Critical Values for Pearson's r

The most common mistake in testing whether a Pearson correlation is statistically significant is treating sample size as the deciding factor. You see a survey of 100 people and reach for a Z critical value. That is wrong. The correct critical value of r comes from the t distribution, not from the normal distribution, because you are always estimating the population standard deviation from your sample. The choice is between known sigma and estimated sigma, not between large n and small n. A critical value for a correlation test is the r threshold that separates a significant result from a non-significant one at a chosen alpha. That threshold depends on degrees of freedom, not on n alone.

For example, a critical value of r at alpha = 0.05 two-tailed is 0.632 for n = 10, 0.444 for n = 20, and 0.349 for n = 30. The number keeps dropping as n rises. That is the t-distribution at work. If you used the Z critical value of 1.96 and tried to convert it to an r threshold, you would get a constant value near 0.632 for any n above 30, which is too strict for moderate samples and too loose for large ones. The Pearson correlation test is a t-test in disguise. The research in OpenIntro Statistics (4th ed.) and Moore, McCabe & Craig's Introduction to the Practice of Statistics both state the rule: use t when sigma is estimated, which is always the case for a sample correlation.

The formula that turns a sample correlation r into a test statistic is t = r × sqrt(n - 2) / sqrt(1 - r²). This is a direct transformation. Under the null hypothesis that the population correlation is zero, this t statistic follows a Student's t distribution with df = n - 2. The critical value of r is the r that makes the absolute t statistic equal to the t critical value for that df and alpha.

Rearrange the formula to solve for r: critical r = t_critical / sqrt(t_critical² + n - 2). Here t_critical is the two-tailed t critical value from the t distribution with df = n - 2. For n = 20 at alpha = 0.05 two-tailed, t_critical is 2.101. Then critical r = 2.101 / sqrt(2.101² + 18) = 2.101 / sqrt(4.414 + 18) = 2.101 / sqrt(22.414) = 2.101 / 4.734 = 0.444. That matches the table. The NIST/SEMATECH e-Handbook of Statistical Methods section 1.3.6.7 provides the underlying t critical values. This is the only correct way to compute the threshold.

The common error is to use the Z critical value of 1.96 in the same formula. For n = 20 that gives r = 1.96 / sqrt(3.842 + 18) = 1.96 / 4.674 = 0.419. That is too small. You would reject the null more often than alpha allows, inflating your Type I error. The Z versus t choice is not about sample size. It is about whether you know the population standard deviation. For a correlation test, you never do.

You have 25 pairs of data. You compute r = 0.45. Your alpha is 0.05, two-tailed. You need to know if 0.45 exceeds the critical threshold. From the table, for n = 25 at alpha = 0.05 two-tailed, the critical r is 0.396. Your r of 0.45 is larger than 0.396. The result is statistically significant.

Now verify with the t formula. Compute t = 0.45 × sqrt(23) / sqrt(1 - 0.2025) = 0.45 × 4.796 / sqrt(0.7975) = 2.158 / 0.893 = 2.417. The t critical value for df = 23 at alpha = 0.05 two-tailed is 2.069. 2.417 exceeds 2.069. Same conclusion. This is the full correlation significance test. If your sample r is smaller than the critical r, you cannot reject the null. The relationship is not significant at that alpha.

What if r = 0.35? That is below 0.396. The test is not significant. You report r = 0.35, p > 0.05, and you move on. Never round a non-significant result to "there is no correlation". A non-significant result means the evidence is not strong enough, not that the true correlation is zero.

What About Software or Spreadsheets?

Excel and Google Sheets give the t critical value directly. For df = 23 and alpha = 0.05 two-tailed, use T.INV(0.975, 23) which returns 2.069. Then plug that into the r formula. Or compute the p-value directly: use a t-distribution function on 2.417 with df = 23 to get p = 0.047. That is below 0.05, so significant. The critical value method and the p-value method give the same decision. Use the one your audience expects.

A two-tailed correlation test asks: is the correlation different from zero in either direction? It splits alpha between both tails. At alpha = 0.05 two-tailed, each tail has 0.025. A one-tailed test asks: is the correlation greater than zero (or less than zero)? It puts all alpha in one tail. At alpha = 0.05 one-tailed, that tail contains the full 0.05.

