Step 1: Understanding the Concept:
This problem requires finding the mode and median of a given discrete frequency distribution. Then, we need to compute the value of the expression .
Step 2: Key Formula or Approach:
1. Mode: The mode is the value that appears most frequently. In a frequency distribution, it is the value with the highest frequency.
2. Median: The median is the middle value of a data set when it is arranged in order. For N observations, the median is the average of the and observations when N is even. We will use a cumulative frequency table to find the median.
Step 3: Detailed Explanation:
Finding the Mode:
From the table, we look for the highest frequency (Number of students).
The frequencies are 6, 20, 24, 28, 15, 4, 2, 1.
The highest frequency is 28, which corresponds to the 'Marks obtained' of 29.
Therefore, Mode = 29.
Finding the Median:
The total number of students (N) is 100. Since N is even, the median is the average of the and the observations.
Let's create a cumulative frequency (cf) table:
\begin{tabular}{|c|c|c|}
\hline
Marks (x) & Frequency (f) & Cumulative Frequency (cf)
\hline
20 & 6 & 6
25 & 20 & 26
28 & 24 & 50
29 & 28 & 78
33 & 15 & 93
38 & 4 & 97
42 & 2 & 99
43 & 1 & 100
\hline
\end{tabular}
The cumulative frequency shows that the students from the 27th to the 50th position all scored 28 marks. So, the 50th observation is 28.
The students from the 51st to the 78th position all scored 29 marks. So, the 51st observation is 29.
Median = .
Median = 28.5.
Calculating the final value:
Value = (3 Mode) - (2 Median)
Value = (3 29) - (2 28.5)
Value = 87 - 57 = 30.
Step 4: Final Answer:
The value of (3 mode - 2 median) is 30.