What is the correlation coefficient for the data shown in the table?
0
1
4
5
Answer (2)
Calculate the means of x and y: x ˉ = 2.5 , y ˉ = 2.5 .
Calculate the standard deviations of x and y: 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. The formula for the correlation coefficient 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.
Calculating the Means First, we need to calculate the means of x and y . The x values are 0, 1, 4, and 5. The y values are 0, 1, 4, and 5. x ˉ = 4 0 + 1 + 4 + 5 = 4 10 = 2.5 y ˉ = 4 0 + 1 + 4 + 5 = 4 10 = 2.5
Calculating the Standard Deviations Next, we calculate the standard deviations of x and y .
s x = n − 1 ∑ ( x i − x ˉ ) 2 = 4 − 1 ( 0 − 2.5 ) 2 + ( 1 − 2.5 ) 2 + ( 4 − 2.5 ) 2 + ( 5 − 2.5 ) 2 s x = 3 ( − 2.5 ) 2 + ( − 1.5 ) 2 + ( 1.5 ) 2 + ( 2.5 ) 2 = 3 6.25 + 2.25 + 2.25 + 6.25 = 3 17 s y = n − 1 ∑ ( y i − y ˉ ) 2 = 4 − 1 ( 0 − 2.5 ) 2 + ( 1 − 2.5 ) 2 + ( 4 − 2.5 ) 2 + ( 5 − 2.5 ) 2 s y = 3 ( − 2.5 ) 2 + ( − 1.5 ) 2 + ( 1.5 ) 2 + ( 2.5 ) 2 = 3 6.25 + 2.25 + 2.25 + 6.25 = 3 17 So, s x = s y = 3 17
Calculating the Covariance Now, we calculate the covariance of x and y .
co v ( x , y ) = n − 1 ∑ ( x i − x ˉ ) ( y i − y ˉ ) 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 ) co v ( x , y ) = 3 ( − 2.5 ) ( − 2.5 ) + ( − 1.5 ) ( − 1.5 ) + ( 1.5 ) ( 1.5 ) + ( 2.5 ) ( 2.5 ) co v ( x , y ) = 3 6.25 + 2.25 + 2.25 + 6.25 = 3 17
Calculating the Correlation Coefficient Finally, we calculate the correlation coefficient. r = s x s y co v ( x , y ) = 3 17 3 17 3 17 = 3 17 3 17 = 1 Therefore, the correlation coefficient is 1.
Final Answer The correlation coefficient for the data shown in the table is 1 .
Examples
Understanding correlation coefficients is crucial in finance when analyzing stock returns. For instance, if two stocks have a correlation coefficient close to 1, they tend to move in the same direction. Investors use this information to diversify their portfolios, aiming to include assets with low or negative correlations to reduce risk. A correlation of 1 indicates a perfect positive linear relationship, meaning the stocks' returns move perfectly in sync.
The correlation coefficient for the data set 0, 1, 4, and 5 is 1, indicating a perfect correlation with itself. This is because the correlation coefficient measures how closely two variables move in relation to each other, and any variable will always perfectly correlate with itself.
;
