x= 70 115 105 82 93 125 88
y= 3 45 21 7 16 62 12
compute r using the computation formula nearest thousandth
To compute the correlation coefficient, often denoted as r, using the computation formula, you would need to follow these steps:
Step 1: Calculate the mean of both sets of data (x and y). Let's denote the means as x̄ and ȳ.
For x:
x̄ = (70 + 115 + 105 + 82 + 93 + 125 + 88) / 7
x̄ ≈ 98.5714
For y:
ȳ = (3 + 45 + 21 + 7 + 16 + 62 + 12) / 7
ȳ ≈ 22.8571
Step 2: Calculate the differences between each data point and the mean for both x and y. Let's denote these differences as dx and dy.
For x:
dx = [70 - x̄, 115 - x̄, 105 - x̄, 82 - x̄, 93 - x̄, 125 - x̄, 88 - x̄]
dx = [-28.5714, 16.4286, 6.4286, -16.5714, -5.5714, 26.4286, -10.5714]
For y:
dy = [3 - ȳ, 45 - ȳ, 21 - ȳ, 7 - ȳ, 16 - ȳ, 62 - ȳ, 12 - ȳ]
dy = [-19.8571, 22.1429, -1.8571, -15.8571, -6.8571, 39.1429, -10.8571]
Step 3: Calculate the sum of the products of the differences for each data point:
Σ(dx * dy) = (-28.5714 * -19.8571) + (16.4286 * 22.1429) + (6.4286 * -1.8571) + (-16.5714 * -15.8571) + (-5.5714 * -6.8571) + (26.4286 * 39.1429) + (-10.5714 * -10.8571)
Σ(dx * dy) ≈ 2996.5714
Step 4: Calculate the sum of the squared differences for each set of data:
Σ(dx^2) = (-28.5714)^2 + (16.4286)^2 + (6.4286)^2 + (-16.5714)^2 + (-5.5714)^2 + (26.4286)^2 + (-10.5714)^2
Σ(dx^2) ≈ 4237.7143
Σ(dy^2) = (-19.8571)^2 + (22.1429)^2 + (-1.8571)^2 + (-15.8571)^2 + (-6.8571)^2 + (39.1429)^2 + (-10.8571)^2
Σ(dy^2) ≈ 3595.4286
Step 5: Calculate the square root of the product of the sum of squared differences:
√(Σ(dx^2) * Σ(dy^2)) = √(4237.7143 * 3595.4286)
√(Σ(dx^2) * Σ(dy^2)) ≈ 3849.6487
Step 6: Divide the sum of the products of the differences (Σ(dx * dy)) by the square root of the product of the sum of squared differences (√(Σ(dx^2) * Σ(dy^2))) to calculate the correlation coefficient (r):
r = Σ(dx * dy) / √(Σ(dx^2) * Σ(dy^2))
r ≈ 2996.5714 / 3849.6487
r ≈ 0.7784
Therefore, the correlation coefficient (r) calculated to the nearest thousandth is approximately 0.778.