Correlation P-Value Calculator Formula

Understand the math behind the correlation p-value calculator. Each variable explained with a worked example.

Use the Correlation P-Value Calculator

Formulas Used

t-Statistic

t_stat = r * sqrt(n - 2) / sqrt(1 - pow(r, 2))

Degrees of Freedom

df = n - 2

R²

r_squared = pow(r, 2)

Variables

VariableDescriptionDefault
rCorrelation Coefficient (r)0.65
nSample Size (n)30

How It Works

Testing Correlation Significance

To determine whether an observed correlation is statistically significant, calculate the t-statistic and compare to the t-distribution.

Formula

t = r × sqrt(n-2) / sqrt(1-r²)

with df = n - 2. If t exceeds the critical value (e.g., ~2.048 for alpha=0.05, df=28), the correlation is statistically significant.

Worked Example

Correlation r = 0.65 from a sample of n = 30.

r = 0.65n = 30
  1. 01t = 0.65 × sqrt(28) / sqrt(1 - 0.4225)
  2. 02t = 0.65 × 5.292 / sqrt(0.5775)
  3. 03t = 3.440 / 0.7599 = 4.527
  4. 04With df = 28, this is highly significant (p < 0.001)

Frequently Asked Questions

What does a significant correlation mean?

It means the observed correlation is unlikely to have occurred by chance if the true correlation is zero. It does NOT mean the correlation is strong or practically important. With large n, even tiny correlations can be significant.

What is the critical value of r for significance?

It depends on n and alpha. For alpha=0.05: n=10 requires |r| > 0.632, n=30 requires |r| > 0.361, n=100 requires |r| > 0.197. Larger samples need smaller r to be significant.

Does significance imply causation?

No. A significant correlation only establishes a linear association. Causation requires experimental manipulation, time ordering, and ruling out confounding variables.

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