Correlation Calculator
Calculate Pearson correlation coefficient, R-squared, and statistical significance between two variables. Analyze the strength and direction of linear relationships.
n = 10 values
n = 10 values
Pearson Correlation (r)
0.9997
R-Squared (R²)
0.9994
Sample Size (n)
10
T-Statistic
116.0577
P-Value
< 0.0001
Covariance
18.3333
Standard Deviations
X: 3.03, Y: 6.06
Interpretation
There is a very strong positive correlation between X and Y (r = 1.000). This correlation is statistically significant (p = <0.0001).
R² = 0.999 means 99.9% of the variance in Y can be explained by its linear relationship with X.
Correlation Strength Guide
|r| ≥ 0.9
Very Strong
0.7 - 0.9
Strong
0.5 - 0.7
Moderate
0.3 - 0.5
Weak
< 0.3
Very Weak
Important Notes
- • Correlation measures linear relationships only
- • Correlation does not imply causation
- • Outliers can significantly affect correlation
- • Check for non-linear patterns in your data
Related Calculators
About This Calculator
Correlation measures the strength and direction of the linear relationship between two variables. The Pearson correlation coefficient (r) ranges from -1 to +1, where -1 indicates a perfect negative relationship, +1 indicates a perfect positive relationship, and 0 indicates no linear relationship.
What is Correlation? Correlation quantifies how two variables move together. When one variable increases, does the other tend to increase (positive correlation), decrease (negative correlation), or show no consistent pattern (no correlation)? This calculator computes the Pearson correlation coefficient and tests its statistical significance.
Why Correlation Matters:
- Identifies relationships between variables in data analysis
- Essential for regression analysis and predictive modeling
- Used in finance for portfolio diversification
- Critical for scientific research and hypothesis testing
- Foundation for many machine learning algorithms
Key Outputs:
- Pearson r: Correlation coefficient (-1 to +1)
- R-squared: Proportion of variance explained
- P-value: Statistical significance of the correlation
- Covariance: Measure of joint variability
This calculator analyzes paired data to find correlations. For predictive models, see our Linear Regression Calculator. For comparing groups, see our T-Test Calculator.
How to Use the Correlation Calculator
- 1Enter your X values separated by commas or spaces.
- 2Enter corresponding Y values (same number of values).
- 3Each X value pairs with the Y value in the same position.
- 4Ensure you have at least 3 data pairs for meaningful results.
- 5Review the Pearson correlation coefficient (r).
- 6Check R-squared for variance explained.
- 7Examine the p-value for statistical significance.
- 8Read the interpretation for practical meaning.
- 9Consider the correlation strength guidelines.
- 10Remember: correlation does not imply causation.
Understanding Pearson Correlation
The Pearson correlation coefficient measures linear relationships.
The Formula
r = Σ(xᵢ - x̄)(yᵢ - ȳ) / √[Σ(xᵢ - x̄)² × Σ(yᵢ - ȳ)²]
Or equivalently: r = Cov(X,Y) / (SD_X × SD_Y)
Interpretation
| r Value | Interpretation |
|---|---|
| +1.0 | Perfect positive |
| +0.7 to +0.9 | Strong positive |
| +0.4 to +0.6 | Moderate positive |
| +0.1 to +0.3 | Weak positive |
| 0 | No correlation |
| -0.1 to -0.3 | Weak negative |
| -0.4 to -0.6 | Moderate negative |
| -0.7 to -0.9 | Strong negative |
| -1.0 | Perfect negative |
Key Properties
- Range: -1 ≤ r ≤ +1
- Symmetric: r(X,Y) = r(Y,X)
- Unit-free: Doesn't depend on measurement scale
- Linear only: Misses curved relationships
R-Squared (Coefficient of Determination)
R² tells you how much variance is explained by the relationship.
Definition
R² = r²
Simply the square of the correlation coefficient.
Interpretation
R² represents the proportion of variance in Y that can be predicted from X.
| R² Value | Meaning |
|---|---|
| 0.9 | 90% of Y variance explained by X |
| 0.5 | 50% explained, 50% unexplained |
| 0.25 | 25% explained |
| 0.1 | Only 10% explained |
Example
If r = 0.8, then R² = 0.64
This means 64% of the variation in Y can be explained by its linear relationship with X. The remaining 36% is due to other factors or random variation.
Caution
High R² doesn't guarantee:
- Causation
- Good predictions outside data range
- Absence of confounding variables
- Correct model specification
Testing Statistical Significance
Is the correlation real or just random chance?
