The problem in the image seems to involve calculating the correlation coefficient (r) between two sets of marks in two subjects: "Roll No. 1" and "Statistics," and "Mathematics." The marks are represented by the variables $x$ (Mathematics) and $y$ (Statistics).

The formula for the Pearson correlation coefficient $r_{xy}$ is given by:

$r_{xy} = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sqrt{\sum (x - \bar{x})^2 \sum (y - \bar{y})^2}}$

This involves several steps. Let's go through them based on the table shown:

Step-by-Step Solution:

Step 1: List the marks for $x$ (Mathematics) and $y$ (Statistics)

From the table:

• $x = [78, 36, 98, 25, 75, 82, 90, 62, 65, 39]$
• $y = [84, 51, 91, 60, 68, 62, 86, 58, 53, 47]$

Step 2: Calculate the mean ($\bar{x}$ and $\bar{y}$)

We calculate the mean for both $x$ and $y$.

$\bar{x} = \frac{\sum x}{n} = \frac{78 + 36 + 98 + 25 + 75 + 82 + 90 + 62 + 65 + 39}{10}$

$\bar{y} = \frac{\sum y}{n} = \frac{84 + 51 + 91 + 60 + 68 + 62 + 86 + 58 + 53 + 47}{10}$

Step 3: Calculate the deviation from the mean

Next, we calculate the deviations $(x - \bar{x})$ and $(y - \bar{y})$ for each data point.

Step 4: Calculate $(x - \bar{x})^2$ and $(y - \bar{y})^2$

Square each deviation for both $x$ and $y$.

Step 5: Calculate the product $(x - \bar{x})(y - \bar{y})$

For each pair of $x$ and $y$, multiply their respective deviations from the mean.

Step 6: Sum the values

Sum the values obtained in steps 4 and 5.

• $\sum (x - \bar{x})(y - \bar{y})$
• $\sum (x - \bar{x})^2$
• $\sum (y - \bar{y})^2$

Step 7: Apply the formula for $r_{xy}$

Finally, apply the formula to get the correlation coefficient $r_{xy}$.

I'll calculate each step based on the data. Let me run these numbers for you.Here is the step-by-step solution based on the calculations:

1. Mean of $x$ (Mathematics marks): $\bar{x} = 65$

2. Mean of $y$ (Statistics marks): $\bar{y} = 66$

3. Sum of squared deviations for $x$: $\sum (x - \bar{x})^2 = 5398$

4. Sum of squared deviations for $y$: $\sum (y - \bar{y})^2 = 2224$

5. Sum of the product of deviations: $\sum (x - \bar{x})(y - \bar{y}) = 2704$

6. Correlation coefficient $r_{xy}$: $r_{xy} = \frac{2704}{\sqrt{5398 \times 2224}} = 0.7804$

So, the Pearson correlation coefficient is approximately 0.780.

Would you like further details or have any questions on these steps?

Follow-up questions:

1. What is the interpretation of the correlation coefficient value?
2. How can this correlation coefficient be applied in real-world scenarios?
3. Can you find the covariance for the same data?
4. What would happen to the correlation if one value were much higher or lower?
5. How does this result compare to a perfectly positive or negative correlation?

Tip:

When interpreting a correlation coefficient, remember that values close to 1 imply a strong positive linear relationship, while values near -1 imply a strong negative relationship.