All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 12 to the entire class. Which of the following statistical measures will not change even after the grace marks were given?

Options

A. Median
B. Mode
C. Variance
D. Mean

Answer

To solve the problem, we need to analyze how the grace marks affect various statistical measures: mean, median, mode, and variance.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a class of students who performed poorly in Mathematics, and each student receives an additional 12 grace marks.
- We need to determine which statistical measure will remain unchanged after adding these grace marks.

2. **Assuming Initial Marks**:
- Let the marks of the students be represented as \( x_1, x_2, x_3, \ldots, x_n \).
- After adding the grace marks, the new marks will be \( x_1 + 12, x_2 + 12, x_3 + 12, \ldots, x_n + 12 \).

3. **Calculating the Mean**:
- The mean of the original marks is given by:
\[ \text{Mean} = \bar{x} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \]
- After adding 12 to each mark, the new mean becomes:
\[ \text{New Mean} = \frac{(x_1 + 12) + (x_2 + 12) + (x_3 + 12) + \ldots + (x_n + 12)}{n} = \frac{x_1 + x_2 + x_3 + \ldots + x_n + 12n}{n} = \bar{x} + 12 \]
- Therefore, the mean changes.

4. **Calculating the Median**:
- The median is the middle value when the data is arranged in ascending order.
- If the original median is \( M \), after adding 12 to each mark, the new median will be \( M + 12 \).
- Thus, the median also changes.

5. **Calculating the Mode**:
- The mode is the value that appears most frequently in the data set.
- If a particular mark \( m \) was the mode before, after adding 12, the new mode will be \( m + 12 \).
- Hence, the mode changes as well.

6. **Calculating the Variance**:
- Variance measures the spread of the data and is calculated as:
\[ \text{Variance} = \sigma^2 = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + (x_3 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n} \]
- After adding 12 to each mark, the deviations from the mean change, but the spread remains the same:
\[ \text{New Variance} = \frac{((x_1 + 12) - (\bar{x} + 12))^2 + ((x_2 + 12) - (\bar{x} + 12))^2 + \ldots}{n} = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots}{n} = \sigma^2 \]
- Therefore, the variance does not change.

### Conclusion:
The statistical measure that will not change after giving grace marks of 12 to the entire class is **Variance**.