Pearson Correlation
Calculate linear dependence ($r$)
[Image of pearson correlation graph]
$$ r = \frac{\sum (x – \bar{x})(y – \bar{y})}{\sqrt{SS_x \cdot SS_y}} $$
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Y Values (Comma sep)
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Correlation Coefficient ($r$)
Detailed Solution
👨🏫
By Prof. David Anderson
Statistics Professor | 20+ Years Exp.
"In 20 years of teaching, the biggest mistake I see students make isn't the math—it's the logic. They see a high correlation and assume causality. They see $r=0.9$ between 'Ice Cream Sales' and 'Shark Attacks' and assume ice cream causes shark attacks (spoiler: it's the summer weather). I built this Pearson Correlation Calculator to not only crunch the numbers for you but to act as your 'Relationship Detective,' ensuring you interpret your data with scientific rigor."
Pearson Correlation Calculator ($r$): Formula, P-Value & Interpretation
Calculate Linear Relationships, Significance, and R-Squared
The Pearson Correlation Calculator is the gold standard statistical tool for measuring the strength and direction of the linear relationship between two continuous variables. The result, known as Pearson's $r$, ranges from -1 to +1.
Whether you are analyzing the link between study hours and exam scores, or marketing spend and revenue, this tool provides the three critical metrics you need: the Correlation Coefficient ($r$), the P-Value (Significance), and the Coefficient of Determination ($R^2$).
[Image of correlation coefficient plots]1. The Pearson Correlation Formula ($r$)
⚠️ Professor's Insight: It's Standardized Covariance
Think of Pearson's $r$ as "Covariance divided by the product of Standard Deviations." Covariance tells you the direction; dividing by standard deviations standardizes it so the result is always between -1 and 1.
The mathematical formula for the Pearson Product-Moment Correlation Coefficient is:
Pearson's r Equation
$$ r = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum(x_i - \bar{x})^2 \sum(y_i - \bar{y})^2}} $$
Where $\bar{x}$ and $\bar{y}$ are the means of the X and Y variables.
2. Interpretation Guide: How Strong is Your $r$?
You calculated $r = 0.65$. Is that strong? Weak? Here is the standard rubric used in social sciences and business analytics.
-1.0 to -0.7
Strong Negative
Strong Negative
-0.7 to -0.3
Moderate Negative
Moderate Negative
-0.3 to +0.3
Weak / None
Weak / None
+0.3 to +0.7
Moderate Positive
Moderate Positive
+0.7 to +1.0
Strong Positive
Strong Positive
| r Value | Strength | Real-World Example |
|---|---|---|
| +1.0 | Perfect Positive | Temperature in Celsius vs Fahrenheit. |
| +0.8 | Strong Positive | Height vs. Shoe Size. |
| 0.0 | No Correlation | IQ Scores vs. Zip Code. |
| -0.6 | Moderate Negative | Hours of TV watched vs. GPA. |
| -1.0 | Perfect Negative | Speed vs. Time taken to travel distance. |
3. How to Calculate Pearson's r (Step-by-Step)
Let's verify the "Study Hours vs. Exam Score" relationship manually.
X (Hours): $\{1, 2, 3, 4, 5\}$
Y (Score): $\{50, 60, 70, 80, 90\}$
Step 1
Find Means ($\bar{x}, \bar{y}$)
Calculate average for X and Y.
$\bar{x} = 3$
$\bar{y} = 70$
$\bar{x} = 3$
$\bar{y} = 70$
Step 2
Calculate Differences
Subtract mean from every data point.
Ex: $1-3=-2$, $50-70=-20$.
Ex: $1-3=-2$, $50-70=-20$.
Step 3
Multiply & Sum
Multiply $(x-\bar{x})(y-\bar{y})$ and sum them. This is the numerator (Covariance).
Result: 200.
Result: 200.
4. The Big Debate: Pearson vs. Spearman
This is the #1 question in my advanced classes. "Professor, which correlation should I use?"
📈 Pearson ($r$)
- Type: Parametric Test.
- Relationship: Measures Linear (Straight Line) relationships only.
- Requirements: Data must be normally distributed. Sensitive to outliers.
- Best For: Physical measurements (Height vs Weight).
📊 Spearman ($\rho$)
- Type: Non-Parametric (Rank) Test.
- Relationship: Measures Monotonic relationships (consistently increasing/decreasing, even if curved).
- Requirements: Can handle outliers and non-normal data.
- Best For: Survey data (Likert scales), Rankings.
5. Beyond r: P-Value & R-Squared
Getting an $r$ value is just the start. You need context.
The P-Value (Significance)
The P-Value answers: "Could this correlation have happened by random luck?"
• If $p < 0.05$: The correlation is statistically significant.
• If $p > 0.05$: The correlation might be random noise. (A high $r$ with a tiny sample size often has a high P-value).
The Coefficient of Determination ($R^2$)
If you square $r$, you get $R^2$. This tells you the percentage of variance explained.
Example: If Correlation $r = 0.9$ between Advertising and Sales:
$R^2 = 0.81$. This means 81% of the fluctuation in Sales can be explained by Advertising. The other 19% is other factors.
6. Professor's FAQ Corner
Does correlation imply causation?
NO. This is the golden rule. Just because "Ice Cream Sales" and "Drowning Deaths" are correlated ($r=0.8$) does not mean ice cream causes drowning. There is a third variable (Summer/Heat) causing both. This is called a Spurious Correlation.
Why is my correlation zero?
A result of $r=0$ only means there is no Linear relationship. The variables could still be related! Imagine a U-shape (Parabola). $Y = X^2$ has a correlation of 0, but is perfectly related. Always plot your data first.
How to calculate Pearson Correlation in Excel?
Use the function
=CORREL(array1, array2) or =PEARSON(array1, array2). Both return the same result. To get $R^2$, use =RSQ(array1, array2).
References
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences. (Standard for interpreting r effect size).
- Pearson, K. (1895). "Note on Regression and Inheritance in the Case of Two Parents."
- NIST/SEMATECH e-Handbook of Statistical Methods. "Correlation Coefficient."
Find Your Relationship
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Calculate Pearson Correlation