Pearson Correlation and Linear Regression Formulas

Posted on May 21, 2026 in Statistics

Pearson Correlation & Linear Regression Cheat Sheet

Pearson Product-Moment Correlation & Linear Regression

Correlation Definition

Correlation measures the strength and direction of a relationship between two variables.

Warning: Correlation does NOT imply causation.

Types of Correlation

Strength of Correlation

Pearson Correlation Coefficient Formula

r=\frac{n\sum xy-(\sum x)(\sum y)}{\sqrt{\left[n\sum x^2-(\sum x)^2\right]\left[n\sum y^2-(\sum y)^2\right]}}

Formula Variables

• r = correlation coefficient
• n = number of observations
• x, y = variables

Interpreting the Correlation Coefficient (r)

Example: Math & English Scores

Given data:

Solution and Interpretation

r=\frac{10(298)-48(50)}{\sqrt{(10(296)-48^2)(10(310)-50^2)}}\approx0.92

Interpretation: There is a very high positive correlation between Math and English scores.

Coefficient of Determination (r²)

Formula

r^2=(0.92)^2=0.8464\approx0.85

Interpretation

• About 85% of the variation in one variable is explained by the other.
• The remaining 15% is caused by other factors.

Scatterplot Patterns and Meanings

Linear Regression Analysis

Linear regression predicts a dependent variable using an independent variable.

General Equation

\hat{y}=a+bx

Where:

• ˆy = predicted value
• a = y-intercept
• b = slope

Slope Formula and Example

b=\frac{n\sum xy-(\sum x)(\sum y)}{n\sum x^2-(\sum x)^2}

Example Calculation

b=\frac{10(298)-48(50)}{10(296)-48^2}=\frac{580}{656}\approx0.88

Interpretation: Every 1-point increase in Math score increases the English score by 0.88.

Y-Intercept Formula and Example

a=\frac{\sum y}{n}-b\frac{\sum x}{n}

Example Calculation

a=\frac{50}{10}-0.88\left(\frac{48}{10}\right)=0.76

Regression Equation Result

\hat{y}=0.76+0.88x

Making Predictions with Regression

Example 1: If x = 95

\hat{y}=0.76+0.88(95)=84.36

Predicted English score = 84.36

Example 2: If x = 80

\hat{y}=0.76+0.88(80)=71.16

Predicted English score = 71.16

Important Limitations and Notes

When Correlation May Fail

• The relationship is non-linear
• Outliers exist in the dataset
• The data range is restricted

Assumptions of Pearson Correlation

Variables must be:

• Continuous
• Normally distributed
• Linearly related

Real-World Applications

Quick Memory Tricks

Formula Summary Reference

Pearson Correlation

r=\frac{n\sum xy-(\sum x)(\sum y)}{\sqrt{\left[n\sum x^2-(\sum x)^2\right]\left[n\sum y^2-(\sum y)^2\right]}}

Coefficient of Determination

r^2

Regression Equation

\hat{y}=a+bx

Slope

b=\frac{n\sum xy-(\sum x)(\sum y)}{n\sum x^2-(\sum x)^2}

Y-Intercept

a=\frac{\sum y}{n}-b\frac{\sum x}{n}

Final Takeaways

Correlation tells us:

• Whether variables are related
• How strong the relationship is
• The direction of the relationship

Linear Regression helps us:

• Model relationships
• Predict future values
• Make data-driven decisions

Strong correlation = r close to ±1
Strong prediction power = high r²