Statistical Calculators
Pearson Correlation & Linear Regression Analysis with Interactive Visualization
Pearson Correlation Coefficient
The Pearson correlation coefficient (r) measures the strength and direction of the linear relationship between two continuous variables. It ranges from -1 to +1, where values closer to -1 or +1 indicate stronger linear relationships.
Formula
Strong Correlation
|r| ≥ 0.7: Strong linear relationship between variables
Moderate Correlation
0.3 ≤ |r| < 0.7: Moderate linear relationship
Weak Correlation
|r| < 0.3: Weak or no linear relationship
Linear Regression Analysis
Linear regression finds the best-fit line through data points using the method of least squares. It allows you to predict Y values based on X values and quantifies how well the line fits the data using R².
Formulas
R² (Coefficient of Determination)
Explains the proportion of variance in Y that is predictable from X. Higher R² indicates better fit.
Prediction
Use the regression equation to predict Y values for new X values within the data range.
Pearson Correlation Calculator
Sample Data
Example dataset showing positive correlation:
Y: 2.1, 3.9, 6.2, 7.8, 10.1, 12.3, 13.8, 16.2, 18.1, 20.0
Make a Prediction
Results
📊 Detailed Statistics
Scatter Plot with Correlation
Statistical Analysis Guide
Interpreting Results
Correlation Strength
Strong (|r| ≥ 0.7)
Variables have a strong linear relationship. Changes in one variable are closely associated with changes in the other.
Moderate (0.3 ≤ |r| < 0.7)
Variables have a moderate linear relationship. Some association exists but with considerable scatter.
Weak (|r| < 0.3)
Little to no linear relationship. Variables are largely independent of each other.
R² Interpretation
R² = 0.8 (80%)
80% of the variance in Y is explained by X. Excellent fit.
R² = 0.5 (50%)
50% of the variance in Y is explained by X. Moderate fit.
R² = 0.2 (20%)
Only 20% of the variance in Y is explained by X. Poor fit.
Real-World Applications
Business & Economics
- • Sales vs. advertising spend
- • Price vs. demand analysis
- • Employee satisfaction vs. productivity
- • Market research correlations
Science & Research
- • Temperature vs. chemical reaction rates
- • Dose-response relationships
- • Environmental factor correlations
- • Experimental data analysis
Social Sciences
- • Education vs. income levels
- • Age vs. technology adoption
- • Survey response correlations
- • Behavioral pattern analysis
Assumptions and Limitations
Important Assumptions
- • Linearity: The relationship between variables is linear
- • Independence: Data points are independent of each other
- • Normality: Variables are approximately normally distributed
- • Homoscedasticity: Constant variance across all levels
Key Limitations
- • Correlation ≠ Causation: Strong correlation doesn't imply one variable causes the other
- • Outliers: Extreme values can significantly affect results
- • Non-linear relationships: May be missed by linear correlation
- • Extrapolation: Predictions outside the data range may be unreliable