Question

question 4 consider a data set of n distinct measurements. if the largest number were increased, what would be the effect on the median and mean? the mean would not change. increase. decrease. be impossible to be determined. the median would not change. increase. decrease. be impossible to be determined.

Explanation:

Step1: Recall mean formula

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. If the largest value $x_{max}$ is increased to $x_{max}+k$ ($k>0$), the new sum $\sum_{i = 1}^{n}x_{i}$ becomes $\sum_{i = 1}^{n}x_{i}+k$. Since $n$ is constant, the new mean $\bar{x}_{new}=\frac{\sum_{i = 1}^{n}x_{i}+k}{n}=\bar{x}+\frac{k}{n}$, so the mean increases.

Step2: Recall median concept

The median is the middle - value when the data is arranged in ascending or descending order. Increasing the largest value does not change the position of the middle - value (because the number of data points and their relative order (except for the largest one) remains the same), so the median does not change.

Answer:

The mean would: B. increase.
The median would: A. not change.