Question

b. when 12 is removed from the data set, does mean remain the same? explain your reasoning.

Explanation:

The mean of a data - set is calculated as the sum of all data points divided by the number of data points. If 12 is removed from the data - set, whether the mean remains the same depends on the value of 12 relative to the original mean. If 12 is equal to the original mean of the data - set, then removing it will not change the mean. Because when we calculate the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, if we remove a value $x_j$ such that $x_j=\bar{x}$, the new sum $\sum_{i
eq j}x_{i}=\sum_{i = 1}^{n}x_{i}-x_j$ and the new number of data points is $n - 1$. Substituting $x_j=\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$ into the new - mean formula $\bar{y}=\frac{\sum_{i
eq j}x_{i}}{n - 1}$, after algebraic manipulation, we can find that $\bar{y}=\bar{x}$. Otherwise, the mean will change.

Answer:

It depends. If 12 is equal to the original mean of the data - set, the mean remains the same; otherwise, it changes.