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To compute the correlation coefficient \( r \) between the two variables \( x \) and \( y \), we can use the formula:

\[
r = \frac{n \sum(xy) - \sum x \sum y}{\sqrt{\left(n \sum x^2 - (\sum x)^2\right) \left(n \sum y^2 - (\sum y)^2\right)}}
\]

where:
- \( n \) is the number of data points,
- \( \sum x \) and \( \sum y \) are the sums of the \( x \)- and \( y \)-values,
- \( \sum(xy) \) is the sum of the products of corresponding \( x \)- and \( y \)-values,
- \( \sum x^2 \) and \( \sum y^2 \) are the sums of squares of the \( x \)- and \( y \)-values.

### Given data
The values provided are:
- \( x: 6, 3, 1, 4, 5 \)
- \( y: 5, 3, 2, 7, 1 \)

### Step-by-Step Calculation
1. **Calculate \( \sum x \), \( \sum y \), \( \sum x^2 \), \( \sum y^2 \), and \( \sum(xy) \):**

   - \( \sum x = 6 + 3 + 1 + 4 + 5 = 19 \)
   - \( \sum y = 5 + 3 + 2 + 7 + 1 = 18 \)
   - \( \sum x^2 = 6^2 + 3^2 + 1^2 + 4^2 + 5^2 = 36 + 9 + 1 + 16 + 25 = 87 \)
   - \( \sum y^2 = 5^2 + 3^2 + 2^2 + 7^2 + 1^2 = 25 + 9 + 4 + 49 + 1 = 88 \)
   - \( \sum(xy) = (6 \cdot 5) + (3 \cdot 3) + (1 \cdot 2) + (4 \cdot 7) + (5 \cdot 1) = 30 + 9 + 2 + 28 + 5 = 74 \)

2. **Plug values into the formula:**

   \[
   r = \frac{5 \cdot 74 - (19 \cdot 18)}{\sqrt{(5 \cdot 87 - 19^2) (5 \cdot 88 - 18^2)}}
   \]

3. **Simplify each part:**

   - \( 5 \cdot 74 = 370 \)
   - \( 19 \cdot 18 = 342 \)
   - \( 5 \cdot 87 = 435 \)
   - \( 19^2 = 361 \)
   - \( 5 \cdot 88 = 440 \)
   - \( 18^2 = 324 \)

   Substitute these into the equation:

   \[
   r = \frac{370 - 342}{\sqrt{(435 - 361)(440 - 324)}}
   \]

4. **Further simplification:**

   - \( 370 - 342 = 28 \)
   - \( 435 - 361 = 74 \)
   - \( 440 - 324 = 116 \)

   So,

   \[
   r = \frac{28}{\sqrt{74 \cdot 116}}
   \]

5. **Calculate the denominator:**

   - \( 74 \cdot 116 = 8584 \)
   - \( \sqrt{8584} \approx 92.65 \)

   Finally,

   \[
   r \approx \frac{28}{92.65} \approx 0.302
   \]

### Answer
The correlation coefficient, rounded to three decimal places, is:

\[
\boxed{0.302}
\]

Would you like a further breakdown or have any questions about this solution?

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### Related Questions:
1. How is the correlation coefficient interpreted in terms of strength and direction?
2. What is the difference between a positive and negative correlation?
3. How would the correlation coefficient change if one of the \( y \)-values was altered?
4. What does a correlation coefficient of zero signify?
5. How can you compute the correlation coefficient using a calculator or software?

### Tip:
When calculating correlation by hand, double-check the sums for each component (like \( \sum x \) and \( \sum y \)) to ensure accuracy, as errors in these values propagate through the entire calculation.

Compute the correlation coefficient between x and y using the given data points: x = [6, 3, 1, 4, 5] and y = [5, 3, 2, 7, 1]. Round the answer to three decimal places.

Calculate the correlation coefficient between the variables x and y with data points x = [6, 3, 1, 4, 5] and y = [5, 3, 2, 7, 1]. This problem includes a step-by-step solution using the correlation formula, providing insights into data analysis concepts. Suitable for students in Grades 10-12.

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