To solve the problem, we need to analyze how the grace marks affect different statistical measures: mean, median, mode, and variance.
### Step 1: Understanding the Statistical Measures
- **Mean**: The average of all the marks.
- **Median**: The middle value when all marks are arranged in order.
- **Mode**: The most frequently occurring mark.
- **Variance**: A measure of how much the marks vary from the mean.
### Step 2: Assigning Marks
Let's assume we have 5 students with the following marks:
- \( x_1, x_2, x_3, x_4, x_5 \) (where \( x_1 < x_2 < x_3 < x_4 \leq x_5 \))
### Step 3: Calculating the Mean
The mean before the grace marks is:
\[
\text{Mean} = \frac{x_1 + x_2 + x_3 + x_4 + x_5}{5}
\]
After adding 12 grace marks to each student, the new marks become:
- \( x_1 + 12, x_2 + 12, x_3 + 12, x_4 + 12, x_5 + 12 \)
The new mean will be:
\[
\text{New Mean} = \frac{(x_1 + 12) + (x_2 + 12) + (x_3 + 12) + (x_4 + 12) + (x_5 + 12)}{5} = \frac{x_1 + x_2 + x_3 + x_4 + x_5 + 60}{5} = \text{Mean} + 12
\]
Thus, the mean changes.
### Step 4: Calculating the Median
The median before the grace marks is:
\[
\text{Median} = x_3
\]
After the grace marks, the new median will be:
\[
\text{New Median} = x_3 + 12
\]
Thus, the median changes.
### Step 5: Calculating the Mode
The mode before the grace marks is the most frequently occurring mark, which we can denote as \( x_4 \) (if it occurs most often). After adding grace marks, the new mode will be:
\[
\text{New Mode} = x_4 + 12
\]
Thus, the mode changes.
### Step 6: Calculating the Variance
Variance is calculated as:
\[
\text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{n}
\]
After adding 12 to each score, the new variance will be:
\[
\text{New Variance} = \frac{\sum ((x_i + 12) - (\text{Mean} + 12))^2}{n} = \frac{\sum (x_i - \text{Mean})^2}{n}
\]
Thus, the variance remains unchanged.
### Conclusion
The statistical measure that does not change after adding grace marks is **Variance**.
