Understanding the Problem: Analyzing Marks Data

The problem provides grouped frequency data for the marks obtained by students in Physics and Mathematics. We are given two statements about the modal value and median of these marks distributions and need to determine which statement(s) are correct.

The data is presented in the following table:

Analyzing Statement 1: Modal Value in Physics

Statement 1 says: "modal value of the marks in Physics less in the interval 30 - 40". For grouped data, the modal value lies in the modal class, which is the class interval with the highest frequency.

Let's look at the frequencies for Physics:

• 10 – 20: 8
• 20 – 30: 11
• 30 – 40: 30
• 40 – 50: 26
• 50 – 60: 15
• 60 - 70: 10

The highest frequency in Physics is 30, which occurs in the class interval 30 – 40. Therefore, the modal class for Physics is 30 – 40.

The statement says the modal value is "less in the interval 30 - 40", which is awkwardly phrased. A common interpretation in this context could be that the modal class is not 30-40, or perhaps it's an interval before 30-40. Since the modal class is indeed 30 – 40, statement 1 implying it is "less in the interval 30-40" (interpreted as being in an interval before 30-40, or the modal class not being 30-40) is incorrect.

Statement 1 is incorrect.

Analyzing Statement 2: Median Comparison

Statement 2 says: "Median of the marks in Physics is less than that of marks in Mathematics." To verify this, we need to calculate the median for both Physics and Mathematics using the formula for grouped data:

\(\text{Median} = L + \frac{\frac{N}{2} - CF}{f} \times h\)

Where:

• \(L\) is the lower limit of the median class.
• \(N\) is the total number of observations.
• \(CF\) is the cumulative frequency of the class preceding the median class.
• \(f\) is the frequency of the median class.
• \(h\) is the class width.

Calculating Median for Physics:

First, find the total number of students (\(N\)).
\(N_{Physics} = 8 + 11 + 30 + 26 + 15 + 10 = 100\).
\(\frac{N}{2} = \frac{100}{2} = 50\).

Next, find the cumulative frequencies (CF) for Physics:

• 10 – 20: 8
• 20 – 30: 8 + 11 = 19
• 30 – 40: 19 + 30 = 49
• 40 – 50: 49 + 26 = 75
• 50 – 60: 75 + 15 = 90
• 60 - 70: 90 + 10 = 100

The median class is the class where the CF first exceeds 50. This is the 40 – 50 class (CF = 75).

Using the formula:

• \(L = 40\)
• \(\frac{N}{2} = 50\)
• \(CF = 49\) (CF of the preceding class 30-40)
• \(f = 26\) (frequency of the median class 40-50)
• \(h = 10\) (class width)

\(Median_{Physics} = 40 + \frac{50 - 49}{26} \times 10 = 40 + \frac{1}{26} \times 10 = 40 + \frac{10}{26} = 40 + \frac{5}{13}\)

Calculating the value:

\(Median_{Physics} \approx 40 + 0.3846 \approx 40.38\)

Calculating Median for Mathematics:

First, find the total number of students (\(N\)).
\(N_{Mathematics} = 10 + 21 + 38 + 15 + 10 + 6 = 100\).
\(\frac{N}{2} = \frac{100}{2} = 50\).

Next, find the cumulative frequencies (CF) for Mathematics:

• 10 – 20: 10
• 20 – 30: 10 + 21 = 31
• 30 – 40: 31 + 38 = 69
• 40 – 50: 69 + 15 = 84
• 50 – 60: 84 + 10 = 94
• 60 - 70: 94 + 6 = 100

The median class is the class where the CF first exceeds 50. This is the 30 – 40 class (CF = 69).

Using the formula:

• \(L = 30\)
• \(\frac{N}{2} = 50\)
• \(CF = 31\) (CF of the preceding class 20-30)
• \(f = 38\) (frequency of the median class 30-40)
• \(h = 10\) (class width)

\(Median_{Mathematics} = 30 + \frac{50 - 31}{38} \times 10 = 30 + \frac{19}{38} \times 10 = 30 + \frac{1}{2} \times 10 = 30 + 5 = 35\)

Comparing Medians:

\(Median_{Physics} \approx 40.38\)
\(Median_{Mathematics} = 35\)

Statement 2 says "Median of the marks in Physics is less than that of marks in Mathematics."
Comparing the calculated values, \(40.38\) is not less than \(35\).

Statement 2 is incorrect.

Conclusion

Based on the analysis of both statements:

• Statement 1 is incorrect because the modal class for Physics is 30 - 40.
• Statement 2 is incorrect because the median of marks in Physics (\(\approx 40.38\)) is greater than the median of marks in Mathematics (35).

Therefore, neither of the statements is correct.