Correlation Coefficient Calculator
Find the strength and direction of a linear relationship between two variables
Correlation Result
Pearson's r:
Data Summary:
Number of pairs:
X mean: , Y mean:
Pearson's r Formula
\[ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2} \sqrt{\sum (y_i - \bar{y})^2}} \]
Where \( \bar{x} \) and \( \bar{y} \) are the means of X and Y respectively.
Computational Formula
\[ r = \frac{n\sum xy - \sum x \sum y}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \]
This form is often easier for calculation.
Scatter Plot
Calculation Steps
How to Use This Tool
Simple Steps to Calculate Correlation
- Enter your X variable values in the first text area (comma or space separated)
- Enter your Y variable values in the second text area (comma or space separated)
- Ensure both datasets have the same number of values
- Click the "Calculate Correlation" button
- View your Pearson's r result and interpretation
- Examine the scatter plot visualization
Understanding the Results
- r = +1: Perfect positive correlation
- 0.7 ≤ r < 1: Strong positive correlation
- 0.3 ≤ r < 0.7: Moderate positive correlation
- 0 < r < 0.3: Weak positive correlation
- r = 0: No correlation
- -0.3 < r < 0: Weak negative correlation
- -0.7 < r ≤ -0.3: Moderate negative correlation
- -1 < r ≤ -0.7: Strong negative correlation
- r = -1: Perfect negative correlation
About Pearson's Correlation Coefficient
What is Pearson's r?
Pearson's correlation coefficient (r) is a measure of the linear correlation between two variables X and Y. It has a value between +1 and -1, where:
- 1 is total positive linear correlation
- 0 is no linear correlation
- -1 is total negative linear correlation
The Pearson correlation measures the strength and direction of the linear relationship between two variables.
When to Use Pearson's r
- When both variables are quantitative (interval or ratio level)
- When the relationship is linear
- When the data is normally distributed
- When there are no significant outliers
If your data doesn't meet these assumptions, you might consider using Spearman's rank correlation instead.
Limitations
- Only measures linear relationships (may miss nonlinear relationships)
- Sensitive to outliers
- Doesn't imply causation
- Requires interval or ratio level data
Frequently Asked Questions
Generally:
- 0.7 to 1.0 (-0.7 to -1.0): Very strong relationship
- 0.5 to 0.7 (-0.5 to -0.7): Strong relationship
- 0.3 to 0.5 (-0.3 to -0.5): Moderate relationship
- 0 to 0.3 (0 to -0.3): Weak or no relationship
However, what's considered "strong" can vary by field. In physics, you might expect correlations above 0.9, while in social sciences, 0.5 might be considered strong.
A negative correlation means that as one variable increases, the other tends to decrease. This is an inverse relationship.
Example: The correlation between hours spent watching TV and exam scores might be negative - as TV time increases, exam scores tend to decrease.
Remember, correlation doesn't imply causation. There might be other factors at play.
No, Pearson's r is mathematically constrained to values between -1 and 1. If you calculate a value outside this range, there must be an error in your calculation.
The values -1 and 1 represent perfect negative and positive linear relationships respectively. In real-world data, you'll rarely see perfect ±1 correlations.
While you can technically calculate correlation with as few as 2 points, more data gives more reliable results:
- n = 5-10: Very unreliable, only for rough estimates
- n = 10-30: Better but still not very reliable
- n = 30-100: Reasonably reliable for most purposes
- n > 100: Good reliability
Also consider the effect size. Strong correlations (near ±1) can be reliably detected with fewer points than weak correlations.
No! Correlation does not imply causation. Just because two variables are correlated doesn't mean one causes the other. There could be:
- A third variable influencing both (confounding factor)
- Pure coincidence
- The relationship might be in the opposite direction
Example: Ice cream sales and drowning incidents are positively correlated, but one doesn't cause the other. The hidden factor is temperature - hot weather increases both.