Question

question #6
a teacher writes a data set on the board.
4, 11, 9, 7, 12, 18
the teacher asks two students to determine which measure will remain the same if each number in the data set is increased by 7. henrik claims the interquartile range will remain the same. lorelei claims the standard deviation will remain the same.
who is correct?
a both henrik and lorelei
b neither henrik nor lorelei
c only henrik
d only lorelei

Explanation:

Step1: Recall properties of inter - quartile range

The inter - quartile range (IQR) is $Q_3 - Q_1$. When each data - point in a dataset is increased by a constant $c$, the first quartile $Q_1$ and the third quartile $Q_3$ are both increased by $c$. So, $IQR_{new}=(Q_3 + c)-(Q_1 + c)=Q_3 - Q_1 = IQR_{old}$.

Step2: Recall properties of standard deviation

The standard deviation $\sigma=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n}}$. When each data - point $x_i$ is increased by a constant $c$, the mean $\bar{x}$ also increases by $c$. Then $(x_i + c)-(\bar{x}+c)=x_i-\bar{x}$. So, the standard deviation remains the same.

Answer:

A. both Henrik and Lorelei