Regression and Correlation
Pearson R to P-Value Calculator
A published correlation and its sample size are converted into a t statistic and a two-sided p-value without asking for the original pairs.
Worksheet
Press Calculate to see the result.
A published correlation and its sample size are converted into a t statistic and a two-sided p-value without asking for the original pairs.
What this calculator answers
Journals often print r and n and leave the p-value out, or they print a p-value you want to recompute. The same t formula used on raw pairs still works, because the test statistic depends on the pairs only through r and n. You type the published correlation and the sample size. The page returns t, the degrees of freedom n minus 2, and a two-sided p-value, then compares that p-value with 0.05. It cannot see outliers that were in the original cloud.
Only the conversion from r and n is performed. No raw data are read and no new r is estimated. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
t conversion from a published r uses this relationship.
t = r * sqrt((n - 2) / (1 - r^2)), df = n - 2
- r. The published Pearson correlation.
- n. The number of pairs behind that r.
- t. The converted test statistic.
- p. The two-sided p-value.
Once r is known, the sums of squares cancel out of the t formula. That is why a secondary reader can recover the test from the published correlation and the sample size alone. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The form is prefilled with a small example so every arithmetic step can be checked by hand. Change one field after you finish the walkthrough and confirm the ticket moves the way the formula predicts.
The published r is 0.45 and the sample size is 25. Degrees of freedom are 25 minus 2 = 23. One minus r squared is 0.7975.
t = 0.45 times sqrt(23 / 0.7975) = 2.4166. The two-sided p-value from the t distribution is 0.024. At 0.05 the result is in the rejection region.
No raw pairs were used. The conversion depends only on r and n. r squared, if you need it beside the test, is 0.2025.
| Two-sided p-value | 0.024 |
|---|---|
| t | 2.4166 |
| Degrees of freedom | 23 |
| Decision | At 0.05 the result is in the rejection region. |
A second situation
The same r with a smaller sample of 8
The published r is 0.45 and the sample size is 8. Degrees of freedom are 8 minus 2 = 6. One minus r squared is 0.7975.
t = 0.45 times sqrt(6 / 0.7975) = 1.2343. The two-sided p-value from the t distribution is 0.2632. At 0.05 the result is outside the rejection region.
No raw pairs were used. The conversion depends only on r and n. r squared, if you need it beside the test, is 0.2025.
| Two-sided p-value | 0.2632 |
|---|---|
| t | 1.2343 |
| Degrees of freedom | 6 |
| Decision | At 0.05 the result is outside the rejection region. |
Holding r at 0.45 and cutting n from 25 to 8 widens the null distribution, so the same correlation produces a larger p-value. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
The published r moved from 0.45 to 0.70 while n stayed at 25. The headline result went from 0.024 to 9.811e−5.
The nudged inputs
The published r is 0.7 and the sample size is 25. Degrees of freedom are 25 minus 2 = 23. One minus r squared is 0.51.
t = 0.7 times sqrt(23 / 0.51) = 4.7009. The two-sided p-value from the t distribution is 9.811e−5. At 0.05 the result is in the rejection region.
No raw pairs were used. The conversion depends only on r and n. r squared, if you need it beside the test, is 0.49.
| Two-sided p-value | 9.811e−5 |
|---|---|
| t | 4.7009 |
| Degrees of freedom | 23 |
| Decision | At 0.05 the result is in the rejection region. |
How to read the result
Treat the result as a recomputation of the linear correlation test, not as a new look at the raw cloud. If the paper used a one-sided test, this two-sided p-value will be about twice as large. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Write the result with its unit, the inputs that produced it, and the name of the formula. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- The published figure really is a Pearson r, not Spearman rho or R from a bigger model.
- n is the number of pairs used to compute that r.
- You want a two-sided test of a zero correlation.
- The original study's assumptions are still the ones you are willing to borrow.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Entering r squared in the r box.
- Using the number of subjects when some pairs were dropped and n was smaller.
- Comparing this two-sided p-value with a one-sided threshold without saying so.
- Inventing raw data in order to get a p-value this converter already returns.
Where a different calculator is the honest choice
Use Pearson's correlation test calculator when you still have the pairs. Use the correlation coefficient calculator to obtain r before you convert it. Use the coefficient of determination calculator when the published number is r squared and you do not need a test. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Pearson's Correlation Test Calculator, Correlation Coefficient Calculator, Coefficient of Determination Calculator.
Check yourself
A paper reports r = 0 with n = 40. What is the two-sided p-value?
t is 0 with 38 degrees of freedom, so the two-sided p-value is 1.
Using the result in a sentence
Reading a table that prints r and n, and checking a citation in a review, are the jobs this converter is built for. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
The converter cannot detect a curved pattern, a misprinted r, or a sample size that counted the wrong rows. It only evaluates the t formula. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Pearson R to P-Value Calculator commit to?
This page uses t conversion from a published r: t = r * sqrt((n - 2) / (1 - r^2)), df = n - 2. Once r is known, the sums of squares cancel out of the t formula. That is why a secondary reader can recover the test from the published correlation and the sample size alone. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Published Pearson r, Sample size n, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the two-sided p-value is 0.024. Treat the result as a recomputation of the linear correlation test, not as a new look at the raw cloud. If the paper used a one-sided test, this two-sided p-value will be about twice as large. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Entering r squared in the r box. The converter cannot detect a curved pattern, a misprinted r, or a sample size that counted the wrong rows. It only evaluates the t formula. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
Use Pearson's correlation test calculator when you still have the pairs. Use the correlation coefficient calculator to obtain r before you convert it. Use the coefficient of determination calculator when the published number is r squared and you do not need a test. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
t is 0 with 38 degrees of freedom, so the two-sided p-value is 1. Name the formula, the sample size or the inputs, and the unit before you compare the figure with someone else's. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.