Correlation coefficient calculator (Pearson and Spearman)

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.

정확도는 입력값과 계산 방법의 가정에 따라 달라집니다. 십진 연산은 유효숫자 50자리를 사용하지만, 추정값·수치해석 방법·원본 데이터의 정밀도는 더 낮을 수 있습니다. 표시값을 반올림해도 이러한 한계는 사라지지 않습니다. 독립적인 출처의 풀이와 대조한 계산 예시: 4. 예를 들어 “Hours studied vs score (default)”은 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)와 대조해 확인합니다.

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).