Paste paired values for two variables and get the correlation coefficient, Pearson's r or Spearman's rho, with an exact p value, confidence interval, and APA 7 write-up. Values are matched by position.
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Get QuantaThree steps here: paste the X values, paste the Y values in the same order, and press calculate. The coefficient appears as r for Pearson or rho for Spearman, alongside the p value and confidence interval.
On a graphing calculator such as the TI-84, the path is different. Enter the two variables as lists under STAT, EDIT, then run STAT, CALC, LinReg(ax+b) on those lists. The r value prints with the regression output. On older operating systems it stays hidden until you turn on DiagnosticOn from the CATALOG menu.
The Pearson formula in computational form:
r = [n∑XY − (∑X)(∑Y)] ÷ √( [n∑X² − (∑X)²] [n∑Y² − (∑Y)²] )
Take five pairs: X = 1, 2, 3, 4, 5 and Y = 2, 4, 5, 4, 5.
| Quantity | Value |
|---|---|
| n | 5 |
| ∑X | 15 |
| ∑Y | 20 |
| ∑XY | 66 |
| ∑X² | 55 |
| ∑Y² | 86 |
Numerator: 5(66) − 15(20) = 30. Denominator: √( [5(55) − 225] [5(86) − 400] ) = √(50 × 30) = 38.73. So r = 30 ÷ 38.73 = .77.
That is a large coefficient by the usual benchmarks, and with n = 5 the p value is .124. Not significant. Five pairs cannot establish much, whatever r says, which is the reason to report both numbers rather than either alone.
The linear correlation coefficient is Pearson's r. It measures how closely the points follow a straight line, and only a straight line. A perfect curve produces a middling r, which is why a scatterplot is worth a look before you trust the number.
Paste the example values into the calculator above (X = 2, 4, 5, 7, 8, 10, 11 and Y = 35, 41, 44, 49, 50, 58, 62) and you get r = .99, r² = .99, n = 7, p < .001, 95% CI [.96, 1.00]. The two variables share about 99% of their variance.
Pearson's r measures the strength of a linear relationship between two continuous variables and is the default when both variables are roughly normally distributed without extreme outliers.
Spearman's rho ranks the data first, so it measures monotonic association. Choose it for ordinal variables (like Likert items), skewed distributions, or data with outliers.
Correlations run from -1 to +1. By common benchmarks, |r| around .10 is small, .30 medium, and .50 large. The p value tests whether the correlation differs from zero; with large samples, even tiny correlations become significant, so always interpret the size of r, not just the p value. r² tells you the proportion of variance the two variables share.
The confidence interval uses the Fisher z transformation. For Spearman, the interval is approximate.
Can a correlation coefficient be negative? Yes. The scale runs from -1 to +1, and a negative value means the variables move in opposite directions. The sign is direction, not strength. An r of -.60 is exactly as strong as an r of +.60.
What does a negative correlation coefficient mean? Higher scores on one variable come with lower scores on the other. Study hours and exam anxiety often move this way. It still says nothing about cause.
What counts as a strong correlation? The .10, .30, .50 benchmarks are conventions and they shift by field. Where measurement is noisy, an r of .30 can be a substantial finding.
A significant correlation says the variables move together. It cannot say which causes which, or whether a third variable drives both. Design, not statistics, answers causal questions.
A correlation answers a narrow question, and the harder work is deciding whether it answers yours: what is confounded, what is mediating, and whether the relationship should have been modeled rather than correlated. That is the kind of question our statistical consulting is built for.
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