Question

which of the following statistics are changed by multiplying each data value by a constant (rescaling)? choose the correct answer below. median mean standard deviation iqr all of the above

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