Calculator
Example Data Table
| Example | r | n | Tail | Approximate Meaning |
|---|---|---|---|---|
| Moderate positive relation | 0.45 | 30 | Two-tailed | Tests whether the correlation differs from zero. |
| Strong negative relation | -0.70 | 18 | Left-tailed | Tests evidence for a negative population correlation. |
| Weak positive relation | 0.20 | 100 | Right-tailed | Tests evidence for a positive population correlation. |
| Near zero relation | 0.04 | 55 | Two-tailed | Usually gives little evidence against zero correlation. |
Formula Used
The calculator converts Pearson correlation coefficient r into a Student t statistic.
t = r × √((n - 2) / (1 - r²))
df = n - 2
The p value is then found from the Student t distribution.
A two-tailed test doubles the smaller tail probability.
A right-tailed test uses P(T ≥ t).
A left-tailed test uses P(T ≤ t).
Fisher confidence limits use:
z = 0.5 × ln((1 + r) / (1 - r))
and SE = 1 / √(n - 3).
How To Use This Calculator
- Enter the Pearson correlation coefficient in the r field.
- Enter the sample size used to calculate that correlation.
- Select two-tailed, right-tailed, or left-tailed testing.
- Choose an alpha level for the significance decision.
- Press Calculate to view the result above the form.
- Use CSV or PDF buttons to save the calculated output.
Understanding P Values From Correlation
A correlation coefficient can look simple, yet it carries a clear test question. The p value tells whether the observed relationship could appear when the true population correlation is zero. This calculator changes r into a t statistic, then reads the probability from the Student t distribution.
The tool is useful for research notes, class work, reports, and quick checks. Enter Pearson r, sample size, and the test direction. A two tailed test asks whether the relationship is different from zero. A right tailed test asks whether it is positive. A left tailed test asks whether it is negative.
Sample size matters a lot. With a small sample, even a strong r may need caution. With a large sample, a modest r can become statistically significant. That is why the calculator shows degrees of freedom, the t statistic, p value, and an alpha comparison together.
The result should not be read alone. A tiny p value does not prove a large practical effect. It only shows that the observed correlation is unlikely under the null model. The size and sign of r still describe the strength and direction of the relationship. Context, measurement quality, and study design also matter.
For deeper review, the calculator includes Fisher confidence limits. These limits estimate a likely range for the population correlation. They are helpful when you want more than a pass or fail decision. Wide limits suggest uncertainty. Narrow limits suggest better precision.
Use clean data before relying on any output. Pearson correlation assumes paired numeric values and a roughly linear pattern. Strong outliers can change r and the p value sharply. Always inspect scatter plots when possible. This page gives a fast statistical answer, but good interpretation still needs judgment.
Advanced users can compare several planned samples before collecting data. Try different n values to see how power improves. Keep the same r and change the tail option only when the research question justifies it. Do not choose a direction after seeing results. That practice weakens inference. Record assumptions before analysis, report exact p values, and include r so readers can judge both evidence and effect. Clear reports are easier to trust.
FAQs
What is a p value from r?
It is the probability of seeing a correlation this extreme, assuming the true population correlation is zero.
What sample size should I enter?
Enter the number of paired observations used to calculate the Pearson correlation coefficient.
Can I use negative r values?
Yes. Negative r values are valid. They show an inverse relationship between the two measured variables.
When should I choose a two-tailed test?
Use it when your question asks whether the correlation differs from zero in either direction.
When should I choose a right-tailed test?
Use it only when your planned hypothesis specifically expects a positive population correlation.
When should I choose a left-tailed test?
Use it only when your planned hypothesis specifically expects a negative population correlation.
What does alpha mean?
Alpha is your chosen cutoff for statistical significance. A common value is 0.05.
Does a small p value prove importance?
No. It shows statistical evidence. Practical importance still depends on r size, context, and data quality.