Question

How does the coefficient of correlation measure the strength of the linear relationship between two variables \(y\) and \(x ?\)

Short answer

Question: Explain the coefficient of correlation and its interpretation in measuring the strength of the linear relationship between two variables, y and x. Answer: The coefficient of correlation, denoted by the symbol 'r', measures the strength and direction of the linear relationship between two variables, y and x. Its value ranges between -1 and 1. To interpret the coefficient of correlation, consider the following: 1. \(r = 1\) represents a perfect positive linear relationship, meaning as x increases, y also increases. 2. \(r = -1\) represents a perfect negative linear relationship, meaning as x increases, y decreases. 3. \(r = 0\) indicates no linear relationship between the variables x and y. 4. \(0<r<1\): A higher value of r indicates a stronger positive linear relationship between the variables. 5. \(-1<r<0\): A more negative value of r indicates a stronger negative linear relationship between the variables. To analyze the strength of the linear relationship between y and x, calculate the correlation coefficient using the associated formula and interpret the resulting value based on the guidelines above.

Step-by-step solution

Define the Coefficient of Correlation

The coefficient of correlation, also known as the Pearson correlation coefficient or simply correlation coefficient, is a measure of the strength and direction of the linear relationship between two variables. The correlation coefficient is denoted by the symbol 'r' and its value ranges between -1 and 1.

Calculate the Coefficient of Correlation

To calculate the correlation coefficient (r) between two variables x and y, use the following formula: \[r = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n} (x_i - \bar{x})^2 \sum_{i=1}^{n} (y_i - \bar{y})^2}}\] Where \(x_i\) and \(y_i\) are individual data points, \(\bar{x}\) and \(\bar{y}\) are the means of the x and y variables, and n is the number of data points for both x and y.

Interpret the Coefficient of Correlation

The value of the correlation coefficient (r) allows you to interpret the strength and direction of the linear relationship between the two variables: 1. \(r = 1\) implies a perfect positive linear relationship, meaning as x increases, y increases. 2. \(r = -1\) implies a perfect negative linear relationship, meaning as x increases, y decreases. 3. \(r = 0\) implies no linear relationship between the variables x and y. 4. \(0<r<1\): With increase in the values of r, the strength of positive linear relationship increases. 5. \(-1<r<0\): As(r) becomes more negative, the strength of negative linear relationship increases. To measure the strength of the linear relationship between two variables y and x, calculate the correlation coefficient using the given formula and interpret the resulting value as described above.