Problem 205

To analyze how the addition of a data point $2$ to the set $0$, $1$, $2$, $2$, $3$, $4$ affects various statistical measures, we'll examine each measure both before and after the addition.

Original Data Set: $0$, $1$, $2$, $2$, $3$, $4$

1. Mean (Average):
- Original: $\frac{0+1+2+2+3+4}{6} = \frac{12}{6} = 2$
- New Data Set (with additional $2$): $\frac{0+1+2+2+2+3+4}{7} = \frac{14}{7} = 2$
- Conclusion: Mean remains unchanged.

2. Median (Middle Value):
- Original: With six numbers, the median is the average of the 3rd and 4th numbers, $\frac{2+2}{2} = 2$
- New Data Set: With seven numbers, the median is the 4th number, which is $2$
- Conclusion: Median remains unchanged.

3. Mode (Most Frequent Value):
- Original: The number $2$ appears most frequently.
- New Data Set: The number $2$ still appears most frequently.
- Conclusion: Mode remains unchanged.

4. Variance (Measure of Dispersion):
- Original Variance Calculation:
$$\frac{1}{6}\left[(0-2)^2 + (1-2)^2 + (2-2)^2 \times 2 + (3-2)^2 + (4-2)^2\right] = \frac{1}{6}\left[4 + 1 + 0 + 1 + 4\right] = \frac{10}{6} = \frac{5}{3}$$
- New Data Set Variance Calculation:
$$\frac{1}{7}\left[(0-2)^2 + (1-2)^2 + (2-2)^2 \times 3 + (3-2)^2 + (4-2)^2\right] = \frac{1}{7}\left[4 + 1 + 0 + 1 + 4\right] = \frac{10}{7}$$
- Conclusion: Variance changes from $\frac{5}{3}$ to $\frac{10}{7}$.

Given these observations, the statistical measure that changes with the addition of another data point $2$ is the Variance.

Therefore, the correct answer is $\boxed{A}$.