Correlation Significance Calculator (p-value from r, n)

What is Pearson's Correlation Coefficient?

Pearson's correlation coefficient (r) measures the strength and direction of the linear relationship between two continuous variables. It ranges from -1 (perfect negative correlation) through 0 (no correlation) to +1 (perfect positive correlation). A positive r indicates that as one variable increases, the other tends to increase; a negative r indicates an inverse relationship.

Testing Significance of r

t = r × √(n-2) / √(1-r²)

The t-statistic tests whether the observed correlation differs significantly from zero. Under the null hypothesis (H₀: ρ = 0), this statistic follows a t-distribution with n-2 degrees of freedom.

One-Sided vs Two-Sided Tests

Two-Sided Test (r ≠ 0)

Tests whether there is any correlation (positive or negative). Use this when you have no prior expectation about the direction.

Right-Tailed Test (r > 0)

Tests specifically for a positive correlation. Use when you hypothesize that as X increases, Y should also increase.

Left-Tailed Test (r < 0)

Tests specifically for a negative correlation. Use when you hypothesize that as X increases, Y should decrease.

Confidence Intervals via Fisher's z Transform

z = 0.5 × ln((1+r)/(1-r))

Fisher's z transformation converts r to a value with an approximately normal distribution, allowing us to construct confidence intervals. The standard error is SE_z = 1/√(n-3).

After computing the CI in z-space, we transform back to r-space to get bounds for the population correlation ρ.

Effect Size Interpretation

Negligible

|r| < 0.1

Small

0.1 - 0.3

Medium

0.3 - 0.5

Large

0.5 - 0.7

Very Large

|r| ≥ 0.7

These guidelines (based on Cohen's conventions) help interpret the practical importance of a correlation, regardless of statistical significance.

Important Considerations

Correlation ≠ Causation: A significant correlation does not prove that one variable causes changes in the other. There may be confounding variables or reverse causation.

Linear relationships only: Pearson's r only measures linear relationships. Non-linear associations may show r ≈ 0 even when variables are strongly related.

Outliers matter: Extreme values can dramatically affect r, especially with small samples. Always visualize your data.

Sample size affects significance: With very large n, even tiny correlations become "significant." Always consider effect size.

Assumptions

Continuous variables:Both X and Y should be measured on interval or ratio scales

Bivariate normality:Ideally, the joint distribution is bivariate normal (less critical with larger n)

Linear relationship:The relationship between variables should be approximately linear

Homoscedasticity:Variance of Y should be roughly constant across values of X

Example Calculation

Testing a correlation between study hours and exam scores

Given: r = 0.42, n = 45

Degrees of freedom: df = 45 - 2 = 43

t-statistic: t = 0.42 × √(43/(1-0.42²)) ≈ 3.03

p-value (two-sided): p ≈ 0.0041

At α = 0.05, p < 0.05, so the correlation is statistically significant. There is evidence of a positive medium-strength relationship between study hours and exam scores.

What is Pearson's Correlation Coefficient?

Testing Significance of r

t = r × √(n-2) / √(1-r²)

The t-statistic tests whether the observed correlation differs significantly from zero. Under the null hypothesis (H₀: ρ = 0), this statistic follows a t-distribution with n-2 degrees of freedom.

One-Sided vs Two-Sided Tests

Two-Sided Test (r ≠ 0)

Tests whether there is any correlation (positive or negative). Use this when you have no prior expectation about the direction.

Right-Tailed Test (r > 0)

Tests specifically for a positive correlation. Use when you hypothesize that as X increases, Y should also increase.

Left-Tailed Test (r < 0)

Tests specifically for a negative correlation. Use when you hypothesize that as X increases, Y should decrease.

Confidence Intervals via Fisher's z Transform

z = 0.5 × ln((1+r)/(1-r))

Fisher's z transformation converts r to a value with an approximately normal distribution, allowing us to construct confidence intervals. The standard error is SE_z = 1/√(n-3).

After computing the CI in z-space, we transform back to r-space to get bounds for the population correlation ρ.

Effect Size Interpretation

Negligible

|r| < 0.1

Small

0.1 - 0.3

Medium

0.3 - 0.5

Large

0.5 - 0.7

Very Large

|r| ≥ 0.7

These guidelines (based on Cohen's conventions) help interpret the practical importance of a correlation, regardless of statistical significance.

Important Considerations

Correlation ≠ Causation: A significant correlation does not prove that one variable causes changes in the other. There may be confounding variables or reverse causation.

Linear relationships only: Pearson's r only measures linear relationships. Non-linear associations may show r ≈ 0 even when variables are strongly related.

Outliers matter: Extreme values can dramatically affect r, especially with small samples. Always visualize your data.

Sample size affects significance: With very large n, even tiny correlations become "significant." Always consider effect size.

Assumptions

Continuous variables:Both X and Y should be measured on interval or ratio scales

Bivariate normality:Ideally, the joint distribution is bivariate normal (less critical with larger n)

Linear relationship:The relationship between variables should be approximately linear

Homoscedasticity:Variance of Y should be roughly constant across values of X

Example Calculation

Testing a correlation between study hours and exam scores

Given: r = 0.42, n = 45

Degrees of freedom: df = 45 - 2 = 43

t-statistic: t = 0.42 × √(43/(1-0.42²)) ≈ 3.03

p-value (two-sided): p ≈ 0.0041

At α = 0.05, p < 0.05, so the correlation is statistically significant. There is evidence of a positive medium-strength relationship between study hours and exam scores.

Understanding Correlation Significance

What is Pearson's Correlation Coefficient?

Testing Significance of r

One-Sided vs Two-Sided Tests

Two-Sided Test (r ≠ 0)

Right-Tailed Test (r > 0)

Left-Tailed Test (r < 0)

Confidence Intervals via Fisher's z Transform

Effect Size Interpretation

Important Considerations

Assumptions

Example Calculation

Frequently Asked Questions

Related Statistical Tools

Regression Analysis

T-Test Calculator

ANOVA Calculator

Descriptive Statistics

Normal Distribution

Error Propagation Calculator

Correlation Significance Calculator

Test Your Correlation

Understanding Correlation Significance

What is Pearson's Correlation Coefficient?

Testing Significance of r

One-Sided vs Two-Sided Tests

Two-Sided Test (r ≠ 0)

Right-Tailed Test (r > 0)

Left-Tailed Test (r < 0)

Confidence Intervals via Fisher's z Transform

Effect Size Interpretation

Important Considerations

Assumptions

Example Calculation

Frequently Asked Questions

Related Statistical Tools

Regression Analysis

T-Test Calculator

ANOVA Calculator

Descriptive Statistics

Normal Distribution

Error Propagation Calculator

Correlation Significance Calculator

Test Your Correlation