Critical r Table (Correlation)

What Is a Critical r Table?

A critical r table displays the minimum absolute value of Pearson's correlation coefficient required to reject the null hypothesis of no correlation. If your calculated |r| exceeds the critical value, the correlation is statistically significant.

The Relationship Between r and t

The significance of a correlation coefficient is tested using the t-distribution:

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

This t-statistic has n - 2 degrees of freedom. The critical r table pre-computes the r values corresponding to critical t values, making it easy to assess significance directly.

How to Use the Table

1. Calculate your sample correlation coefficient r
2. Find the row for your sample size n
3. Find the column for your significance level α
4. Compare |r| to the critical value

Decision: If |r| ≥ r_critical, the correlation is statistically significant.

Example: With n = 25 at α = 0.05 (two-tailed):

• Critical r = 0.396
• If r = 0.52, then |0.52| > 0.396, so reject H₀

One-Tailed vs. Two-Tailed Tests

• Two-tailed (r ≠ 0): Tests whether any correlation exists (positive or negative)
• One-tailed (r > 0 or r < 0): Tests for correlation in a specific direction

For a one-tailed test, use the column for α. For two-tailed, use α/2 (or look for a two-tailed table).

Common Critical Values (Two-Tailed, α = 0.05)

Sample Size and Significance

Notice how critical r decreases as sample size increases. This means:

• Small samples: Need very strong correlations to be significant
• Large samples: Even weak correlations can be statistically significant

This is why statistical significance ≠ practical importance. A correlation of r = 0.10 might be significant with n = 500, but explains only 1% of variance.

Effect Size Interpretation

Cohen's guidelines for interpreting correlation strength:

| |r| Value | Interpretation | |-----------|----------------| | 0.10 | Small effect | | 0.30 | Medium effect | | 0.50 | Large effect |

Coefficient of Determination

Remember that r² (r-squared) indicates the proportion of variance explained:

• r = 0.50 → r² = 0.25 (25% of variance explained)
• r = 0.70 → r² = 0.49 (49% of variance explained)
• r = 0.90 → r² = 0.81 (81% of variance explained)

Assumptions for Valid Testing

The correlation significance test assumes:

1. Both variables are continuous
2. The relationship is linear
3. Data are approximately bivariate normal
4. Observations are independent
5. No significant outliers

Violating these assumptions may invalidate the test.

Alternative: Fisher's z Transformation

For comparing correlations or constructing confidence intervals, use Fisher's z:

$z = \frac{1}{2}\ln\left(\frac{1+r}{1-r}\right) = \text{arctanh}(r)$

The standard error of z is approximately 1/√(n-3).

Related Tools

• Pearson Correlation Calculator — Calculate r
• Correlation P-Value Calculator — Get exact p-values
• Spearman Correlation Calculator — Rank correlation
• Linear Regression Calculator — Predict relationships

Summary

The critical r table provides a quick way to assess whether a correlation is statistically significant. By comparing your calculated correlation to the critical value for your sample size and significance level, you can make an immediate decision about rejecting the null hypothesis. However, always consider effect size and practical significance alongside statistical significance—a "significant" correlation in a large sample may still represent a trivially small relationship.