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Correlation Coefficient Calculator

Free correlation coefficient calculator. Pearson r, Spearman rho, Kendall tau. Scatter plot, regress

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Why This Statistical Analysis Matters

Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

r
STATISTICSDescriptive Statistics

Correlation Coefficient — Pearson r, Spearman ρ, Kendall τ

Compute r, R², p-value, 95% CI. Scatter plot, correlation matrix, r significance curve. Sources: NIST, Pearson 1895, OpenIntro.

Linear Regression →R² Calculator →

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Real-World Scenarios — Click to Load

xy
correlation_results.sh
CALCULATED
$ correlation --pearson --spearman --kendall --ci=0.95
Pearson r
0.9948
Strong correlation
R²
0.9897
p-value
0.00e+0
95% CI for r
[0.921, 1.000]
Spearman ρ
1.0000
Kendall τ
1.0000
n
5 pairs
Regression
ŷ = 1.70x + 0.30
Share:
Correlation Coefficient Result
Pearson r = 0.995
R² = 99.0%
p = 0.00e+0Strong correlationn = 5
numbervibe.com/calculators/statistics/correlation-coefficient-calculator

Scatter Plot with Regression Line

Correlation Matrix (2×2)

1.00
0.995
0.995
1.00

X (left) vs Y (right). Diagonal = 1 (self-correlation).

Critical |r| for Significance (α=0.05, two-tailed)

For your n=5, need |r| > 0.749 for significance. Your |r|=0.995 → Significant.

Calculation Breakdown

INPUT
Sample size n
5
n = 5 pairs
COMPUTATION
Mean x̄
3.0000
Σx/n = 3.0000
Mean ȳ
5.4000
Σy/n = 5.4000
Covariance numerator
17.0000
\text{Sigma} (xᵢ-x̄)(yᵢ-ȳ)
Σ(x−x̄)²
10.0000
ext{Sum} ext{of} ext{squared} ext{deviations} (x)
Σ(y−ȳ)²
29.2000
ext{Sum} ext{of} ext{squared} ext{deviations} (y)
Pearson r
0.9948
r = ext{cov} / √(\text{Sigma} (x-x̄)^{2} imes \text{Sigma} (y-ȳ)^{2})
R²
0.9897
r^{2} = ext{coefficient} ext{of} ext{determination}
SIGNIFICANCE
t-statistic
17.0000
t = r√(n−2)/√(1−r²), df=3
p-value
0.00e+0
ext{Two}- ext{tailed} ext{test} H_{0}: \text{rho} =0
CONFIDENCE INTERVAL
95% CI for r
[0.9207, 0.9997]
ext{Fisher} z- ext{transform}
ALTERNATIVE MEASURES
Spearman ρ
1.0000
ext{Rank}- ext{based} ext{correlation}
Kendall τ
1.0000
ext{Concordant}/ ext{discordant} ext{pairs}

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

  • • Pearson r: Measures linear correlation. r ∈ [−1, 1]. r = ±1 means perfect linear relationship.
  • • Spearman ρ: Pearson applied to ranks. Robust to outliers; measures monotonic relationship.
  • • Kendall τ: Based on concordant/discordant pairs. Good for small samples and ties.
  • • R² = r²: Coefficient of determination — fraction of variance in y explained by x.
  • • Interpretation: |r| < 0.3 weak, 0.3–0.7 moderate, > 0.7 strong.
  • • Correlation ≠ causation. A high r does not imply x causes y.

Pearson vs Spearman: When to Use Each

CriterionPearson rSpearman ρ
MeasuresLinear relationshipMonotonic relationship (rank-based)
Data typeInterval/ratio, continuousOrdinal, or when outliers present
AssumptionsBivariate normality, linearityNo normality assumption
OutliersSensitive to outliersRobust to outliers
Use whenData is roughly linear and normalData is ordinal, skewed, or has outliers
Formular = Σ((x−x̄)(y−ȳ)) / √(Σ(x−x̄)² Σ(y−ȳ)²)ρ = 1 − 6Σd²/(n(n²−1)) on ranks

Did You Know?

