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Correlation significance test

P-Value for Correlation Coefficient

Quick answer

The p-value for a correlation tells you the probability that the observed relationship occurred by chance. A p-value < 0.05 means the correlation is statistically significant.

Enter your r value and sample size. Find out if your correlation is statistically significant, where the p-value comes from, and what that result actually means.

p-value + t statistic

95% and 99% CI

t distribution visual

Confidence Intervals and Next Steps

95% Confidence Interval for r

[0.291, 0.848]

99% Confidence Interval for r

[0.149, 0.885]

Your correlation is significant. Now decide whether it is large enough to matter.

Interpret the strength of your r →Calculate correlation from raw data →Understand r squared and explained variance →

t Distribution Visual

Red area shows the p-value tail area; the blue line is your observed t statistic.

df = 18

Same r, different sample size

The core intuition behind a correlation hypothesis test is simple: the same r value becomes more convincing as the sample size grows.

Small class: not significant

r = 0.45, n = 8

p ≈ 0.263

Could be chance

In a class of only 8 people, the same-looking relationship is too unstable to trust as real evidence.

Large sample: significant

r = 0.45, n = 200

p < 0.0001

Unlikely to be chance

With 200 observations, the same r value becomes strong evidence that the population correlation is not zero.

The key intuition

same r, different n

larger n -> smaller p

Sample size matters

A p-value is not only about the size of r. It also reflects how much evidence the sample size provides.

How the p-value is calculated

The calculator converts a correlation coefficient into a t statistic, then reads the tail probability from a t distribution with $n - 2$ degrees of freedom.

Step 1

Convert r to a t statistic

t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}

t = 0.650 × √18 / √(1 - 0.650²) = 3.629

Step 2

Set degrees of freedom

df = n - 2

df = 20 - 2 = 18

Step 3

Read the tail probability

p = 2P(T_{df} \ge |t|)

p = 0.00192 (two-tailed)

Critical values at a glance

Your absolute r must exceed the critical value in your sample-size row to be significant at that level. These values use a two-tailed Pearson correlation test.

Sample Size (n)	df	Critical r (α=0.05)	Critical r (α=0.01)	Critical r (α=0.001)
5	3	0.878	0.959	0.991
10	8	0.632	0.765	0.872
20	18	0.444	0.561	0.679
30	28	0.361	0.463	0.570
50	48	0.279	0.361	0.451
100	98	0.197	0.256	0.324

Example: with n = 20, a two-tailed α = 0.05 test needs |r| above about 0.444.

What p-value actually means

The p-value for a correlation tells you the probability that the observed relationship occurred by chance if the true population correlation were zero. A p-value < 0.05 means the correlation is statistically significant by the common 95% confidence convention.

p value	Conclusion
p < 0.01	Highly significant, roughly a 99% confidence standard
p < 0.05	Significant, the common 95% confidence standard
p >= 0.05	Not significant; the result may be due to chance

Three common mistakes

Mistake

“p < 0.05 means the correlation is strong”

p-value only tells you about statistical significance, not strength. Strength is read from the r value.

Interpret r value →

Mistake

“p > 0.05 means there is no relationship”

It may only mean the sample is too small or noisy. A real relationship can fail to reach significance in an underpowered study.

Mistake

“A significant correlation proves causation”

Significance says the pattern is unlikely under a zero-correlation null. It does not prove that one variable causes the other.

Correlation vs causation →

Use p-value and effect size together

A correlation result needs two readings: the p-value tells you whether the pattern is statistically reliable, while $r$ and $r^2$ tell you whether it is practically meaningful.

Result

Small effect (r)

Large effect (r)

Significant

Real but tiny

Real and meaningful

Not significant

Likely noise

Possibly underpowered

FAQ

What p-value is acceptable for correlation?

In most fields, p < 0.05 is the standard threshold for statistical significance. In medical research or stricter settings, p < 0.01 is often preferred.

Can I have a high r but high p-value?

Yes. If your sample size is very small, even a high correlation such as r = 0.70 may not reach statistical significance.

My calculator shows r = 0.3, p = 0.001. Is this useful?

It is statistically significant, but r = 0.3 is usually a weak to moderate effect. Significance does not automatically mean practical importance.

What is the difference between one-tailed and two-tailed p-value?

A two-tailed test asks whether r is different from zero in either direction. A one-tailed test asks whether the relationship is in one specific direction. Use two-tailed unless you had a strong directional hypothesis before seeing the data.

How does sample size affect the p-value?

Larger samples make it easier to detect statistical significance. With n = 100, even r = 0.20 is usually significant in a two-tailed Pearson correlation test.

Now that you know whether the correlation is significant

Move from the p-value to the full interpretation: strength, explained variance, raw-data calculation, and assumptions.

Interpret the strength of your r →Calculate correlation from raw data →Understand r² and explained variance →