Question

roger claims that the two statistics most likely to change greatly in a data set are the mean and the median. is roger’s claim correct? 1) yes, both the mean and median are likely to change greatly. 2) no, only the mean is likely to change greatly. 3) no, only the median is likely to change greatly. 4) no, neither the mean nor the median are likely to change greatly. 5) there is not enough information to determine if the mean or the median are likely to change greatly.

Explanation:

Step1: Understand mean and median properties

The mean is the sum of all data - values divided by the number of data - values. It is sensitive to extreme values. For example, in the data set \(1, 2, 3\), the mean is \(\frac{1 + 2+3}{3}=2\). If we change the data set to \(1, 2, 100\), the mean becomes \(\frac{1 + 2+100}{3}=\frac{103}{3}\approx34.33\), a large change.
The median is the middle - value when the data is arranged in ascending or descending order. For a small data set with an odd number of values like \(1, 2, 3\), the median is \(2\). If we change the data set to \(1, 2, 100\), the median is still \(2\). For a data set with an even number of values, the median is the average of the two middle - values. Extreme values do not affect the median as much as they affect the mean.

Step2: Evaluate Roger's claim

Since the mean is highly affected by extreme values and outliers, while the median is relatively robust to extreme values and outliers, only the mean is likely to change greatly when there are significant changes in the data set (such as the addition of extreme values).

Answer:

1. No, only the mean is likely to change greatly.