Coefficient of Correlation: Understanding and Proving Its Range Between -1 and 1

The coefficient of correlation is a statistical measure that quantifies the degree of linear relationship between two variables. It is denoted by r and falls between -1 and 1. This article aims to explore the concept of the correlation coefficient and provide a theoretical proof of its range.

Introduction to Correlation Coefficient

A correlation coefficient is a ratio, therefore it is independent of any units. It measures the degree of relationship between two variables, where one variable (Y) is dependent on the values of another variable (X). The variables are labeled as positively correlated or negatively correlated based on the direction of the relationship.

Positive and Negative Correlation

- Positive Correlation: If the values of Y increase with the increase in the values of X, the variables are said to be positively correlated.

- Negative Correlation: If the values of Y decrease as X increases, the variables are said to be negatively correlated.

Theoretical Proof of the Range -1 to 1

To understand how the coefficient of correlation can prove its range between -1 and 1, we will explore the mathematical derivation:

Definition and Notation

Karl Pearson's coefficient of correlation r between two variables x and y is defined as:

r Cov(x, y) / (√Variance(x) × √Variance(y))

or

r E(x - x', y - y') / (√σx × √σy)

where

Cov(x, y) is the covariance of x and y E(x - x', y - y') is the expected value of the product of the standardized variables X and Y σx and σy are the standard deviations of x and y, respectively.

Standardization and Properties

Let X (x - x') / σx and Y (y - y') / σy. By definition, any standard deviation of a standardized random variable is 1. Therefore, E[X2] 1 and E[Y2] 1.

Now consider the expression E[(X Y)2], we can expand it as follows:

E[(X Y)2] E[X2 2XY Y2]

Simplifying further:

E[X2] 2E[XY] E[Y2] 1 2E[XY] 1 2 2E[XY] 2(1 E[XY]

Since E[(X Y)2] is always non-negative:

2(1 E[XY]) ≥ 0

1 E[XY] ≥ 0

0 ≤ E[XY] ≤ 1

Hence, we have:

-1 ≤ r ≤ 1

This proves that the coefficient of correlation lies between -1 and 1, inclusive.

Intuitive Understanding

Intuitively, the largest and smallest possible values for the correlation coefficient can be demonstrated as follows:

Largest Value of 1

The largest value of the correlation coefficient (1) is achieved when there is a perfect positive linear relationship between two variables. For example, if the value of Y is exactly the same as the value of X, the correlation coefficient will be 1.

Smallest Value of -1

The smallest value of the correlation coefficient (-1) is achieved when there is a perfect negative linear relationship between two variables. For example, if the value of Y is the exact opposite of the value of X, the correlation coefficient will be -1.

Examples and Scope

- Exact Correlation (r 1): If your cousin ages one year for every year you age, the correlation is exact and equals 1.

- Perfect Negative Correlation (r -1): If your bank account balance decreases by the same amount each time you make a withdrawal, the correlation coefficient will be -1.

- Slightly Imperfect Correlation (r ≈ -0.93): In real-world scenarios, correlations may not be exact. For example, if your bank account balance decreases by nearly the same amount you withdraw, the correlation coefficient might be around -0.93.

Conclusion

The range of the coefficient of correlation between -1 and 1 is not arbitrary. It is mathematically proven by the properties of the standardized variables and the expectation of their product. Understanding this range is crucial to interpreting the strength and direction of the relationship between two variables.

By grasping the theoretical underpinnings of the correlation coefficient, you can better apply it to real-world data analysis and make informed decisions based on the strength and direction of linear relationships between variables.