Regression and Correlation
Correlation Coefficient Calculator
Typed pairs are turned into Pearson r and r squared, and the absolute value of r is described with a verbal strength band and no p-value.
Worksheet
Press Calculate to see the result.
Typed pairs are turned into Pearson r and r squared, and the absolute value of r is described with a verbal strength band and no p-value.
What this calculator answers
Pearson r rescales covariance so the result sits between minus one and plus one. Dividing by both standard deviations removes the units, which is why an r from centimetres can be compared with an r from dollars. The square, r squared, is the share of variance lined up with a straight fit. This page reports those two figures and a plain-language band. It does not run a significance test. The page keeps the definition fixed while the ticket updates, so a hand check can match every intermediate term. The worksheet asks for Pairs, one x y per line, and it refuses a blank that the formula cannot interpret. Treat the loaded example as a rehearsal for the live numbers you will type next, not as a claim about your own data.
The page reports Pearson r, r squared, and a verbal band for the absolute value. It does not report a t statistic or a p-value. 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
Pearson product-moment correlation uses this relationship.
r = sum((x - mean x)(y - mean y)) / sqrt(sum (x - mean x)^2 * sum (y - mean y)^2)
- r. Pearson correlation, from minus 1 to plus 1.
- r squared. The square of r, from 0 to 1 when r is defined.
- n. The number of typed pairs.
- band. A verbal label for the absolute size of r.
The numerator is the same sum of paired products used in covariance. Dividing by the two spread terms cancels the units. Squaring r drops the sign and leaves the straight-line share of variance. 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.
There are 6 pairs. The mean of x is 3.5 and the mean of y is 5. The sum of paired deviation products is 19. The sum of squared x deviations is 17.5 and the sum of squared y deviations is 24.
Pearson r is 19 / sqrt(17.5 times 24) = 0.9271. r squared is 0.9271 squared, which equals 0.8595. The absolute value 0.9271 falls in the very strong band, and the sign is positive.
Bands used here: under 0.20 very weak, under 0.40 weak, under 0.60 moderate, under 0.80 strong, otherwise very strong. This page does not convert r into a t statistic or a p-value.
| Pearson r | 0.9271 |
|---|---|
| r squared | 0.8595 |
| Verbal band | very strong positive |
| Pairs | 6 |
A second situation
The last pair is moved to 6 3
There are 6 pairs. The mean of x is 3.5 and the mean of y is 4.1667. The sum of paired deviation products is 6.5. The sum of squared x deviations is 17.5 and the sum of squared y deviations is 14.8333.
Pearson r is 6.5 / sqrt(17.5 times 14.8333) = 0.4034. r squared is 0.4034 squared, which equals 0.1628. The absolute value 0.4034 falls in the moderate band, and the sign is positive.
Bands used here: under 0.20 very weak, under 0.40 weak, under 0.60 moderate, under 0.80 strong, otherwise very strong. This page does not convert r into a t statistic or a p-value.
| Pearson r | 0.4034 |
|---|---|
| r squared | 0.1628 |
| Verbal band | moderate positive |
| Pairs | 6 |
Dropping the last Y from 8 toward the middle lowers |r|. The pair count stays 6, so the change is in the alignment, not in the sample size. 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 last response fell from 8 to 5, which loosens the straight-line pattern. The headline result went from 0.9271 to 0.7482.
The nudged inputs
There are 6 pairs. The mean of x is 3.5 and the mean of y is 4.5. The sum of paired deviation products is 11.5. The sum of squared x deviations is 17.5 and the sum of squared y deviations is 13.5.
Pearson r is 11.5 / sqrt(17.5 times 13.5) = 0.7482. r squared is 0.7482 squared, which equals 0.5598. The absolute value 0.7482 falls in the strong band, and the sign is positive.
Bands used here: under 0.20 very weak, under 0.40 weak, under 0.60 moderate, under 0.80 strong, otherwise very strong. This page does not convert r into a t statistic or a p-value.
| Pearson r | 0.7482 |
|---|---|
| r squared | 0.5598 |
| Verbal band | strong positive |
| Pairs | 6 |
How to read the result
Say the sign and the band together. A strong negative r is a strong relationship in the opposite direction. The band is a description, not a significance test. 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
- Each line is one pair in the order you intend.
- Both variables actually vary. A constant column has no r.
- You want the linear association, not a rank association.
- You are not asking for a p-value on this page.
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
- Reading r squared as if it still carried a positive or negative sign.
- Calling a moderate r significant without a test and a sample size.
- Using Pearson r for a clearly curved cloud and treating a small r as no relationship of any kind.
- Comparing r across studies that used different ranges of X.
Where a different calculator is the honest choice
Use the covariance calculator when you need the unit-bearing product. Use the coefficient of determination calculator when r squared is the only input you have. Use Spearman's calculator when the ranks matter more than the raw scale. 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: Covariance Calculator, Coefficient of Determination Calculator, Spearman Correlation Coefficient Calculator.
Check yourself
Pairs (1, 2), (2, 4), and (3, 6) lie on a straight line through the origin with positive slope. What is r?
Every point sits on one rising line, so Pearson r is 1 and r squared is 1.
Using the result in a sentence
Exam scores against study hours, instrument agreement, and two raters on the same scale are common Pearson r questions. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
Pearson r only hears a straight line. A perfect U-shape can have an r near zero. The verbal bands are conventions for speech, not cut-offs from a test. 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 Correlation Coefficient Calculator commit to?
This page uses Pearson product-moment correlation: r = sum((x - mean x)(y - mean y)) / sqrt(sum (x - mean x)^2 * sum (y - mean y)^2). The numerator is the same sum of paired products used in covariance. Dividing by the two spread terms cancels the units. Squaring r drops the sign and leaves the straight-line share of variance. 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 Pairs, one x y per line, 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 pearson r is 0.9271. Say the sign and the band together. A strong negative r is a strong relationship in the opposite direction. The band is a description, not a significance test. 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?
Reading r squared as if it still carried a positive or negative sign. Pearson r only hears a straight line. A perfect U-shape can have an r near zero. The verbal bands are conventions for speech, not cut-offs from a test. 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 the covariance calculator when you need the unit-bearing product. Use the coefficient of determination calculator when r squared is the only input you have. Use Spearman's calculator when the ranks matter more than the raw scale. 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?
Every point sits on one rising line, so Pearson r is 1 and r squared 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.