Find the value for the correlation coefficient r.
x - 21 30 25 24 29
y - 49 41 31 36 39
To find the correlation coefficient (r), we need to calculate several values first:
1. Find the arithmetic mean of x and y:
- x̄ (x-bar): (21 + 30 + 25 + 24 + 29) / 5 = 129 / 5 = 25.8
- ȳ (y-bar): (49 + 41 + 31 + 36 + 39) / 5 = 196 / 5 = 39.2
2. Calculate the deviation for each value of x and y:
- For each x value, compute x - x̄:
21 - 25.8 = -4.8
30 - 25.8 = 4.2
25 - 25.8 = -0.8
24 - 25.8 = -1.8
29 - 25.8 = 3.2
- For each y value, compute y - ȳ:
49 - 39.2 = 9.8
41 - 39.2 = 1.8
31 - 39.2 = -8.2
36 - 39.2 = -3.2
39 - 39.2 = -0.2
3. Calculate the product of the deviations (x - x̄) and (y - ȳ) for each pair of values:
- For each pair, multiply the respective deviations:
(-4.8) * 9.8 = -47.04
(4.2) * 1.8 = 7.56
(-0.8) * (-8.2) = 6.56
(-1.8) * (-3.2) = 5.76
(3.2) * (-0.2) = -0.64
4. Calculate the sum of the products of deviations:
- Sum of the product deviations: -47.04 + 7.56 + 6.56 + 5.76 - 0.64 = -27.8
5. Calculate the sum of the squared deviations for both x and y:
- For x, calculate the squared deviation for each value and sum them:
(-4.8)^2 + (4.2)^2 + (-0.8)^2 + (-1.8)^2 + (3.2)^2 = 23.76 + 17.64 + 0.64 + 3.24 + 10.24 = 55.52
- For y, calculate the squared deviation for each value and sum them:
(9.8)^2 + (1.8)^2 + (-8.2)^2 + (-3.2)^2 + (-0.2)^2 = 96.04 + 3.24 + 67.24 + 10.24 + 0.04 = 176.8
6. Calculate the square root of the product of the sum of squared deviations for x and y:
- √(55.52 * 176.8) = √9808.74 = 99.04
7. Finally, calculate the correlation coefficient (r):
- r = (sum of the product deviations) / √((sum of squared deviations for x) * (sum of squared deviations for y))
- r = -27.8 / 99.04 ≈ -0.28
Therefore, the value of the correlation coefficient (r) is approximately -0.28.