Use this tool for Pearson correlation significance testing when the null hypothesis assumes population correlation equals zero.
The line shows how the t statistic changes as r changes for the selected sample size.
| Example | r | n | α | Tail | t Statistic | p Value | Decision |
|---|---|---|---|---|---|---|---|
| Classroom sample | 0.62 | 28 | 0.05 | Two-tailed | 4.0293 | 0.0004 | Reject the null hypothesis |
| Study group sample | 0.41 | 35 | 0.01 | Two-tailed | 2.5823 | 0.0144 | Fail to reject the null hypothesis |
| Revision quiz sample | -0.53 | 24 | 0.05 | Left-tailed | -2.9315 | 0.0039 | Reject the null hypothesis |
Main test statistic:
t = r × √((n − 2) / (1 − r²))
Degrees of freedom: df = n − 2
Two-tailed p value: p = 2 × min[P(T ≤ t), P(T ≥ t)]
Variance explained: r² × 100%
This calculator tests whether a sample Pearson correlation differs significantly from zero.
It also estimates a confidence interval using Fisher’s z transformation.
That interval is most reliable with larger samples.
- Enter the sample correlation coefficient, r.
- Enter the total sample size, n.
- Choose the significance level, such as 0.05 or 0.01.
- Select two-tailed, left-tailed, or right-tailed testing.
- Pick how many decimal places you want displayed.
- Press the calculate button to see the test statistic, p value, decision, interval, and graph above the form.
1. What does the r test statistic measure?
It measures whether an observed Pearson correlation is strong enough to be statistically different from zero in the population.
2. When should I use a two-tailed test?
Use a two-tailed test when any meaningful relationship matters, whether the correlation is positive or negative.
3. What sample size is required?
You need at least three paired observations to compute the statistic, because degrees of freedom equal n minus 2.
4. What does a small p value mean?
A small p value suggests the sample correlation would be unlikely if the true population correlation were zero.
5. Why does the calculator show r²?
r² shows the proportion of variance explained by the relationship. It helps students interpret practical importance, not only significance.
6. Can this be used for Spearman correlation?
No. This page is designed for Pearson correlation testing. Spearman methods need a different approach and assumptions.
7. Why is my confidence interval missing?
The interval needs a sample size above 3 because Fisher’s z standard error depends on n minus 3.
8. Does statistical significance prove a strong relationship?
No. A result can be significant but still weak. Always read r, r², and the effect label together.