To calculate the correlation coefficient $r$ for the data provided, I'll use the formula shown in the second image. Here’s a step-by-step breakdown of the calculation:

Step 1: Set Up the Data

Let's define:

• $x$ as the number of absences.
• $y$ as the final grades.

Data for each student:

Step 2: Calculate $\sum x$, $\sum y$, $\sum x^2$, $\sum y^2$, and $\sum xy$

1. Calculate $\sum x$ and $\sum y$: $\sum x = 6 + 2 + 15 + 9 + 12 + 5 + 8 = 57$ $\sum y = 82 + 86 + 43 + 74 + 58 + 90 + 78 = 511$

2. Calculate $\sum x^2$ and $\sum y^2$: $\sum x^2 = 6^2 + 2^2 + 15^2 + 9^2 + 12^2 + 5^2 + 8^2 = 36 + 4 + 225 + 81 + 144 + 25 + 64 = 579$ $\sum y^2 = 82^2 + 86^2 + 43^2 + 74^2 + 58^2 + 90^2 + 78^2 = 6724 + 7396 + 1849 + 5476 + 3364 + 8100 + 6084 = 38993$

3. Calculate $\sum xy$: $\sum xy = (6 \times 82) + (2 \times 86) + (15 \times 43) + (9 \times 74) + (12 \times 58) + (5 \times 90) + (8 \times 78)$ $= 492 + 172 + 645 + 666 + 696 + 450 + 624 = 3745$

Step 3: Calculate SS(x), SS(y), and SS(xy)

Using the formulas:

1. SS(x): $SS(x) = \sum x^2 - \frac{(\sum x)^2}{n}$ $SS(x) = 579 - \frac{57^2}{7} = 579 - \frac{3249}{7} = 579 - 464.14 = 114.86$

2. SS(y): $SS(y) = \sum y^2 - \frac{(\sum y)^2}{n}$ $SS(y) = 38993 - \frac{511^2}{7} = 38993 - \frac{261121}{7} = 38993 - 37231.57 = 1761.43$

3. SS(xy): $SS(xy) = \sum xy - \frac{\sum x \cdot \sum y}{n}$ $SS(xy) = 3745 - \frac{57 \times 511}{7} = 3745 - \frac{29127}{7} = 3745 - 4153.86 = -408.86$

Step 4: Calculate $r$

The correlation coefficient $r$ is given by: $r = \frac{SS(xy)}{\sqrt{SS(x) \cdot SS(y)}}$ $r = \frac{-408.86}{\sqrt{114.86 \times 1761.43}}$ Calculating the denominator: $\sqrt{114.86 \times 1761.43} = \sqrt{202162.37} = 449.63$ Thus, $r = \frac{-408.86}{449.63} = -0.91$

Interpretation

The correlation coefficient $r \approx -0.91$ indicates a strong negative correlation between the number of absences and the final grade. This suggests that as the number of absences increases, the final grade tends to decrease significantly.

Would you like any further details or clarification on this calculation?

1. What is the effect of outliers on the correlation coefficient?
2. How does the interpretation of a negative correlation differ from a positive one?
3. What does a correlation of $r = 0$ signify about the relationship between two variables?
4. How could we test the statistical significance of this correlation coefficient?
5. What other measures can describe the strength of the relationship between two variables besides the correlation coefficient?

Tip: Remember that correlation does not imply causation; it only shows a relationship between two variables.