Correlation Significance Calculator: Is r Significant? p-Value, Confidence Interval and Critical r
Decide if a correlation is real or just chance.
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How this calculator works
Enter the correlation coefficient and the number of pairs.
Choose the confidence level.
The calculator gives the t statistic, the p-value, the confidence interval and the critical r.
Formulas used
- t = r\sqrt{\frac{n - 2}{1 - r^2}}
- z = \operatorname{artanh}r,\ SE = \frac{1}{\sqrt{n - 3}}
- r_c = \frac{t^*}{\sqrt{n - 2 + t^{*2}}}
Worked example
With the default numbers in the calculator above:
- Test statistic t for the null hypothesis that the true correlation is zero: 4.1021 with 23 degrees of freedom
- Two-tailed p-value: 0.0004365
- Confidence interval for the true correlation (Fisher's z transformation): 0.343 to 0.8316
- Smallest correlation that would be significant with this many pairs (two-tailed, at the same level): 0.3961
- Coefficient of determination: the share of the variation in y explained by x: 42.25 %
Change the numbers in the calculator to see the full working, every step, for your own case.
Common questions
How do I know if a correlation is significant?
Test it with t. If the p-value is below your significance level, the correlation is unlikely to be zero.
Does a significant correlation mean a strong one?
No. With many pairs, even a very weak correlation can be significant.
Does correlation mean causation?
No. Two things can be correlated because of a third factor or by coincidence.
Related calculators
Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Any statistics text (inference for the correlation coefficient).
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