Question

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 greater 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 less 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 less than the IQR of all $10$IQ scores.

Short answer

The correct option is (e).

Step-by-step solution

Step 1. Given information

Given IQ score:

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

Step 2. Explanation

For option (a), order the data values from smallest to largest:

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

The median is the middle value of the data. And since the number of data is even then:

$$\begin{gathered} M=Q_2=\frac{122+125}{2} \\ =\frac{247}{2} \\ =123.5 \end{gathered}$$

The first quartile is the median of the data values below the median i.e.

$Q_1=118$

The third quartile is the median of the data values above the median i.e.

$Q_3=130$

Now, the interquartile range will be:

$$\begin{gathered} IQR=Q_3-Q_1 \\ =130-118 \\ =12 \end{gathered}$$

Outliers are observations that are more than $1.5$times the IQR above $Q_{3}orbelow{Q}_1.$

$$\begin{gathered} Q_3+1.5IQR=130+1.5\left(12\right)=148 \\ {Q}_1-1.5IQR=118-1.5\left(12\right)=100 \end{gathered}$$

Since$96$ lies below $100$,$96$ is considered an outlier by the$1.5$ times interquartile range and thus the option (a) is true.

Step 3. Explanation

For option(b), the five summary consists of the minimum, first quartile, median, third quartile and the maximum, which are all the same thus the option (b) is true.

Step 4. Explanation

For option (c), $96$is the smallest value in the data set. We then expect the mean to increase if$96$ is removed from the data set, as the centre of the data will become higher when the minimum is removed and thus, option (c) is true.

Step 5. Explanation

For option (d), the standard deviation measures the amount of variation or the spread of the data set. $96$is the smallest value in the data set. We then expect the standard deviation to increase if$96$ is removed from the data set, as the spread decrease when the minimum is removed and thus, option (d) is true.

Step 6. Explanation

For option (e), $96$is the smallest value in the data set and from part (a) we know that$96$ is an outlier. We then expect the IQR to not be affected if $96$is removed from the data as the IQR should not be influenced by the outlier and thus option (e) is not true.

Thus, we will select the option (e) as correct answer.