Correlation coefficient calculator: Pearson's r or Spearman's rank correlation for paired data, with r², a t statistic and a two-sided p-value.
Cập nhật Ví dụ đã kiểm tra: 4
r = 0.9816 is a strong positive linear association; r² = 0.9636, so a straight line accounts for 96.36% of the variation in y. p = 4.86 × 10⁻⁷: if there were no correlation in the population, a sample of 10 pairs would show one at least this strong with that probability. Correlation alone does not show that one variable causes the other.
Assumes the pairs are independent and roughly bivariate normal.
Pearson's r measures how closely paired values follow a straight line: r = Sxy/√(Sxx·Syy), the sum of cross-products of deviations from the two means divided by the square root of the product of the two sums of squares. It runs from −1, a perfect falling line, through 0 to +1, a perfect rising line. Spearman's ρ is Pearson's r computed on the ranks, so it measures any steadily rising or falling trend and is less affected by outliers. The p-value comes from t = r√((n − 2)/(1 − r²)) with n − 2 degrees of freedom.
The default pairs hours studied with exam scores for 10 students: r = 0.9816, r² = 0.9636 and p = 4.9 × 10⁻⁷, a strong positive association.
A correlation near 0 rules out only a straight-line relationship: y = x² for x from −3 to 3 has r = 0 exactly.
Nguồn đối chiếu: Python fractions for Sxx, Syy, Sxy with a decimal square root; p from the closed-form Student t CDF for integer df (Abramowitz & Stegun 26.7.4)
Nguồn đối chiếu: Python fractions with decimal square roots; p = 2·(1 − F_t(|t|; 4)) from the A&S 26.7.4 closed form
Nguồn đối chiếu: Python: ties averaged by hand (x ranks 1, 2.5, 2.5, 4, 5, 6.5, 6.5, 8), Pearson r of the ranks with fractions (Sxy = 95/4, Sxx = 41, Syy = 83/2), p from the A&S 26.7.4 t CDF
Nguồn đối chiếu: y = 2x + 1 exactly, so r = 1 by definition and no sample could be more extreme
A fairly strong positive linear association: as x rises, y tends to rise, and r² = 0.49 says a straight line accounts for 49% of the variation in y. Cohen (1988) called r = 0.1 small, 0.3 medium and 0.5 large, and the Strength output uses those cut-offs. They are rough conventions from the behavioural sciences, so what counts as strong still depends on the field.
Pearson's r measures linear association using the values themselves; Spearman's ρ applies the same formula to their ranks, so it measures any monotonic trend. For y = x³ with x from 1 to 10, Spearman gives exactly 1 but Pearson gives 0.928. Spearman also resists outliers and suits ordinal data such as ratings, while Pearson's r is the one that matches a least-squares line.
No. A correlation shows that two variables move together, not why. A third variable can drive both, as hot weather raises both ice-cream sales and drownings. The cause can also run the other way, or the pattern can be chance: test 20 unrelated pairs at α = 0.05 and about one will look significant. Showing cause takes a randomized experiment or a careful causal design.
Convert r to t = r√((n − 2)/(1 − r²)) and compare it with a t distribution with n − 2 degrees of freedom. The default data give r = 0.9816 with n = 10, so t = 14.56 on 8 df and p = 4.9 × 10⁻⁷. Significance depends heavily on n: with 1,000 pairs, r = 0.07 already gives p = 0.027, although it explains under 0.5% of the variation.
Độ chính xác phụ thuộc vào dữ liệu nhập và giả định của phương pháp. Phép tính thập phân dùng 50 chữ số có nghĩa, nhưng ước lượng, phương pháp số và dữ liệu nguồn có thể kém chính xác hơn; làm tròn khi hiển thị không loại bỏ các giới hạn đó. Ví dụ có lời giải đã đối chiếu với nguồn độc lập: 4. Ví dụ, “Hours studied vs score (default)” được kiểm tra bằng Python fractions for Sxx, Syy, Sxy with a decimal square root; p from the closed-form Student t CDF for integer df (Abramowitz & Stegun 26.7.4).
NIST/SEMATECH e-Handbook of Statistical Methods, §7.1.3 / Dataplot CORRELATION (Pearson and rank correlation); Spearman (1904), The proof and measurement of association between two things, American Journal of Psychology 15; Cohen (1988), Statistical Power Analysis for the Behavioral Sciences, 2nd ed., §3.2 (r = 0.1, 0.3, 0.5).
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