Here are the IQ scores of $10$randomly chosen fifth-grade students:

Which of the following statements about this data set is not true?

(a) The student with an IQ of $96$is considered an outlier by the $1.5\times IQR$rule.

(b) The five-number summary of the $10$IQ scores is $96,118,123.5,130,145$

(c) If the value $96$were removed from the data set, the mean of the remaining $9$ IQ scores would be higher than the mean of all $10$IQ scores.

(d) If the value $96$ were removed from the data set, the standard deviation of the remaining $9$ IQ scores would be lower than the standard deviation of all $10$IQ scores.

(e) If the value $96$were removed from the data set, the IQR of the remaining $9$IQ scores would be lower than the IQR of all $10$IQ scores.

Number of IQ scores$=10$

The IQ scores are$145,139,126,122,125,130,96,110,118,118$

Inter quartile range

$$\begin{gathered} IQR=Q_3-Q_1 \\ {Q}_3=3\times {(\frac{n+1}{4})}^{th}term \\ {Q}_1={(\frac{n+1}{4})}^{th}term \end{gathered}$$

Arranging in ascending order

$96,110,118,118,122,125,126,130,139,145$

The IQR for the $10$scores can be calculated as follows

$$\begin{gathered} Q_3=3\times {(\frac{10+1}{4})}^{th}term \\ {Q}_3=3\times {2.75}^{th}term={8.25}^{th}term \\ {Q}_3=8^{th}term+\frac{1}{4}\times (9^{th}term-8^{th}term) \\ {Q}_3=130+\frac{1}{4}\times (139-130)=132.25 \end{gathered}$$

The lower quartile is

$$\begin{gathered} Q_1={(\frac{10+1}{4})}^{th}term \\ {Q}_1={2.75}^{th}term \\ {Q}_1=2^{th}term+\frac{3}{4}\times (3^{th}term-2^{th}term) \\ {Q}_1=110+\frac{3}{4}\times (118-110)=116 \end{gathered}$$

Therefore the IQR for $10$scores is

$132.25-116=16.25$

After deleting $96$, find the IQR.

The updated set of results is as follows:

$110,118,118,122,125,126,130,139,145$

The IQR for the $9$scores can be calculated as follows

$$\begin{gathered} Q_3=3\times {(\frac{9+1}{4})}^{th}term \\ {Q}_3=3\times {2.5}^{th}term={7.5}^{th}term \\ {Q}_3=7^{th}term+\frac{1}{2}\times (8^{th}term-7^{th}term) \\ {Q}_3=130+\frac{1}{2}\times (139-130)=134.5 \end{gathered}$$

The lower quartile is

$$\begin{gathered} Q_1={(\frac{9+1}{4})}^{th}term \\ {Q}_1={2.5}^{th}term \\ {Q}_1=2^{th}term+\frac{1}{2}\times (3^{th}term-2^{th}term) \\ {Q}_1=118+\frac{1}{2}\times (118-118)=118 \end{gathered}$$

Therefore the IQR for $9$scores is

$134.5-118=16.5$

As a result, the IQR of scores after deleting $96(=16.5)$ is somewhat higher than the original$(=16.25)$ invalidating the subpart statement (e).