Question

The marks out of 50 obtained by 100 students in a test are given below:

Marks obtained	20	25	28	29	33	38	42	43
Number of students	6	20	24	28	15	4	2	1

Find: \(3\text{ mode} - 2\text{ median}\)
 

• A. 27.5
• B. 31
• C. 30
• D. 28.8

Hint:

For discrete data, the mode is simply the data point with the highest frequency. To find the median for N data points, use a cumulative frequency table to locate the middle value(s). For even N, it's the average of the (N/2)th and (N/2 + 1)th values.

Answer 1

The Correct Option is C

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 \( (3 \times \text{Mode}) - (2 \times \text{Median}) \).
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 \( \left(\frac{N}{2}\right)^{th} \) and \( \left(\frac{N}{2} + 1\right)^{th} \) 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 \( \frac{100}{2} = 50^{th} \) and the \( \frac{100}{2} + 1 = 51^{st} \) 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 = \( \frac{50^{th} \text{ observation} + 51^{st} \text{ observation}}{2} = \frac{28 + 29}{2} = \frac{57}{2} = 28.5 \).
Median = 28.5.
Calculating the final value:
Value = (3 \(\times\) Mode) - (2 \(\times\) Median)
Value = (3 \(\times\) 29) - (2 \(\times\) 28.5)
Value = 87 - 57 = 30.
Step 4: Final Answer:
The value of (3 mode - 2 median) is 30.