How do I interpret correlation coefficient?

r = +1 is perfect positive correlation, r = -1 is perfect negative, r = 0 means no linear relationship. Generally: |r| > 0.7 is strong, 0.3-0.7 is moderate, < 0.3 is weak.

What is R-squared in correlation?

R-squared (r²) is the coefficient of determination. It represents the proportion of variance in one variable explained by the other. If r = 0.8, then r² = 0.64, meaning 64% of variance is explained.

Does correlation prove causation?

No! Correlation only shows two variables move together. Causation requires controlled experiments. The relationship could be reverse causation, due to a third variable, or coincidental.

Correlation Coefficient Calculator - Pearson's r Calculator

Correlation Calculator

Calculate the Pearson correlation coefficient (r) to measure the strength and direction of the linear relationship between two quantitative variables.

Variable X (comma or space separated)

Variable Y (comma or space separated)

How to Use This Calculator

Enter X values: First variable’s data (comma or space separated)
Enter Y values: Second variable’s data (must have same number of values as X)
Click Calculate: Get correlation coefficient, r², and significance test
Interpret results: Understand the strength and direction of the relationship

Input Example

X values: 2, 3, 4, 5, 6
Y values: 65, 70, 75, 80, 85

Understanding the Results

Pearson Correlation Coefficient (r)

The correlation coefficient ranges from -1 to +1:

Value	Interpretation
+1.0	Perfect positive correlation
+0.7 to +1.0	Strong positive correlation
+0.3 to +0.7	Moderate positive correlation
0 to +0.3	Weak positive correlation
0	No linear correlation
-0.3 to 0	Weak negative correlation
-0.7 to -0.3	Moderate negative correlation
-1.0 to -0.7	Strong negative correlation
-1.0	Perfect negative correlation

Coefficient of Determination (r²)

r² tells you what proportion of the variation in one variable can be explained by the other:

r = 0.8 → r² = 0.64 → 64% of variation explained
r = 0.5 → r² = 0.25 → 25% of variation explained
r = -0.9 → r² = 0.81 → 81% of variation explained

Statistical Significance

The calculator performs a hypothesis test:

H₀: No correlation exists in the population (ρ = 0)
Hₐ: Correlation exists in the population (ρ ≠ 0)

Significant result (p < 0.05): The correlation is unlikely due to chance

Not significant (p ≥ 0.05): The correlation could be due to random variation

Interpreting Correlation Direction

Positive Correlation (r > 0)

As X increases, Y tends to increase
Points slope upward on a scatter plot
Example: Study hours and test scores

Negative Correlation (r < 0)

As X increases, Y tends to decrease
Points slope downward on a scatter plot
Example: Exercise and resting heart rate

No Correlation (r ≈ 0)

No clear linear relationship
Points are randomly scattered
Example: Shoe size and intelligence

Important: Correlation ≠ Causation

Just because two variables are correlated doesn’t mean one causes the other!

Common Scenarios:

Reverse causation: Y might cause X instead
Third variable: Z causes both X and Y
Coincidence: The correlation is spurious

Classic Example

Ice cream sales and drowning deaths are positively correlated.

❌ Does ice cream cause drowning? No!

✅ Both are caused by hot weather (a third variable)

When to Use Correlation Analysis

✅ Appropriate Uses:

Exploring relationships between variables
Identifying potential predictors for regression
Validating measurement instruments
Analyzing trends in financial data

❌ Inappropriate Uses:

Proving causation
When relationship is clearly non-linear
With categorical data (use chi-square instead)
When data has severe outliers

Assumptions

Pearson correlation assumes:

Linearity: The relationship is linear (not curved)
Continuous variables: Both X and Y are quantitative
Normal distribution: Data is approximately normally distributed
No outliers: Extreme values can distort correlation
Homoscedasticity: Constant variance across the range

Tip: Always visualize your data with a scatter plot first!

Example Interpretations

Strong Positive Correlation

r = 0.85, r² = 0.72, p < 0.001

“There is a strong positive correlation between study hours and test scores (r = 0.85, p < 0.001). Study hours explain 72% of the variation in test scores. Students who study more tend to score higher.”

Weak Negative Correlation

r = -0.25, r² = 0.06, p = 0.12

“There is a weak negative correlation between age and reaction time (r = -0.25, p = 0.12). However, this correlation is not statistically significant, suggesting the relationship may be due to chance. Age explains only 6% of the variation in reaction time.”

Other Correlation Types:

Spearman’s rho: For ordinal data or non-normal distributions
Kendall’s tau: For small samples with tied ranks
Point-biserial: When one variable is binary

Next Steps:

After finding correlation, you might want to:

Create a scatter plot to visualize the relationship
Perform linear regression to model the relationship
Test for causation with experimental design

Common Applications

Science

Temperature and plant growth
Dose and response in pharmacology
Age and cognitive performance

Finance

Stock price movements
Economic indicators
Risk and return relationships

Income and education level
Screen time and sleep quality
Personality traits and behavior

Health

BMI and blood pressure
Exercise and heart rate
Age and bone density

Correlation Analysis Lesson - Learn the theory
Scatter Plot Generator - Visualize your data
Linear Regression Calculator - Build predictive models