Based on the data shown below, calculate the correlation coefficient (to three decimal places)

x	y
3	80.8
4	81.2
5	73.4
6	70
7	67.4
8	69.2
9	61
10	60.4


To calculate the correlation coefficient \( r \) between \( x \) and \( y \), we 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 points,
- \( \sum x \), \( \sum y \), \( \sum xy \), \( \sum x^2 \), and \( \sum y^2 \) are the sums of the respective variables.

Let's calculate this step by step:

| \( x \) | \( y \)  | \( xy \)    | \( x^2 \) | \( y^2 \)  |
|--------|--------|------------|--------|------------|
| 3      | 80.8   | 242.4      | 9      | 6528.64    |
| 4      | 81.2   | 324.8      | 16     | 6593.44    |
| 5      | 73.4   | 367.0      | 25     | 5388.36    |
| 6      | 70     | 420.0      | 36     | 4900       |
| 7      | 67.4   | 471.8      | 49     | 4542.76    |
| 8      | 69.2   | 553.6      | 64     | 4788.64    |
| 9      | 61     | 549.0      | 81     | 3721       |
| 10     | 60.4   | 604.0      | 100    | 3648.16    |

Now, let's compute the sums:

\[
\sum x = 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 52
\]
\[
\sum y = 80.8 + 81.2 + 73.4 + 70 + 67.4 + 69.2 + 61 + 60.4 = 563.4
\]
\[
\sum xy = 242.4 + 324.8 + 367 + 420 + 471.8 + 553.6 + 549 + 604 = 3532.6
\]
\[
\sum x^2 = 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 380
\]
\[
\sum y^2 = 6528.64 + 6593.44 + 5388.36 + 4900 + 4542.76 + 4788.64 + 3721 + 3648.16 = 40111
\]

Finally, we'll substitute these values into the formula for \( r \).

I'll calculate that for you.The correlation coefficient \( r \) is \( -0.960 \), rounded to three decimal places.

This indicates a strong negative correlation between \( x \) and \( y \), meaning as \( x \) increases, \( y \) tends to decrease.

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the correlation coefficient affect data interpretation?
2. What is the significance of a negative correlation?
3. How is a correlation coefficient interpreted in real-world scenarios?
4. Can correlation imply causation between two variables?
5. What is the difference between correlation and regression analysis?

**Tip:** Correlation coefficients range from -1 to 1, where values closer to -1 or 1 indicate stronger relationships.

Based on the data shown below, calculate the correlation coefficient (to three decimal places)

x   y
3   80.8
4   81.2
5   73.4
6   70
7   67.4
8   69.2
9   61
10  60.4

Learn how to calculate the correlation coefficient between two variables from the given data set. This example includes step-by-step calculations for ∑x, ∑y, ∑xy, ∑x², and ∑y², resulting in a correlation coefficient of -0.960. Suitable for students in Grades 10-12.

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