To compute Pearson's Product Moment Correlation (often denoted as $r$), we can use 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:

• $n$ is the number of data pairs.
• $x$ and $y$ are the values of Data A and Data B, respectively.
• $\sum xy$ is the sum of the products of corresponding values in Data A and Data B.
• $\sum x$ is the sum of all values in Data A.
• $\sum y$ is the sum of all values in Data B.
• $\sum x^2$ is the sum of the squares of each value in Data A.
• $\sum y^2$ is the sum of the squares of each value in Data B.

Given Data

• Data A: $[34, 22, 10, 11, 23, 20, 16, 18, 21, 15]$
• Data B: $[12, 23, 21, 24, 10, 30, 28, 29, 32, 25]$

Step-by-Step Calculation

1. Compute the required sums:

   $\sum x = 34 + 22 + 10 + 11 + 23 + 20 + 16 + 18 + 21 + 15 = 190$ $\sum y = 12 + 23 + 21 + 24 + 10 + 30 + 28 + 29 + 32 + 25 = 234$

2. Compute the sum of squares:

   $\sum x^2 = 34^2 + 22^2 + 10^2 + 11^2 + 23^2 + 20^2 + 16^2 + 18^2 + 21^2 + 15^2$ $= 1156 + 484 + 100 + 121 + 529 + 400 + 256 + 324 + 441 + 225 = 4036$

   $\sum y^2 = 12^2 + 23^2 + 21^2 + 24^2 + 10^2 + 30^2 + 28^2 + 29^2 + 32^2 + 25^2$ $= 144 + 529 + 441 + 576 + 100 + 900 + 784 + 841 + 1024 + 625 = 5964$

3. Compute the sum of products:

   $\sum xy = (34 \cdot 12) + (22 \cdot 23) + (10 \cdot 21) + (11 \cdot 24) + (23 \cdot 10) + (20 \cdot 30) + (16 \cdot 28) + (18 \cdot 29) + (21 \cdot 32) + (15 \cdot 25)$ $= 408 + 506 + 210 + 264 + 230 + 600 + 448 + 522 + 672 + 375 = 4235$

4. Substitute into the correlation formula:

   [ r = \frac{10(4235) - (190)(234)}{\sqrt{[10(4036) - 190^2][10(5964) - 234^2]}} ] [ r = \frac{42350 - 44460}{\sqrt{[40360 - 36100][59640 - 54756]}} ] $r = \frac{-1110}{\sqrt{4260 \times 4884}}$ $r = \frac{-1110}{\sqrt{20780640}}$ $r = \frac{-1110}{4558.01}$ $r \approx -0.243$

Conclusion

The Pearson's Product Moment Correlation coefficient $r \approx -0.243$, indicating a weak negative correlation between Data A and Data B.

Would you like more details or have any questions?

Relative Questions

1. What does a Pearson correlation of -0.243 imply about the relationship between two datasets?
2. How can the correlation coefficient be used to interpret data in real-world scenarios?
3. What is the significance of the correlation coefficient being negative?
4. How would outliers in Data A or Data B affect the correlation coefficient?
5. What other statistical measures can be used to assess the relationship between two datasets?

Tip

When interpreting the Pearson correlation coefficient, remember that values close to -1 or 1 indicate a strong relationship, while values near 0 indicate a weak relationship.