# Correlation coefficient calculator (Pearson and Spearman)

> 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.

ইন্টারঅ্যাকটিভ সংস্করণ: https://www.calcopenly.com/bn/statistics/correlation-coefficient-calculator
বিষয়: পরিসংখ্যান ও সম্ভাবনার ক্যালকুলেটর

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.

## ইনপুট

- **x values**: Hours studied (default example).
- **y values**: Exam score for the same students, in the same order.
- **Coefficient** (বিকল্প: Pearson r, Spearman ρ)

## ফলাফল

- Correlation coefficient — প্রধান ফলাফল
- Strength
- Coefficient of determination r²
- t statistic
- Degrees of freedom
- p-value (two-sided)
- Pairs n

## সূত্র

$$
r = \frac{\sum (x_i-\bar x)(y_i-\bar y)}{\sqrt{\sum (x_i-\bar x)^2 \sum (y_i-\bar y)^2}},\qquad t = r\sqrt{\frac{n-2}{1-r^2}}
$$

## সমাধান করা উদাহরণ

### Hours studied vs score (default)

- x values: 1, 2, 2, 3, 4, 5, 5, 6, 7, 8
- y values: 52, 55, 61, 58, 66, 70, 68, 75, 79, 84
- Coefficient: Pearson r
- **Correlation coefficient: 0.981638**
- **Coefficient of determination r²: 0.963613**
- **t statistic: 14.555426**
- **Degrees of freedom: 8**
- **p-value (two-sided): 4.865 × 10⁻⁷**
- **Strength: Strong positive**
- যাচাইয়ের উৎস: 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)

### Negative association

- x values: 10, 20, 30, 40, 50, 60
- y values: 8.1, 7.4, 7.9, 6.2, 5.8, 6.0
- Coefficient: Pearson r
- **Correlation coefficient: -0.891042**
- **t statistic: -3.925982**
- **p-value (two-sided): 0.017161**
- **Strength: Strong negative**
- যাচাইয়ের উৎস: Python fractions with decimal square roots; p = 2·(1 − F_t(|t|; 4)) from the A&S 26.7.4 closed form

### Spearman with tied ranks

- x values: 1, 2, 2, 3, 4, 5, 5, 6
- y values: 3, 1, 4, 4, 5, 9, 2, 6
- Coefficient: Spearman ρ
- **Correlation coefficient: 0.575768**
- **t statistic: 1.724946**
- **p-value (two-sided): 0.135297**
- **Degrees of freedom: 6**
- **Strength: Strong positive**
- যাচাইয়ের উৎস: 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

### Perfect straight line

- x values: 1, 2, 3, 4
- y values: 3, 5, 7, 9
- Coefficient: Pearson r
- **Correlation coefficient: 1**
- **Coefficient of determination r²: 1**
- **p-value (two-sided): 0**
- **Strength: Strong positive**
- যাচাইয়ের উৎস: y = 2x + 1 exactly, so r = 1 by definition and no sample could be more extreme

## প্রশ্ন

### What does a correlation coefficient of 0.7 mean?

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.

### What is the difference between Pearson and Spearman correlation?

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.

### Does correlation imply causation?

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.

### How do you test whether a correlation is significant?

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.

### “Correlation coefficient calculator (Pearson and Spearman)” কতটা নির্ভুল?

নির্ভুলতা আপনার ইনপুট ও পদ্ধতির অনুমানের ওপর নির্ভর করে। দশমিক গণনায় 50টি সার্থক অঙ্ক ব্যবহৃত হয়, কিন্তু আনুমানিক হিসাব, সংখ্যাগত পদ্ধতি ও উৎসের তথ্য কম নির্ভুল হতে পারে; প্রদর্শিত মান রাউন্ড করলে এই সীমাবদ্ধতাগুলি দূর হয় না। স্বতন্ত্র উৎসের সমাধানের সঙ্গে যাচাই করা উদাহরণ: ৪। যেমন, “Hours studied vs score (default)” উদাহরণটি 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)-এর সঙ্গে যাচাই করা হয়।

### এই পদ্ধতির উৎস কী?

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).

## উৎস

- [NIST/SEMATECH e-Handbook of Statistical Methods, §7.1.3 / Dataplot CORRELATION (Pearson and rank correlation)](https://www.itl.nist.gov/div898/handbook/prc/section1/prc13.htm)
- 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)