Compute Pearson’s Product Moment Correlation for the following data:
Individuals	A	B	C	D	E	F	G	H	I	J
Data A	34	22	10	11	23	20	16	18	21	15
Data B	12	23	21	24	10	30	28	29	32	25

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.

Learn how to compute Pearson’s Product Moment Correlation coefficient for Data Sets A and B with a step-by-step solution. Understand the formula and its application to these specific data sets. Suitable for advanced high school students and beyond.

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