91Ó°ÊÓ

The percent of the observations that are classified as outliers by the rule $=1.5×IQR$

In a standard Normal distribution, the first and third quartiles are $Â\pm 0.6745.forN(\mathrm{μ},\mathrm{σ})$.

The first and third quartiles of a $N(\mathrm{μ},\mathrm{σ})$distribution are defined as follows:

$Q_1=\mathrm{μ}-0.6745\mathrm{σ}And{Q}_3=\mathrm{μ}+0.6745\mathrm{σ}$

IQR is defined as the interquartile range

localid="1649998096448" $$\begin{gathered} IQR=Q_3-Q_1 \\ =1.349 \end{gathered}$$

Hence, $1.5\left(IQR\right)=2.0235\mathrm{σ}$

The suspected outliers are below $Q_1-1.5\left(IQR\right)=\mathrm{μ}-2.023\mathrm{σ}$or above $Q_1+1.5\left(IQR\right)=\mathrm{μ}+2.023\mathrm{σ}$

Within the range of $-2.0235to2.0235$the proportion outside of this range is approximately equal to the area under the standard Normal distribution outside of this range.

The corresponding $z-$values to $-2.0235to2.0235are0.2151$.

This means that there are $2.151\%$of observations on the extreme left and right of the distribution.

localid="1649998118762" $$\begin{gathered} 2(0.02151)=0.043 \\ =4.30\% \end{gathered}$$

Hence, there exists $4.30\%$of observations are considered as outliers.