Understanding Correlation Coefficient for Points on a Straight Line

The question asks about the possible values of the correlation coefficient between two variables, x and y, given a set of bivariate observations \((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\). We are told that all these observations are distinct and that they all fall on a straight line with a non-zero slope. Let's break down what this means for the correlation coefficient.

What is the Correlation Coefficient?

The correlation coefficient, often denoted by \(r\) (for sample) or \(\rho\) (for population), is a measure of the linear relationship between two variables. Its value ranges from -1 to +1.

• A value of \(+1\) indicates a perfect positive linear relationship. As one variable increases, the other increases proportionally.
• A value of \(-1\) indicates a perfect negative linear relationship. As one variable increases, the other decreases proportionally.
• A value of \(0\) indicates no linear relationship. The variables may still be related in a non-linear way, but there is no straight-line association.
• Values between -1 and 1 indicate varying degrees of linear association. For example, \(0.8\) suggests a strong positive linear relationship, while \(-0.3\) suggests a weak negative linear relationship.

Points Falling on a Straight Line

When a set of bivariate observations \((x_i, y_i)\) all fall exactly on a straight line, it means there is a perfect linear relationship between \(x\) and \(y\). A straight line can be represented by the equation \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

Since all points lie perfectly on this line, their relationship is perfectly linear. This means the correlation coefficient will be either \(+1\) or \(-1\), depending on the slope of the line.

Impact of Non-Zero Slope

The question specifies that the straight line has a non-zero slope. The slope \(m\) determines the direction of the linear relationship:

• If the slope \(m\) is positive (\(m > 0\)), then as \(x\) increases, \(y\) also increases. This corresponds to a positive linear relationship. When the points lie perfectly on such a line, the correlation coefficient is \(+1\).
• If the slope \(m\) is negative (\(m < 0\)), then as \(x\) increases, \(y\) decreases. This corresponds to a negative linear relationship. When the points lie perfectly on such a line, the correlation coefficient is \(-1\).

If the slope were zero (\(m = 0\)), the line would be horizontal (\(y = c\)). In this case, \(y\) remains constant regardless of the value of \(x\). There would be no linear relationship between \(x\) and \(y\), and the correlation coefficient would be \(0\). However, the question explicitly states the slope is non-zero, ruling out this possibility.

Distinct Observations

The condition that all values are distinct ensures that we have a meaningful set of points defining a line and calculating correlation, avoiding degenerate cases like all points being the same.

Conclusion on Possible Values

Given that the bivariate observations lie exactly on a straight line, the linear relationship is perfect, meaning the correlation coefficient is either +1 or -1.

Given that the slope is non-zero, the line is not horizontal, which rules out a correlation of 0.

Therefore, the only possible values for the correlation coefficient between x and y are -1 and 1.

Summary of Analysis

• The bivariate observations fall on a straight line \(\implies\) Perfect linear relationship.
• Perfect linear relationship \(\implies\) Correlation coefficient is either +1 or -1.
• The straight line has a non-zero slope \(\implies\) The line is not horizontal.
• Not a horizontal line \(\implies\) Correlation coefficient is not 0.
• Combining these points, the possible values of the correlation coefficient must be -1 or 1.

Thus, the possible values of the correlation coefficient between x and y are -1 and 1 only.

Revision Table: Key Concepts

Additional Information: Why Correlation is Exactly 1 or -1

The mathematical formula for the sample correlation coefficient \(r\) is:

$$r = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^n (x_i - \bar{x})^2 \sum_{i=1}^n (y_i - \bar{y})^2}}$$

If all points \((x_i, y_i)\) lie on a straight line \(y_i = mx_i + c\), then \(y_i - \bar{y} = m(x_i - \bar{x})\) (since \(\bar{y} = m\bar{x} + c\)). Substituting this into the formula for \(r\):

$$r = \frac{\sum_{i=1}^n (x_i - \bar{x})[m(x_i - \bar{x})]}{\sqrt{\sum_{i=1}^n (x_i - \bar{x})^2 \sum_{i=1}^n [m(x_i - \bar{x})]^2}}$$

$$r = \frac{m \sum_{i=1}^n (x_i - \bar{x})^2}{\sqrt{\sum_{i=1}^n (x_i - \bar{x})^2 \cdot m^2 \sum_{i=1}^n (x_i - \bar{x})^2}}$$

$$r = \frac{m \sum_{i=1}^n (x_i - \bar{x})^2}{\sqrt{m^2 (\sum_{i=1}^n (x_i - \bar{x})^2)^2}}$$

$$r = \frac{m \sum_{i=1}^n (x_i - \bar{x})^2}{|m| \sum_{i=1}^n (x_i - \bar{x})^2}$$

Since the observations are distinct and lie on a line with non-zero slope, not all \(x_i\) values can be the same, so \(\sum_{i=1}^n (x_i - \bar{x})^2 > 0\). Also, \(m \neq 0\). Thus, we can simplify:

$$r = \frac{m}{|m|}$$

• If \(m > 0\), then \(|m| = m\), so \(r = m/m = 1\).
• If \(m < 0\), then \(|m| = -m\), so \(r = m/(-m) = -1\).

This confirms that when points lie exactly on a straight line with a non-zero slope, the correlation coefficient is exactly +1 or -1.