Correlation Significance Test Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/9/2026
The significance test for a correlation is calculated with t = r × √(n−2) ÷ √(1−r²), with n−2 degrees of freedom. For a correlation of 0.6 in a sample of 20 observations, the t-statistic is about 3.18, with 18 degrees of freedom.
Explanation
Our Pearson correlation calculator gives a coefficient r that measures the strength and direction of a linear relationship between two variables, but this coefficient alone doesn't say whether the correlation observed in a sample reflects a genuine relationship in the overall population, or whether it could simply be explained by sampling chance — especially in a small sample, where a relatively strong apparent correlation can easily arise by chance. This calculator tests exactly this question, by turning the coefficient r and the sample size n into a t-statistic, which follows a Student's t-distribution with n−2 degrees of freedom under the assumption that the true correlation in the population is zero. The larger the absolute value of this t-statistic, the less plausible it is that the observed correlation is due to chance: a large sample size amplifies the t-statistic for the same value of r, reflecting the fact that a moderate correlation measured over many observations is more reliable than the same correlation measured on a very small sample. Like our two-sample t-test calculator, this calculator deliberately stops at the t-statistic and degrees of freedom, without converting this result into a p-value: that last step requires comparing the obtained t-statistic to a Student's t-distribution table for this exact number of degrees of freedom, generally done with statistical software.
Example: a correlation of 0.6 in a sample of 20 observations
Inputs
Correlation coefficient (r): 0.6. Sample size (n): 20.
Calculation
t = 0.6 × √(20−2) ÷ √(1−0.6²) = 0.6 × √18 ÷ √0.64 = 0.6 × 4.2426 ÷ 0.8 ≈ 3.182. Degrees of freedom = 20−2 = 18.
Result
The t-statistic is about 3.18, with 18 degrees of freedom.
Frequently asked questions
Why can the same correlation be significant in a large sample but not a small one?
Because the formula includes the sample size (n) in the numerator, under the square root: the larger n is, the higher the t-statistic for the same value of r, much like a measurement repeated more times inspires more confidence. A correlation of 0.3, for example, might not be judged significant with 15 observations but clearly would be with 200 observations, even though the strength of the relationship itself hasn't changed.
Does this calculator directly give a p-value or a significance conclusion?
No, deliberately: this calculator gives the t-statistic and degrees of freedom, but no p-value or automatic conclusion. Interpreting this result requires comparing the obtained t-statistic to a critical value of Student's t-distribution for this number of degrees of freedom and the chosen significance threshold (generally 5%), a step that requires a statistical table or dedicated software.
Does a statistically significant correlation mean it is strong or important in practice?
No, these are two distinct notions: in a very large sample, even a numerically weak correlation (for example r=0.1) can come out as statistically significant, without representing a strong or practically relevant relationship. Statistical significance indicates the correlation is probably not zero, but it's the value of r itself that tells you about the actual strength of the relationship.