Correlation P-Value Calculator

Q: What null hypothesis is tested?

H0: the true population correlation (rho) equals zero, meaning no linear relationship.

Q: Can a weak correlation be significant?

Yes, with large enough n. With n = 1,000, even r = 0.06 can be significant. Always consider magnitude alongside p-value.

Q: What sample size do I need?

Depends on r. For r = 0.50, about n = 15. For r = 0.30, about n = 45. For r = 0.10, about n = 400.

Q: Why the t-distribution, not z?

Because r is an estimate with uncertainty. The t-distribution accounts for this. With large n, t and z converge.

Q: What if r is exactly 1 or -1?

The formula produces division by zero (1 - r squared = 0). This is trivially significant with more than 2 data points.

Use this free calculator to test whether a Pearson correlation coefficient is statistically significant. Enter the correlation value and sample size to get the t-statistic, p-value, and significance decision.

Enter Values

Correlation Coefficient (r)

The Pearson r value (between -1 and 1)

Sample Size (n)

The number of paired observations

Significance Level (alpha)

The threshold for statistical significance

Result

Enter values above and click Calculate to see your result.

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Formula

t = r x sqrt((n - 2) / (1 - r squared))

Convert the correlation coefficient to a t-statistic using the sample size. Under H0 (true correlation = 0), this follows a t-distribution with n - 2 degrees of freedom.

Worked Example

r = 0.45, n = 40, alpha = 0.05 Step 1: t = 0.45 x sqrt(38 / 0.7975) = 0.45 x 6.903 = 3.106 Step 2: df = 38 Step 3: Two-tailed p-value = approximately 0.0035 Step 4: Since p < 0.05, the correlation is statistically significant Result: t = 3.106, p = 0.0035. Strong evidence the true correlation is not zero.

Why Test the Significance of a Correlation?

A correlation coefficient alone does not tell you whether the relationship is real or due to chance. Even random data produces nonzero correlations, especially with small samples. The significance test tells you how likely it is to see this strong a correlation if the true value is zero.

Small samples can produce large correlations by chance. With n = 5, even r = 0.80 might not be significant
Large samples can make tiny correlations significant. With n = 10,000, even r = 0.02 might be significant but practically meaningless
Statistical significance is not the same as practical significance. Always consider the magnitude of r alongside the p-value
The null hypothesis is H0: rho = 0 (no linear relationship)

Always report both r (effect size) and p-value (significance). A weak but significant correlation may not be useful; a strong but non-significant one may just need more data.

You can also calculate changes using our Correlation Coefficient Calculator, T-Test Calculator or Degrees of Freedom Calculator.

Frequently Asked Questions

What null hypothesis is tested?

H0: the true population correlation (rho) equals zero, meaning no linear relationship.

Can a weak correlation be significant?

Yes, with large enough n. With n = 1,000, even r = 0.06 can be significant. Always consider magnitude alongside p-value.

What sample size do I need?

Depends on r. For r = 0.50, about n = 15. For r = 0.30, about n = 45. For r = 0.10, about n = 400.

Why the t-distribution, not z?

Because r is an estimate with uncertainty. The t-distribution accounts for this. With large n, t and z converge.

What if r is exactly 1 or -1?

The formula produces division by zero (1 - r squared = 0). This is trivially significant with more than 2 data points.

AI Assistant

Ask about this calculator

I can help you understand the correlation p-value calculator formula, interpret your results, and answer follow-up questions.

Try asking

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