Question
When all the points fall on the regression line, the value of the correlation coefficient is
Ask by Burns Chang. in the United States
Mar 20,2025
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Answer
When all points lie exactly on the regression line, the correlation coefficient is 1 or -1.
Solution
-
The correlation coefficient, denoted by
, measures the strength and direction of the linear relationship between two variables. -
When all data points fall exactly on the regression line, the relationship between the variables is perfectly linear.
-
In a perfect linear relationship:
- If the line has a positive slope,
. - If the line has a negative slope,
.
- If the line has a positive slope,
-
Thus, when all the points fall on the regression line, the absolute value of the correlation coefficient is
.
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Simplify this solution Beyond the Answer
When all the points fall perfectly on the regression line, the correlation coefficient is exactly +1 or -1, depending on the direction of the relationship. A value of +1 indicates a perfect positive correlation, meaning as one variable increases, the other does too in a linear fashion. Conversely, a value of -1 signifies a perfect negative correlation, where one variable decreases as the other increases. In essence, this scenario represents full predictability between the two variables!
In practice, achieving such a perfect correlation is rare in real-world data; most datasets will show some degree of variability and noise. However, striving for high correlation can provide meaningful insights in fields like finance, social sciences, and natural sciences, guiding decision-making and strategies based on the relationships observed in the data!
