What is the $r$-value of the following data, to three decimal places?
| x | y |
| -- | -- |
| 4 | 2 |
| 5 | 9 |
| 8 | 10 |
| 9 | 12 |
| 13 | 23 |
A. 0.953
B. -0.953
C. -0.908
D. 0.908
Answer (1)
Calculate the mean of x values: x ˉ = 7.8 .
Calculate the mean of y values: y ˉ = 11.2 .
Calculate the Pearson correlation coefficient: r = ( n − 1 ) s x s y ∑ i = 1 n ( x i − x ˉ ) ( y i − y ˉ ) .
The r -value of the data, rounded to three decimal places, is 0.953 .
Explanation
Understanding the Problem We are given a table of x and y values and asked to find the r -value (correlation coefficient) to three decimal places. The r -value measures the strength and direction of a linear relationship between two variables. A positive r -value indicates a positive correlation, while a negative r -value indicates a negative correlation. The closer the absolute value of r is to 1, the stronger the linear relationship is.
Formula for Correlation Coefficient The formula for calculating the Pearson correlation coefficient r is: r = ( n − 1 ) s x s y ∑ i = 1 n ( x i − x ˉ ) ( y i − y ˉ ) where:
n is the number of data points
x i and y i are the individual data points
x ˉ and y ˉ are the means of the x and y values, respectively
s x and s y are the sample standard deviations of the x and y values, respectively.
Calculating the Means First, we calculate the means of the x and y values: x ˉ = 5 4 + 5 + 8 + 9 + 13 = 5 39 = 7.8 y ˉ = 5 2 + 9 + 10 + 12 + 23 = 5 56 = 11.2
Calculating Standard Deviations Next, we calculate the sample standard deviations of the x and y values: s x = n − 1 ∑ i = 1 n ( x i − x ˉ ) 2 = 5 − 1 ( 4 − 7.8 ) 2 + ( 5 − 7.8 ) 2 + ( 8 − 7.8 ) 2 + ( 9 − 7.8 ) 2 + ( 13 − 7.8 ) 2 s x = 4 ( − 3.8 ) 2 + ( − 2.8 ) 2 + ( 0.2 ) 2 + ( 1.2 ) 2 + ( 5.2 ) 2 = 4 14.44 + 7.84 + 0.04 + 1.44 + 27.04 = 4 50.8 = 12.7 ≈ 3.564 s y = n − 1 ∑ i = 1 n ( y i − y ˉ ) 2 = 5 − 1 ( 2 − 11.2 ) 2 + ( 9 − 11.2 ) 2 + ( 10 − 11.2 ) 2 + ( 12 − 11.2 ) 2 + ( 23 − 11.2 ) 2 s y = 4 ( − 9.2 ) 2 + ( − 2.2 ) 2 + ( − 1.2 ) 2 + ( 0.8 ) 2 + ( 11.8 ) 2 = 4 84.64 + 4.84 + 1.44 + 0.64 + 139.24 = 4 230.8 = 57.7 ≈ 7.65. C a l c u l a t in g t h e N u m er a t or N o w , w ec a l c u l a t e t h e n u m er a t oro f t h ecorre l a t i o n coe ff i c i e n t : \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y}) = (4-7.8)(2-11.2) + (5-7.8)(9-11.2) + (8-7.8)(10-11.2) + (9-7.8)(12-11.2) + (13-7.8)(23-11.2) = (-3.8)(-9.2) + (-2.8)(-2.2) + (0.2)(-1.2) + (1.2)(0.8) + (5.2)(11.8) = 34.96 + 6.16 - 0.24 + 0.96 + 61.36 = 103.2 6. Calculating the Correlation Coefficient Finally, we calculate the correlation coefficient $r$: r = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{(n-1)s_x s_y} = \frac{103.2}{(5-1)(3.564)(7.6)} = \frac{103.2}{4(3.564)(7.6)} = \frac{103.2}{108.40} \approx 0.952
Final Answer Rounding the calculated r -value to three decimal places, we get r ≈ 0.953 .
Examples
Understanding the correlation coefficient is extremely useful in many real-world applications. For instance, in finance, it can help determine the relationship between the returns of two different stocks. If the correlation is high, the stocks tend to move in the same direction, which can inform decisions about portfolio diversification. Similarly, in marketing, the correlation between advertising spend and sales revenue can help businesses understand the effectiveness of their campaigns. By calculating and analyzing the r -value, businesses and individuals can make more informed decisions based on the relationships between different variables.
