pearson's r correlation formula

3 min read 14-03-2025

Pearson's r, also known as the Pearson product-moment correlation coefficient, is a fundamental statistical measure that quantifies the linear association between two continuous variables. This article will delve into the formula, its interpretation, and its applications. Understanding Pearson's r is crucial for anyone working with data analysis, from researchers to data scientists.

What is Pearson's r?

Pearson's r measures the strength and direction of a linear relationship. The value of r ranges from -1 to +1:

r = +1: Indicates a perfect positive linear correlation. As one variable increases, the other increases proportionally.
r = -1: Indicates a perfect negative linear correlation. As one variable increases, the other decreases proportionally.
r = 0: Indicates no linear correlation between the variables. Note that this doesn't necessarily mean there's no relationship, just no linear one. A non-linear relationship could exist.
Values between -1 and +1: Represent varying degrees of correlation. The closer the absolute value of r is to 1, the stronger the correlation.

The Pearson's r Formula

The formula for calculating Pearson's r might seem daunting at first, but breaking it down makes it manageable. The formula is:

r = Σ[(xi - x̄)(yi - ȳ)] / √[Σ(xi - x̄)² * Σ(yi - ȳ)²]

Where:

r: Pearson's correlation coefficient.
xi: The individual values of the first variable (X).
x̄: The mean (average) of the first variable (X).
yi: The individual values of the second variable (Y).
ȳ: The mean (average) of the second variable (Y).
Σ: Represents the summation (adding up all the values).

Let's break down the formula step-by-step:

(xi - x̄): This calculates the deviation of each X value from the mean of X.
(yi - ȳ): This calculates the deviation of each Y value from the mean of Y.
(xi - x̄)(yi - ȳ): This multiplies the corresponding deviations of X and Y. Positive values indicate that X and Y deviate in the same direction (both above or both below their means). Negative values indicate that they deviate in opposite directions.
Σ[(xi - x̄)(yi - ȳ)]: This sums up all the products of the deviations. This sum is often called the "sum of cross-products."
Σ(xi - x̄)²: This calculates the sum of squared deviations for X. This is the numerator of the variance of X.
Σ(yi - ȳ)²: This calculates the sum of squared deviations for Y. This is the numerator of the variance of Y.
√[Σ(xi - x̄)² * Σ(yi - ȳ)²]: This takes the square root of the product of the sums of squared deviations for X and Y. This represents the product of the standard deviations of X and Y.

Finally, the sum of cross products is divided by the product of the standard deviations to normalize the correlation coefficient to the range -1 to +1.

Calculating Pearson's r: A Step-by-Step Example

Let's illustrate with a simple example. Suppose we have the following data:

X	Y
1	2
2	4
3	6
4	8

Calculate the means: x̄ = 2.5, ȳ = 5
Calculate deviations: (xi - x̄) and (yi - ȳ)
Calculate the products of deviations: (xi - x̄)(yi - ȳ)
Sum the products of deviations: Σ[(xi - x̄)(yi - ȳ)] = 10
Calculate the sum of squared deviations: Σ(xi - x̄)² = 5, Σ(yi - ȳ)² = 20
Calculate the denominator: √[Σ(xi - x̄)² * Σ(yi - ȳ)²] = √(5 * 20) = 10
Calculate Pearson's r: r = 10 / 10 = 1

In this example, r = 1, indicating a perfect positive linear correlation. As X increases, Y increases proportionally.

Interpreting Pearson's r

While the value of r indicates the strength and direction of the relationship, it's crucial to consider:

Statistical Significance: A statistically significant r indicates that the correlation is unlikely due to chance. This is often determined using a p-value.
Causation vs. Correlation: Correlation does not imply causation. Even a strong correlation doesn't prove that one variable causes changes in the other. There could be other factors involved.
Outliers: Extreme values can heavily influence the correlation coefficient.

Applications of Pearson's r

Pearson's r finds applications across numerous fields, including:

Social Sciences: Examining relationships between variables like education level and income.
Healthcare: Studying the correlation between blood pressure and age.
Finance: Analyzing the relationship between stock prices and economic indicators.
Engineering: Investigating the correlation between material properties and performance.

Limitations of Pearson's r

Pearson's r is best suited for linear relationships. It may not accurately represent non-linear relationships. Furthermore, it's sensitive to outliers, which can skew the results.

Conclusion

Pearson's r is a powerful tool for understanding the linear relationship between two continuous variables. By carefully applying the formula and interpreting the results, you can gain valuable insights from your data. Remember to consider statistical significance, potential confounding variables, and the limitations of the method. Mastering Pearson's r is a crucial step in becoming proficient in statistical analysis.