📊Karl Pearson developed the Pearson correlation in 1895. It's the most widely used measure of linear association in statistics.Source: Pearson, 1895
📈R² tells you how much of the variation in y is 'explained' by x. R² = 0.81 means 81% of variance is explained by the linear model.Source: Regression Analysis
🔬The p-value tests H₀: ρ = 0. A small p-value suggests the correlation is statistically significant (unlikely due to chance).Source: Hypothesis Testing
📐Spearman and Kendall are non-parametric — they don't assume normality. Use them when data is ordinal or has outliers.Source: Nonparametric Statistics
🌊Fisher's z-transform stabilizes the variance of r, making it suitable for constructing confidence intervals and meta-analysis.Source: NIST Handbook
⚖️For n = 30, you need |r| > 0.36 to be significant at α = 0.05. For n = 100, you need |r| > 0.20.Source: Critical Values

How It Works

Pearson r: r = Σ((x−x̄)(y−ȳ)) / √(Σ(x−x̄)² × Σ(y−ȳ)²). Covariance divided by product of standard deviations.

Spearman ρ: Replace x, y with their ranks, then compute Pearson on ranks. ρ = 1 − 6Σd²/(n(n²−1)) when no ties.

Kendall τ: τ = (C−D)/(n(n−1)/2). C = concordant pairs (same order), D = discordant (opposite order).

95% CI: Fisher z-transform: z = 0.5·ln((1+r)/(1−r)). CI: z ± 1.96/√(n−3). Back-transform to get r interval.

Expert Tips

Correlation ≠ Causation

A high correlation does not mean x causes y. Both could be caused by a third variable, or the relationship could be coincidental.

When to Use Spearman/Kendall

Use when data is ordinal, has outliers, or is not bivariate normal. Spearman is more common; Kendall is better for small n and many ties.

Check for Linearity

Always plot your data. Pearson r only captures linear relationships. A U-shaped curve can have r ≈ 0 despite a strong nonlinear relationship.

Sample Size Matters

With large n, even tiny correlations become significant. Report both p-value and effect size (r or R²) for a complete picture.

Formulas Reference

Pearson: r = Σ((x−x̄)(y−ȳ)) / √(Σ(x−x̄)² Σ(y−ȳ)²)

R² = r²

t-test: t = r√(n−2)/√(1−r²), df = n−2

Fisher z: z = 0.5·ln((1+r)/(1−r))

95% CI: z ± 1.96/√(n−3), back-transform

Frequently Asked Questions

What is the difference between Pearson, Spearman, and Kendall?

Pearson measures linear correlation. Spearman and Kendall measure monotonic association (rank-based). Use Pearson for linear, normal data; Spearman/Kendall for ordinal data or when outliers are present.

What does a negative correlation mean?

r < 0 means as x increases, y tends to decrease. The relationship is inverse. Example: study time vs errors (more study, fewer errors).

How do I interpret the p-value?

p < 0.05 typically means the correlation is statistically significant — unlikely to have occurred by chance if the true correlation were zero.

What is the confidence interval for r?

The 95% CI gives a range of plausible values for the true population correlation. If it includes 0, the correlation may not be significant.

Why use Fisher z-transform for the CI?

The sampling distribution of r is skewed when r ≠ 0. Fisher's z stabilizes the variance, making the CI symmetric and more accurate.

Can I use Pearson for ordinal data?

Technically yes, but Spearman or Kendall are preferred for ordinal data because they use ranks and don't assume equal intervals.

Step-by-Step: Pearson r

Step 1: Compute means x̄ and ȳ.

Step 2: For each pair, compute (xᵢ − x̄)(yᵢ − ȳ). Sum to get covariance numerator.

Step 3: Compute Σ(xᵢ − x̄)² and Σ(yᵢ − ȳ)². Multiply and take square root for denominator.

Step 4: r = numerator / denominator. Always between −1 and 1.

Residuals and Model Fit

Residuals = observed y − predicted ŷ. A good linear fit has residuals scattered randomly around zero with no pattern. If residuals show a curve (e.g., U-shape), the relationship may be nonlinear — consider transforming variables or using a different model.

Interpretation Guide

|r|Strength
0 - 0.3Weak
0.3 - 0.7Moderate
0.7 - 1.0Strong

Disclaimer: This calculator provides correlation analysis for educational purposes. Correlation does not imply causation. Verify results for research or professional use. For small samples (n < 30), consider the t-distribution for more accurate p-values.

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