What is the correlation coefficient for the data shown in the table?
Answer (2)
Calculate the mean of x and y values: x ˉ = 2.5 , y ˉ = 2.5 .
Calculate the standard deviation of x and y values: s x = 3 17 , s y = 3 17 .
Calculate the covariance of x and y: co v ( x , y ) = 3 17 .
Calculate the correlation coefficient: r = s x s y co v ( x , y ) = 1 . The final answer is 1 .
Explanation
Understanding the Problem We are given a table of x and y values and asked to find the correlation coefficient. The correlation coefficient, denoted by r , measures the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, where 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no linear correlation.
Formula for Correlation Coefficient The formula for the correlation coefficient r is given by: r = s x s y co v ( x , y ) where co v ( x , y ) is the covariance between x and y , and s x and s y are the standard deviations of x and y , respectively. An alternative formula is: r = ∑ i = 1 n ( x i − x ˉ ) 2 ∑ i = 1 n ( y i − y ˉ ) 2 ∑ i = 1 n ( x i − x ˉ ) ( y i − y ˉ ) where x ˉ and y ˉ are the means of x and y , respectively.
Calculating Correlation Coefficient Given the data points (0, 0), (1, 1), (4, 4), and (5, 5), we can observe that y = x for all data points. This indicates a perfect positive linear relationship between x and y . Therefore, the correlation coefficient should be 1. Let's verify this by calculating the means, standard deviations, and covariance.
The mean of x values is: x ˉ = 4 0 + 1 + 4 + 5 = 4 10 = 2.5
The mean of y values is: y ˉ = 4 0 + 1 + 4 + 5 = 4 10 = 2.5
The standard deviation of x values is: s x = 4 − 1 ( 0 − 2.5 ) 2 + ( 1 − 2.5 ) 2 + ( 4 − 2.5 ) 2 + ( 5 − 2.5 ) 2 = 3 6.25 + 2.25 + 2.25 + 6.25 = 3 17 = 5.666... ≈ 2.38
The standard deviation of y values is: s y = 4 − 1 ( 0 − 2.5 ) 2 + ( 1 − 2.5 ) 2 + ( 4 − 2.5 ) 2 + ( 5 − 2.5 ) 2 = 3 6.25 + 2.25 + 2.25 + 6.25 = 3 17 = 5.666... ≈ 2.38
The covariance of x and y is: co v ( x , y ) = 4 − 1 ( 0 − 2.5 ) ( 0 − 2.5 ) + ( 1 − 2.5 ) ( 1 − 2.5 ) + ( 4 − 2.5 ) ( 4 − 2.5 ) + ( 5 − 2.5 ) ( 5 − 2.5 ) = 3 6.25 + 2.25 + 2.25 + 6.25 = 3 17 = 5.666...
Therefore, the correlation coefficient is: r = s x s y co v ( x , y ) = 3 17 3 17 3 17 = 3 17 3 17 = 1
Final Answer Since the y values are exactly the same as the x values, there is a perfect positive correlation. Thus, the correlation coefficient is 1.
Examples
Understanding correlation coefficients is crucial in finance for analyzing the relationship between different stocks. For instance, if two stocks have a correlation coefficient close to 1, they tend to move in the same direction, indicating that they might be affected by similar market factors. Conversely, a correlation coefficient close to -1 suggests that the stocks move in opposite directions, which could be useful for creating a diversified portfolio to mitigate risk. A correlation coefficient of 0 suggests no linear relationship, implying that the stocks' movements are independent of each other.
The correlation coefficient measures the strength and direction of a linear relationship between two variables, calculated using means, standard deviations, and covariance. It ranges from -1 to 1, where 1 indicates a perfect positive correlation. To find the value, apply the relevant formulas to your specific data set.
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