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

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

Find the value of (3 mode - 2 median).

The correct answer is 30.

To find the value of $3\times \text{mode}-2\times \text{median}$, we need to determine the mode and the median of the given data distribution. Let's write down the marks and their corresponding frequencies (number of students):

Step 1: Find the Mode
The mode is the value that appears most frequently in the data set (the mark with the highest frequency).
Looking at the frequency table, the maximum frequency is 28, which corresponds to the marks 29.
Therefore, $\text{Mode}=29$.

Step 2: Find the Median
The total number of students is $N=100$, which is an even number.
For an even number of observations, the median is the average of the ${\left(\frac{N}{2}\right)}^{\text{th}}$ and ${\left(\frac{N}{2}+1\right)}^{\text{th}}$ observations.
Here, $\frac{N}{2}=50$ and $\frac{N}{2}+1=51$.
We need to find the 50th and 51st observations:
- From the cumulative frequency column, we see that the cumulative frequency up to marks 28 is exactly 50. This means the 50th observation is 28.
- The cumulative frequency for marks 29 starts from 51 up to 78. This means the 51st observation is 29.
Thus, the median is calculated as:
$\text{Median}=\frac{{50}^{\text{th}}\text{ observation}+{51}^{\text{st}}\text{ observation}}{2}$
$\text{Median}=\frac{28+29}{2}=28.5$

Step 3: Calculate the value of (3 mode - 2 median)
Substitute the values of mode and median into the expression:
$3\times \text{mode}-2\times \text{median}=3\left(29\right)-2\left(28.5\right)$
$3\times \text{mode}-2\times \text{median}=87-57=30$