the Null and Alternative Hypotheses?

For a Pearson correlation test, the standard two-tailed hypotheses are:

This setup tests whether any linear population relationship exists, whether positive or negative.

A directional research question may use a one-tailed alternative.

Positive Correlation

Negative Correlation

For a Spearman rank correlation test, the corresponding parameter can be written as .

Use a directional alternative only when the research question specifies the expected direction before the sample results are examined.

How Does the Parametric Pearson Correlation Test Work?

The Pearson correlation test evaluates whether two quantitative variables have a nonzero linear relationship in the population.

Exact inference using the Pearson t-test relies on paired observations from a bivariate normal population. The observations should also be independent, and influential outliers should not dominate the calculated coefficient.

The test statistic is:

The degrees of freedom are:

Where:

• = calculated test statistic

• = sample Pearson correlation coefficient

• = number of paired observations

• = degrees of freedom

• = population Pearson correlation coefficient

The formula converts the sample correlation into a standardized statistic.

Two elements have a direct effect:

• A larger moves the test statistic farther from zero.

• A larger also moves the statistic farther from zero for a given .

This explains why the same sample correlation can lead to different decisions at different sample sizes.

Pearson Correlation Decision Rule

For a two-tailed test, reject when:

You can reach the same decision by comparing the p-value with .

How Does the Nonparametric Spearman Rank Correlation Test Work?

The Spearman rank correlation test evaluates whether two variables have a monotonic relationship.

A monotonic relationship moves consistently in one general direction. As one variable increases, the other tends to increase or tends to decrease, although the rate of change may vary.

Spearman correlation works with the ranks of the observations rather than their original numerical magnitudes. This makes it useful when:

• One or both variables are ordinal.

• The observations are already provided as rankings.

• The relationship is monotonic but curved.

• Influential outliers would distort Pearson correlation.

• Bivariate normality cannot be reasonably supported.

Ranking reduces the influence of extreme values because the largest observation receives the highest rank regardless of how far it lies from the other observations.

Spearman remains a statistical procedure with its own conditions. The paired observations should be independent, and the ranks must represent an ordered relationship.

For exam questions, apply the critical value, p-value, or testing procedure specified for the Spearman coefficient. Do not automatically reuse the Pearson t-statistic formula unless the question explicitly provides or permits an approximation.

Pearson vs Spearman Correlation Tests

How Do You Test Whether a Correlation Equals Zero?

The following process works for both Pearson and Spearman tests.

Step 1: State the Hypotheses

For a standard two-tailed Pearson test:

For Spearman, replace with .

Step 2: Select the Appropriate Method

Use Pearson when:

• Both variables are quantitative.

• The relationship is linear.

• The parametric assumptions are reasonable.

• Influential outliers are absent.

Use Spearman when:

• The data are ordinal or ranked.

• The relationship is monotonic but not linear.

• Extreme observations would distort Pearson correlation.

• The required Pearson assumptions cannot be supported.

Step 3: Select the Significance Level

Choose before evaluating the sample evidence.

Step 4: Calculate the Test Statistic

For Pearson, apply:

Step 5: Determine the Critical Value or P-Value

Use the appropriate distribution, degrees of freedom, and test direction.

Step 6: Make the Statistical Decision

Reject when the calculated statistic falls in the rejection region or when the p-value is less than or equal to .

Step 7: Interpret the Result

State whether the evidence supports a nonzero population association. Keep the conclusion focused on association rather than causation.

Worked Example: Testing a Pearson Correlation

An analyst collects 30 paired monthly observations for the excess returns of a commodity index and an emerging-market equity index.

The sample information is:

• Sample Pearson correlation:

• Number of paired observations:

• Significance level:

The analyst wants to determine whether the population correlation differs from zero.

Step 1: State the Hypotheses

The alternative is non-directional, so the test is two-tailed.

Step 2: Determine the Degrees of Freedom

Step 3: Calculate the Test Statistic

Calculate the numerator:

Calculate the denominator:

The test statistic is:

Step 4: Apply the Critical-Value Rule

At the 5% significance level for a two-tailed test with 28 degrees of freedom, the critical values are approximately:

Because:

the calculated statistic falls inside the rejection region.

Decision: Reject .

Step 5: Confirm With the P-Value

The two-tailed p-value is approximately:

Because:

the p-value method produces the same decision.

Step 6: Interpret the Result

At the 5% significance level, the sample provides evidence of a nonzero population correlation between the commodity and emerging-market equity return series.

The result supports an association between the returns. It does not establish that changes in one index cause changes in the other. Both could respond to a shared factor such as global growth expectations, inflation, or investor risk appetite.

When Would Spearman Be More Appropriate?

Suppose the analyst instead compares 30 countries using:

• Sovereign credit-quality ranks

• Equity-market valuation ranks

Both variables are ordinal. A plot shows that the rankings generally rise together, but the relationship flattens among the highest-ranked countries.

Spearman rank correlation would be more suitable because the data are ranks and the relationship is monotonic rather than clearly linear.

Common Exam Traps

• Confusing with . The sample Pearson coefficient is . The population parameter in the hypothesis is .

• Using and interchangeably. The first is the sample Spearman coefficient, while the second represents population rank correlation.

• Writing the null hypothesis using the sample value. The usual null is , not .

• Using degrees of freedom. The Pearson correlation test uses .

• Placing in the numerator. It belongs in the denominator.

• Using Pearson for ordinal data. The numerical distance between ranks does not represent a meaningful interval.

• Using Spearman only because the sample is small. The measurement scale, outliers, relationship shape, and assumptions determine the method.

• Applying Pearson’s t-formula automatically to Spearman. Use the testing method or reference distribution specified for the rank correlation.

• Interpreting significance as causation. A significant coefficient supports association only.

• Using a one-tailed critical value for a two-tailed alternative. Match the rejection region to the hypotheses.

Practice Question

An analyst calculates a sample Pearson correlation coefficient of 0.35 using 42 paired observations.

She tests:

The two-tailed critical value at the 5% significance level and the appropriate degrees of freedom is 2.021.

The calculated test statistic and decision are closest to:

1. 2.21, and fail to reject

2. 2.36, and reject

3. 2.42, and reject

FAQs About Tests of a Population Correlation Coefficient