The Null Hypothesis
H₀: ρ = 0 (population correlation is zero) H₁: ρ ≠ 0 (population correlation is not zero)
T-Test for Correlation
t = r × √(n-2) / √(1-r²)
Degrees of freedom: df = n - 2
P-Value Interpretation
- p < 0.05: Correlation is statistically significant
- p ≥ 0.05: Cannot conclude correlation exists
Important Considerations
Sample size matters:
- With n = 10, r = 0.63 needed for significance
- With n = 100, r = 0.20 is significant
- With n = 1000, even r = 0.06 is significant
Significant ≠ Important: A tiny correlation can be "significant" with large samples but practically meaningless.
Critical Values (α = 0.05, two-tailed)
| n | Critical r |
|---|---|
| 10 | 0.632 |
| 20 | 0.444 |
| 30 | 0.361 |
| 50 | 0.279 |
| 100 | 0.197 |
Correlation vs. Causation
This is the most important concept in correlation analysis.
The Golden Rule
Correlation does NOT imply causation.
Just because X and Y move together doesn't mean X causes Y (or vice versa).
Why Correlation Exists Without Causation
1. Reverse causation
- Correlation: Ice cream sales and drowning deaths
- Not: Ice cream causes drowning
- Reality: Hot weather causes both
2. Confounding variables
- Correlation: Shoe size and reading ability (in children)
- Not: Big feet make you read better
- Reality: Age affects both
3. Coincidence
- Correlation: Nicolas Cage films and pool drownings
- Not: Nicolas Cage movies are deadly
- Reality: Random chance in finite datasets
Establishing Causation Requires
- Temporal precedence (cause before effect)
- Correlation (statistical association)
- No confounding variables
- Mechanism (plausible explanation)
- Experimental evidence (randomized controlled trial)
Limitations and Alternatives
Pearson correlation has specific assumptions and limitations.
Assumptions
- Linearity: Relationship is approximately linear
- Normality: Variables roughly normally distributed
- Homoscedasticity: Constant variance
- No outliers: Extreme values can distort r
When Pearson Fails
Curved relationships:
- X and Y² have a perfect relationship
- But Pearson r might be near 0
Outliers:
- One extreme point can dramatically change r
- Always visualize your data
Alternative Correlation Measures
Spearman's Rank Correlation (ρ):
- Non-parametric
- Works with ordinal data
- Robust to outliers
- Detects monotonic relationships
Kendall's Tau (τ):
- Non-parametric
- Better for small samples
- Handles ties well
Point-Biserial:
- One continuous, one binary variable
When to Use Each
| Situation | Use |
|---|---|
| Linear, normal data | Pearson |
| Ordinal data | Spearman |
| Non-linear monotonic | Spearman |
| Many outliers | Spearman |
| Small sample, many ties | Kendall |
Practical Applications
How correlation is used across different fields.
Finance
Portfolio Diversification:
- Low correlation between assets reduces risk
- r close to 0 or negative is desirable
- Example: Stocks and bonds often have low correlation
Risk Management:
- Correlation breakdown during crises
- "All correlations go to 1 in a crash"
Medicine and Healthcare
Diagnostic Tests:
- Correlation between test results and disease
- Sensitivity and specificity analysis
Epidemiology:
- Risk factor identification
- But remember: correlation ≠ causation
Psychology and Social Sciences
Scale Validation:
- Correlation between related measures
- Test-retest reliability
Research:
- Initial exploration before experiments
- Identifying variables for further study
Business and Marketing
Customer Analysis:
- Purchase patterns
- Satisfaction and loyalty
Operations:
- Quality metrics and outcomes
- Process optimization
Common Pitfalls
- Ecological fallacy: Group correlations don't apply to individuals
- Restriction of range: Limited data range reduces correlation
- Aggregation: Combining groups can create spurious correlations
- Time series: Auto-correlation complicates analysis
Pro Tips
- 💡Always visualize your data with a scatter plot before calculating correlation.
- 💡Correlation ranges from -1 to +1; the sign indicates direction, magnitude indicates strength.
- 💡r = 0 means no LINEAR relationship, not no relationship at all.
- 💡Large samples can make tiny correlations statistically significant.
- 💡Correlation does NOT prove causation - this is the #1 mistake.
- 💡Check for outliers - they can dramatically affect correlation.
- 💡R-squared shows proportion of variance explained (r² = 0.49 means 49%).
- 💡Report both the correlation coefficient and its p-value.
- 💡Use Spearman correlation for non-normal data or ordinal variables.
- 💡Restriction of range (limited variability) weakens correlation.
- 💡Consider practical significance, not just statistical significance.
- 💡Multiple correlations increase chance of false positives.
Frequently Asked Questions
It depends on the field. In physics, r > 0.99 might be expected. In social sciences, r > 0.5 is often considered strong. In psychology, r > 0.3 may be meaningful. Context matters: a "weak" correlation in one field might be "strong" in another. Always consider practical significance alongside statistical significance.
