Which of the following statements is true about the correlation coefficient?
Multiple Choice
A. A negative correlation coefficient indicates a direct relationship between two quantitative variables.
B. A positive correlation coefficient indicates an indirect relationship between two quantitative variables.
C. A correlation coefficient of zero means that two quantitative variables are not linearly related to each other.
D. The correlation coefficient ranges in value from -3 to +3.
The correct answer and explanation is :
Correct Answer:
C. A correlation coefficient of zero means that two quantitative variables are not linearly related to each other.
Explanation:
The correlation coefficient, commonly denoted as r, measures the strength and direction of the linear relationship between two quantitative variables. It takes values between -1 and +1.
- A correlation coefficient of +1 indicates a perfect positive linear relationship, meaning that as one variable increases, the other increases proportionally.
- A correlation coefficient of -1 indicates a perfect negative linear relationship, meaning that as one variable increases, the other decreases proportionally.
- A correlation coefficient of 0 suggests no linear relationship between the two variables, though there may still be a non-linear relationship.
Let’s analyze the given answer choices:
- Option A (Incorrect): A negative correlation coefficient does not indicate a direct relationship. Instead, it shows an inverse relationship, where one variable increases while the other decreases.
- Option B (Incorrect): A positive correlation coefficient does not indicate an indirect relationship. Instead, it shows a direct relationship, where both variables increase or decrease together.
- Option C (Correct): A correlation coefficient of zero means that there is no linear relationship between the two variables, making this the correct answer.
- Option D (Incorrect): The correlation coefficient ranges from -1 to +1, not from -3 to +3.
Thus, the correct answer is C, as a correlation coefficient of zero means no linear relationship exists between the two variables.
The image above visually represents different types of correlation:
- Positive Correlation (Left Plot, Blue): As one variable increases, the other also increases, indicating a direct relationship.
- Negative Correlation (Middle Plot, Red): As one variable increases, the other decreases, showing an inverse relationship.
- No Correlation (Right Plot, Green): The points are scattered randomly, indicating no linear relationship.
This visualization supports the explanation that a correlation coefficient of zero means no linear relationship, making option C the correct answer.