r = 0.996232, a large positive linear association. r² = 0.992479, so about 99.2% of the variation in y is linearly associated with x.
| n (pairs) | 7 |
| Σx, Σy | 37.5000, 33.7000 |
| Σxy, Σx², Σy² | 211.6500, 234.8700, 190.9500 |
| Sxy = Σxy − (Σx·Σy)/n | 31.114286 |
| Pearson r = Sxy / √(Sxx·Syy) | 0.996232 |
| r² | 0.992479 |
| t = r √(n − 2) / √(1 − r²), df = n − 2 | 25.6863, df 5 |
| Two-tailed p, H₀: ρ = 0 | 0.000002 |
r is always between −1 and +1. The sign gives the direction, and the size gives how tightly the points follow a straight line. Cohen (1988, Statistical Power Analysis for the Behavioral Sciences, ch. 3) suggests about 0.1, 0.3 and 0.5 as small, medium and large effects. Those are rules of thumb for research, not cut-offs for every field. The p-value tests whether the population correlation could be zero. It says nothing about whether one variable causes the other, and a curved relationship can give a small r even when the two variables are strongly related.
Find r, r² and the significance of a correlation. The Pearson correlation coefficient r measures how closely paired measurements follow a straight line, from −1 for a perfect negative line to +1 for a perfect positive line. This calculator computes r from the sums of deviations, squares it to get r², and tests whether the population correlation could be zero with a t statistic on n − 2 degrees of freedom.
r measures linear association only. Two variables can be strongly related along a curve and still show r near zero, and a large r between two variables says nothing about cause. Plot the data before you interpret the number. 100% free, no registration, and complete privacy — everything runs locally in your browser, so your data never touches a server.
r = Sxy / √(Sxx·Syy), where Sxy = Σxy − (Σx·Σy)/n. The result always lies between −1 and +1.
Σx, Σy, Σxy, Σx² and Σy² are listed so each step can be reproduced on a calculator or in a spreadsheet.
t = r·√(n − 2) / √(1 − r²), with n − 2 degrees of freedom, gives a two-tailed p-value for H0: ρ = 0.
The strength labels follow Cohen's conventional 0.1, 0.3 and 0.5 bands, which are rules of thumb for research rather than cut-offs.
Slope, intercept and the slope test for the same paired data.
Sample and population standard deviation and variance for a list.
One-tailed and two-tailed p-values for a z or t test statistic.
One-sample, Welch and paired t tests with the p-value.
There is no universal cut-off. Cohen (1988) suggested about 0.1 as small, 0.3 as medium and 0.5 as large for behavioural research. In instrument calibration, 0.95 may be weak by the standards of that field. The right threshold depends on the measurement and the question.
No. A high r shows that the two variables move together in the data. The link can come from a third variable, from reverse causation or from chance. A randomized experiment is needed to test causation.
r² is the proportion of the variation in y that a straight line in x accounts for. With r = 0.5, r² = 0.25, so a quarter of the variation is linearly associated with x. It makes the strength easier to compare across studies than r alone.
Pearson assumes a linear relationship and is most useful when the data have no extreme outliers. For ranked data, or for a monotonic but curved relationship, Spearman's rank correlation is usually the better choice. This page computes Pearson only.
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