Calculate the Pearson product-moment correlation for the data below.
X 3 4 2 1
Y 5 5 3 4
Using an online calculator:
4 data pairs (x,y):
( 3.00 , 5.00 ); ( 4.00 , 5.00 ); ( 2.00 , 3.00 ); ( 1.00 , 4.00 );
Regression equation:
y = 3 + .5x
Correlation Coefficient:
r = 0.674
Check with your own calculations.
To calculate the Pearson product-moment correlation for the given data, you need to follow these steps:
Step 1: Calculate the mean (average) of X and Y.
Mean of X = (3 + 4 + 2 + 1) / 4 = 2.5
Mean of Y = (5 + 5 + 3 + 4) / 4 = 4.25
Step 2: Calculate the deviation of each value from the mean (X - X̄ and Y - Ȳ).
Deviation of X: (-0.5, 1.5, -0.5, -1.5)
Deviation of Y: (0.75, 0.75, -1.25, -0.25)
Step 3: Multiply the deviations of X and Y for each corresponding pair.
(-0.5 * 0.75, 1.5 * 0.75, -0.5 * -1.25, -1.5 * -0.25) = (-0.375, 1.125, 0.625, 0.375)
Step 4: Calculate the sum of the products obtained in step 3 and the sum of the squares of the deviations.
Sum of products = -0.375 + 1.125 + 0.625 + 0.375 = 1.75
Sum of squared deviations of X = (-0.5)² + 1.5² + (-0.5)² + (-1.5)² = 4.5
Sum of squared deviations of Y = 0.75² + 0.75² + (-1.25)² + (-0.25)² = 3.625
Step 5: Calculate the square root of the product of the sums of squared deviations for X and Y.
√(4.5 * 3.625) = √16.3125 = 4.04
Step 6: Divide the sum of products (step 4) by the product of the square roots of the sums of squared deviations (step 5).
1.75 / 4.04 = 0.433
Therefore, the Pearson product-moment correlation for the given data is 0.433.
To calculate the Pearson product-moment correlation, follow these steps:
Step 1: Find the mean (average) for each set of data.
- For X: (3 + 4 + 2 + 1) / 4 = 2.5
- For Y: (5 + 5 + 3 + 4) / 4 = 4.25
Step 2: Subtract the mean from each value in the X and Y sets.
- For X: 3 - 2.5 = 0.5, 4 - 2.5 = 1.5, 2 - 2.5 = -0.5, 1 - 2.5 = -1.5
- For Y: 5 - 4.25 = 0.75, 5 - 4.25 = 0.75, 3 - 4.25 = -1.25, 4 - 4.25 = -0.25
Step 3: Multiply the differences for each pair of X and Y values.
- For the first pair: 0.5 * 0.75 = 0.375
- For the second pair: 1.5 * 0.75 = 1.125
- For the third pair: -0.5 * -1.25 = 0.625
- For the fourth pair: -1.5 * -0.25 = 0.375
Step 4: Square each difference obtained in Step 3.
- For the first pair: 0.375 * 0.375 = 0.140625
- For the second pair: 1.125 * 1.125 = 1.265625
- For the third pair: 0.625 * 0.625 = 0.390625
- For the fourth pair: 0.375 * 0.375 = 0.140625
Step 5: Sum up all the squared differences obtained in Step 4.
- 0.140625 + 1.265625 + 0.390625 + 0.140625 = 1.9375
Step 6: Calculate the product-moment correlation using the formula:
- Pearson's correlation coefficient (r) = Σ(xy) / √(Σ(x^2) * Σ(y^2))
To do this, we need the sum of x^2 (Σ(x^2)) and the sum of y^2 (Σ(y^2)).
Step 7: Square each value in the X and Y sets and calculate their sums.
- For X: 3^2 + 4^2 + 2^2 + 1^2 = 9 + 16 + 4 + 1 = 30
- For Y: 5^2 + 5^2 + 3^2 + 4^2 = 25 + 25 + 9 + 16 = 75
Step 8: Plug in the values from Step 5 and Step 7 into the correlation coefficient formula.
- r = 1.9375 / √(30 * 75)
Step 9: Simplify the equation by calculating the square root of the product in the denominator.
- r = 1.9375 / √(2250) ≈ 0.3903
So, the Pearson product-moment correlation for the given data is approximately 0.3903.