which of the following statistics are changed by multiplying each data value by a constant (rescaling)…

Explanation:

Step1: Recall median property

If we multiply each data - value (x_i) in a data - set by a constant (c), the new data - set is (y_i = c\times x_i). The median of the original data - set is the middle value (or the average of the two middle values for an even - sized data - set). For the new data - set, the middle value(s) will also be multiplied by (c). So the median changes.

Step2: Recall mean property

The mean (\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}). The new mean (\bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}=\frac{\sum_{i = 1}^{n}(c\times x_i)}{n}=c\times\frac{\sum_{i = 1}^{n}x_i}{n}=c\times\bar{x}). So the mean changes.

Step3: Recall standard deviation property

The standard deviation (s_x=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}). The new standard deviation (s_y=\sqrt{\frac{\sum_{i = 1}^{n}(y_i-\bar{y})^2}{n - 1}}=\sqrt{\frac{\sum_{i = 1}^{n}(c\times x_i - c\times\bar{x})^2}{n - 1}}=\sqrt{\frac{c^2\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}=|c|\times s_x). So the standard deviation changes.

Step4: Recall IQR property

The inter - quartile range (IQR) is the difference between the third quartile (Q_3) and the first quartile (Q_1), (IQR = Q_3 - Q_1). When we multiply each data - value by (c), (Q_1) and (Q_3) are also multiplied by (c). The new (IQR_{new}=cQ_3 - cQ_1=c(Q_3 - Q_1)=c\times IQR_{old}). So the IQR changes.

Answer:

E. All of the above