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?

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.