All the students of a class performed poorly in physics. The teacher decided to give grace marks of 15 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 determine which statistical measure will not change after the teacher gives grace marks of 15 to all students, we will analyze the effects of this addition on various statistical measures: mean, median, mode, and standard deviation.

### Step-by-Step Solution:

1. **Understanding the Situation**:
- Let’s denote the original marks of the students as \( x_1, x_2, x_3, \ldots, x_n \).
- After adding 15 grace marks to each student, the new marks will be \( x_1 + 15, x_2 + 15, x_3 + 15, \ldots, x_n + 15 \).

2. **Calculating the Mean**:
- The original mean \( \mu \) is given by:
\[ \mu = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \]
- After adding 15 to each score, the new mean \( \mu' \) becomes:
\[ \mu' = \frac{(x_1 + 15) + (x_2 + 15) + (x_3 + 15) + \ldots + (x_n + 15)}{n} \]
\[ = \frac{(x_1 + x_2 + x_3 + \ldots + x_n) + 15n}{n} \]
\[ = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} + 15 \]
\[ = \mu + 15 \]
- Thus, the mean changes.

3. **Calculating the Median**:
- The median is the middle value when the data is sorted. Since we are adding the same number (15) to each score, the relative order of the scores remains unchanged. Therefore, the median will also increase by 15.
- Thus, the median changes.

4. **Calculating the Mode**:
- The mode is the most frequently occurring value in the dataset. Since we are adding the same number to all scores, the mode will also shift by 15.
- Thus, the mode changes.

5. **Calculating the Standard Deviation**:
- The standard deviation measures the spread of the data. It is calculated using the formula:
\[ \sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2} \]
- When we add a constant (15) to each score, the distances between the scores remain the same. Therefore, the standard deviation does not change.
- Thus, the standard deviation remains the same.

### Conclusion:
The statistical measure that will **not change** after the grace marks are given is the **standard deviation**.