Chapter 2.2, Problem 60E

To determine

To find: the percent of observations which are considered as outliers.

Answer to Problem 60E

There exist $4.30\%$ of observations which are considered as outliers.

Explanation of Solution

Given:

  $1.5\times IQR$ rule

Calculation:

The quartiles for a standard Normal distribution are $\pm 0.6745$ . For a $N\left(\mu ,\sigma \right)$ distribution, the first and third quartiles are defined as follows:

  $Q_1=\mu -0.6745\sigma$ and $Q_3=\mu +0.6745\sigma$

And the inter-quartile range ( IQR ) is given by

  $\begin{array}{l}IQR=Q_3-Q_1\\ =\left(\mu +0.6745\sigma \right)-\left(\mu -0.6745\sigma \right)\\ =1.349\sigma \end{array}$

Therefore, $15\left(IQR\right)=2.0235\sigma$ ,

Then, the suspected outliers are below $Q_1-1.5\left(IQR\right)=\mu -2.0235\sigma$ or above $Q_1+1.5\left(IQR\right)=\mu +2.0235\sigma$

The proportion outside of this range is approximately the same as the area under the standard Normal distribution outside of the range from $-2.0235$ to $2.0235$ . The z values corresponding to $-2.0235$ to $2.0235$ are $0.02151$ . That is, there exists 2.151% of observations to the extreme left and right side of the distribution.

  $\begin{array}{l}2\left(0.02151\right)=0.043\\ =4.30\%\end{array}$

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

Conclusion:

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