The one-tailed critical r is smaller. For n = 20 at alpha = 0.05 one-tailed, the t critical value is 1.734 (from T.INV(0.95, 18)). The two-tailed t critical value was 2.101. Plug 1.734 into the r formula: critical r = 1.734 / sqrt(3.007 + 18) = 1.734 / 4.583 = 0.378. The two-tailed critical r was 0.444. You can reject the null with a smaller r in a one-tailed test, provided you have a directional hypothesis in advance. Do not choose a one-tailed test after seeing the data. That is p-hacking.

The confusion pair here is common: using a one-tailed critical value for a two-tailed test. That makes the threshold too small and rejects the null too often. Using a two-tailed critical value for a one-tailed test makes the threshold too large and fails to reject when you should. Always match the test to the research question before collecting data.

The table in this section gives critical r values for common sample sizes and alpha levels. Each entry is computed from the t relationship. To use it, find your n and your alpha. If your r exceeds the table value, the correlation is significant. If it does not, it is not. No calculator needed for standard cases.

For non-standard alpha levels, compute the t critical value using T.INV in Excel or invT on the TI-84. Then solve for r. For alpha = 0.10 two-tailed with n = 15, T.INV(0.95, 13) = 1.771. Critical r = 1.771 / sqrt(3.136 + 13) = 1.771 / 4.015 = 0.441. That is the threshold. If your sample r is above 0.441, the correlation is significant at the 0.10 level. For publication, always report the actual p-value alongside the critical value decision. The ASA Statement on Statistical Significance and P-Values (Wasserstein & Lazar 2016) reminds you that statistical significance is not the same as practical importance.

R Critical Value Table for One-Tailed Tests

For a one-tailed test at alpha = 0.05 with n = 20, the critical r is 0.378. For n = 30, it is 0.306. For n = 50, it is 0.237. For n = 100, it is 0.165. These are smaller than the two-tailed values. Use them only when your hypothesis specifies a direction, such as "hours studied correlates positively with exam score". If you have no directional prediction, use the two-tailed values from the main table.

The table below is generated from the t relationship and covers the most common sample sizes. Each row gives the critical r for alpha 0.10, 0.05, and 0.01, all two-tailed. For a test at alpha = 0.05, find your n in the left column and compare your r to the value in the "0.05" column. If your r is larger, the correlation is significant. If it is smaller, it is not. No interpolation is needed for these standard alphas because the table includes ten n values from 5 to 1000.

For n = 10, the 0.05 critical r is 0.632. That means a sample correlation of 0.63 barely fails significance. A correlation of 0.64 passes. This is why reporting the exact p-value, not just a binary significant/not-significant, is better practice. For n = 1000, the critical r at 0.05 is just 0.062. A very small correlation can be statistically significant with a large sample, but that does not mean it is practically important. The ASA statement from 2016 makes this distinction: a critical value decision is not a substitute for effect size reporting.

What to Do When Your Alpha Is Not in the Table

If you need alpha = 0.025 two-tailed, or alpha = 0.005, use software. In Excel, compute T.INV(1 - alpha/2, n - 2) to get the t critical value. Then plug into r = t / sqrt(t² + n - 2). For n = 30 and alpha = 0.025 two-tailed, T.INV(0.9875, 28) = 2.380. Then r = 2.380 / sqrt(5.664 + 28) = 2.380 / 5.800 = 0.410. The table above gives 0.361 for alpha = 0.05, so the 0.025 threshold is larger. That is expected: a smaller alpha means a larger critical value.

For right-tail only tests like when you are testing r against a non-zero null, the formula shifts. You need to adjust the t critical value to reflect the non-zero null. That case is rare. Most introductory correlation tests use the zero null. Stick with the table above for those.

The critical value of r from the t relationship is the correct tool for testing a Pearson correlation against a null of zero. It avoids the n ≥ 30 trap and the Z-versus-t confusion. But it assumes bivariate normality. With a sample of 5 pairs, a single outlier can push r to 0.9 even when the true correlation is zero. The critical r table does not protect against that. Always plot your data. If the scatterplot shows a non-linear pattern or extreme outliers, the Pearson r and its critical value test are misleading. Use Spearman's rank correlation instead. For small samples, the Fisher z transformation and a bootstrap confidence interval give a more honest picture. The table is a starting point, not a final verdict.

The single thing that most often goes wrong is using the table without checking for non-linearity. A correlation of 0.7 can look significant but may reflect a single influential point, not a real relationship. Before you declare a correlation significant, look at the graph. If the relationship is curved, the Pearson r underestimates the strength. If it is linear and clean, the critical value test works. If it is anything else, stop and use a different method.