Step 1: Understanding the Concept:
The concept of the "average" (or arithmetic mean) is one of the most fundamental tools in statistics used to represent a central value of a dataset.
Mathematically, the average is defined as the sum of all numerical values in a set divided by the total number of items or observations in that set.
In this specific problem, we are dealing with a dynamic change in a group. When one member (a student) is removed or "excluded" from the group, two things happen simultaneously: the total sum of the marks decreases by the amount of that student's marks, and the total number of students decreases by exactly one.
The change in the average value provides us with a hint about the relative value of the excluded item. If the average drops after an exclusion, it signifies that the excluded value was higher than the original average, effectively "pulling down" the mean of the remaining group once it left.
This problem requires us to use the "Total Sum" method, which relies on the principle that the total sum of a group is conserved through its components. By calculating the total marks before and after the student left, we can isolate the individual value of the excluded student through simple subtraction.
Step 2: Key Formula or Approach:
To solve this, we utilize the following standard mathematical relationships:
1. \[ \text{Total Sum} = \text{Average} \times \text{Number of Observations} \]
2. \[ \text{Excluded Value} = \text{Initial Total Sum} - \text{New Total Sum} \]
Step 3: Detailed Explanation:
Let's perform the calculation in a detailed, stepwise manner to ensure no errors in computation.
First, we analyze the "Initial State" of the class.
The number of students is given as 12, and their collective average mark is 68.
To find the total marks scored by all these 12 students combined, we multiply the count by the average:
\[ \text{Initial Total Sum} = 12 \times 68 \]
To calculate this, we can break it down: \(12 \times 60 = 720\) and \(12 \times 8 = 96\). Summing these gives:
\[ \text{Initial Total Sum} = 720 + 96 = 816 \]
Next, we analyze the "New State" after one student is excluded.
The number of remaining students is now \(12 - 1 = 11\).
The problem states that the average for these remaining 11 students has dropped to 66.
Now, we calculate the total marks for this new group:
\[ \text{New Total Sum} = 11 \times 66 \]
Using the multiplication rule for 11 (\(6 \dots (6+6) \dots 6\)), we find:
\[ \text{New Total Sum} = 726 \]
The difference between the original total sum (with 12 students) and the new total sum (with 11 students) must be exactly equal to the marks of the student who was removed from the calculation.
\[ \text{Excluded Student's Marks} = 816 - 726 \]
Subtracting 726 from 816:
\[ 816 - 700 = 116 \]
\[ 116 - 26 = 90 \]
Thus, the specific marks belonging to the excluded student are 90.
Step 4: Final Answer:
The excluded student’s marks are 90, which corresponds to option (c).