The question asks us to find the mean, median, and mode of the marks obtained by 100 students in an examination, given in a frequency distribution table. We will calculate each of these measures of central tendency step-by-step.

Marks Obtained: Calculating the Mean

The mean (average) for grouped data is calculated using the formula:

$$\text{Mean} (\bar{x}) = \frac{\sum fx}{\sum f}$$

where $\sum fx$ is the sum of the products of marks (x) and their corresponding frequencies (f), and $\sum f$ is the total number of students (total frequency).

Let's create a table to calculate $\sum fx$:

Now, substitute the values into the mean formula:

$$\text{Mean} (\bar{x}) = \frac{3300}{100} = 33$$

So, the mean marks obtained by the students is 33.

Marks Obtained: Determining the Median

The median is the middle value in an ordered dataset. For grouped discrete data, we first find the cumulative frequency.

The total number of students (N) is 100. Since N is an even number, the median is the average of the values at the $\left(\frac{N}{2}\right)^{th}$ and $\left(\frac{N}{2} + 1\right)^{th}$ positions.

• $\frac{N}{2} = \frac{100}{2} = 50^{th}$ position
• $\left(\frac{N}{2} + 1\right) = \left(\frac{100}{2} + 1\right) = 51^{st}$ position

Let's create a cumulative frequency table:

Now, we locate the $50^{th}$ and $51^{st}$ observations:

• The cumulative frequency for marks 30 is 40. This means the first 40 students have marks up to 30.
• The cumulative frequency for marks 35 is 80. This means students from the $41^{st}$ position to the $80^{th}$ position have marks 35.

Since both the $50^{th}$ and $51^{st}$ observations fall within the range where marks are 35, the value of the $50^{th}$ observation is 35, and the value of the $51^{st}$ observation is also 35.

$$\text{Median} = \frac{50^{th} \text{ observation} + 51^{st} \text{ observation}}{2} = \frac{35 + 35}{2} = \frac{70}{2} = 35$$

So, the median marks obtained by the students is 35.

Marks Obtained: Identifying the Mode

The mode is the value that appears most frequently in a dataset. In a frequency distribution table, the mode is the value of 'x' that has the highest frequency (f).

Let's look at the frequencies from the given table:

• Marks 25: Frequency 20
• Marks 30: Frequency 20
• Marks 35: Frequency 40
• Marks 40: Frequency 20

The highest frequency is 40, which corresponds to the marks value of 35.

Therefore, the mode of the marks obtained by the students is 35.

Summary of Measures

Based on our calculations:

• Mean: 33
• Median: 35
• Mode: 35

This matches the option stating Mean 33, Median 35, Mode 35.