Math Problem Statement
Solution
To calculate the coefficient of correlation using Karl Pearson's formula, we can follow these steps:
Formula for Pearson's Correlation Coefficient (r):
[ r = \frac{n\sum{XY} - \sum{X}\sum{Y}}{\sqrt{[n\sum{X^2} - (\sum{X})^2][n\sum{Y^2} - (\sum{Y})^2]}} ]
Where:
- and are the variables in the given data,
- is the number of pairs,
- , , , , and are the sums of the corresponding terms.
Given the table:
X | 10 | 12 | 15 | 25 | 20 | 27 | 19 |
---|---|---|---|---|---|---|---|
Y | 14 | 17 | 23 | 24 | 18 | 21 | 14 |
Let’s proceed step by step.
- Step 1: Find the sum of X, Y, X^2, Y^2, and XY.
I will calculate these values to help compute the correlation coefficient. numpy as np
Given data
X = np.array([10, 12, 15, 25, 20, 27, 19]) Y = np.array([14, 17, 23, 24, 18, 21, 14])
Calculate the required sums
n = len(X) sum_X = np.sum(X) sum_Y = np.sum(Y) sum_X_squared = np.sum(X2) sum_Y_squared = np.sum(Y2) sum_XY = np.sum(X * Y)
(sum_X, sum_Y, sum_X_squared, sum_Y_squared, sum_XY)The calculated sums are:
Step 2: Apply the Pearson's Correlation Coefficient Formula
Now, we can substitute these values into the formula:
[ r = \frac{n\sum{XY} - \sum{X}\sum{Y}}{\sqrt{[n\sum{X^2} - (\sum{X})^2][n\sum{Y^2} - (\sum{Y})^2]}} ]
Where .
Let's compute the correlation coefficient . Compute Pearson's correlation coefficient r_numerator = n * sum_XY - sum_X * sum_Y r_denominator = np.sqrt((n * sum_X_squared - sum_X2) * (n * sum_Y_squared - sum_Y2))
r = r_numerator / r_denominator rThe Pearson correlation coefficient () is approximately 0.556.
This indicates a moderate positive linear relationship between the variables and .
Would you like further clarification on this calculation or have any additional questions?
Here are some related questions you may find useful:
- How can you interpret different values of the Pearson correlation coefficient?
- What are some assumptions behind Pearson’s correlation?
- How can Pearson’s correlation be used in data analysis?
- What are the limitations of Pearson’s correlation coefficient?
- How do you calculate the coefficient of determination (r²)?
Tip: Always check for linearity and normality of the data before using Pearson’s correlation, as the formula assumes these conditions.
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Math Problem Analysis
Mathematical Concepts
Statistics
Correlation
Karl Pearson's Coefficient
Formulas
Pearson's Correlation Coefficient: r = [nΣXY - ΣXΣY] / √{[nΣX^2 - (ΣX)^2][nΣY^2 - (ΣY)^2]}
Theorems
Karl Pearson's Correlation Coefficient Formula
Suitable Grade Level
Undergraduate level or high school statistics
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