The task involves calculating the coefficient of correlation between the marks obtained in Physics and Chemistry by twelve students.

The coefficient of correlation (denoted as $r$) measures the strength and direction of the linear relationship between two variables. It can be calculated using the Pearson’s 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:

• $n$ = number of students (12 in this case)
• $x$ = marks in Physics
• $y$ = marks in Chemistry

We need to compute the following:

1. $\sum x$ = Sum of Physics marks
2. $\sum y$ = Sum of Chemistry marks
3. $\sum x^2$ = Sum of squares of Physics marks
4. $\sum y^2$ = Sum of squares of Chemistry marks
5. $\sum xy$ = Sum of products of corresponding Physics and Chemistry marks

Extracted Data from the Image:

• Marks in Physics: 78, 36, 98, 25, 75, 82, 90, 62, 65, 39, 45, 68
• Marks in Chemistry: 84, 51, 91, 60, 68, 62, 86, 58, 53, 47, 59, 78

Let’s proceed with these calculations step-by-step.Here are the detailed calculations:

1. Sum of Physics marks ($\sum x$): 763
2. Sum of Chemistry marks ($\sum y$): 797
3. Sum of squares of Physics marks ($\sum x^2$): 54,297
4. Sum of squares of Chemistry marks ($\sum y^2$): 55,349
5. Sum of products of corresponding Physics and Chemistry marks ($\sum xy$): 53,563

Using the Pearson’s correlation formula:

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

$r = \frac{12 \times 53563 - 763 \times 797}{\sqrt{(12 \times 54297 - 763^2) \times (12 \times 55349 - 797^2)}}$

After computing:

$r \approx 0.773$

Result:

The coefficient of correlation between the marks in Physics and Chemistry is approximately 0.773. This indicates a strong positive correlation between the two subjects.

Do you need any further details or clarifications?
Here are five related questions for practice:

1. What does a correlation coefficient of 0.773 imply about the relationship between two variables?
2. How would the value of $r$ change if outliers were present in the data?
3. Can the correlation coefficient be used to predict one variable from the other? Why or why not?
4. How would the formula change if one dataset had missing values?
5. What is the significance of a correlation value close to +1 or -1?

Tip: A correlation coefficient closer to +1 indicates a strong positive relationship, while a value near 0 suggests no linear relationship.