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: Problem Definition: This task involves determining the mode and median from a given discrete frequency distribution. Subsequently, the expression \( (3 \times \text{Mode}) - (2 \times \text{Median}) \) must be computed. Step 2: Methodologies: 1. Mode Determination: The mode is identified as the data value exhibiting the highest frequency. 2. Median Determination: The median represents the central value in an ordered dataset. For an even number of observations (N), it is the average of the \( \left(\frac{N}{2}\right)^{th} \) and \( \left(\frac{N}{2} + 1\right)^{th} \) observations. A cumulative frequency table will be utilized to facilitate median calculation. Step 3: Execution: Mode Calculation: The frequencies provided are 6, 20, 24, 28, 15, 4, 2, 1. The maximum frequency observed is 28, which corresponds to the 'Marks obtained' value of 29. Therefore, Mode = 29. Median Calculation: The total count of students (N) is 100. As N is even, the median is the mean of the \( \frac{100}{2} = 50^{th} \) and the \( \frac{100}{2} + 1 = 51^{st} \) observations. The cumulative frequency (cf) table is as follows: \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 frequencies indicate that observations from the 27th to the 50th position have a score of 28 marks. Thus, the 50th observation is 28. Observations from the 51st to the 78th position have a score of 29 marks. Thus, the 51st observation is 29. Median = \( \frac{50^{th} \text{ observation} + 51^{st} \text{ observation}}{2} = \frac{28 + 29}{2} = \frac{57}{2} = 28.5 \). Therefore, Median = 28.5. Final Value Computation: Value = (3 \(\times\) Mode) - (2 \(\times\) Median) Value = (3 \(\times\) 29) - (2 \(\times\) 28.5) Value = 87 - 57 = 30. Step 4: Conclusion: The calculated value of (3 mode - 2 median) is 30.