91Ó°ÊÓ

A datum that significantly differs from other observations is known as an outlier. The consideration or rejection of such a datum from the entire observation is found through the Grubbs test.

The Grubbs statistic G is mathematically presented as:

${\mathbf{G}}_{\mathbf{calculated}}\mathbf{=}\frac{\mathbf{âˆ\pounds }\mathbf{questionablevalue}\mathbf{-}\overline{\mathbf{x}}\mathbf{âˆ\pounds }}{\mathbf{s}}$

Here,

$\overline{\mathbf{x}}$is the mean of entire data set.

s is the standard deviation of entire data set.

If ${\mathbf{G}}_{\mathbf{calculated}}\mathbf{>}{\mathbf{G}}_{\mathbf{table}}$, then the questionable point should be discarded.

If${\mathbf{G}}_{\mathbf{calculated}}\mathbf{>}{\mathbf{G}}_{\mathbf{table}}$, then the questionable point should be retained.

Given data:

The given sets of results are 192, 216, 202, 195, and 204.

The questionable value is 216:

The mean of the given set of results is calculated as follows:

$$\begin{gathered} \overline{x}=\frac{{\displaystyle \underset{i}{∑x_i}}}{n} \\ =\frac{192+216+202+195+204}{5} \\ =\frac{1009}{5} \\ =201.8 \end{gathered}$$

The Standard deviation is calculated as follows:

$$\begin{gathered} s=\sqrt{\frac{{\displaystyle \underset{i}{∑{\left(x_i-\overline{x}\right)}^2}}}{n-1}} \\ =\sqrt{\frac{{\left(201.8-192\right)}^2+...........{\left(201.8-204\right)}^2}{\left(5-1\right)}} \\ =9.34 \end{gathered}$$

The G value is calculated by using the Grubbs test as shown below:

$$\begin{gathered} G_{calculated}=\frac{\left|questionablevalue-\overline{x}\right|}{s} \\ =\frac{\left|216-201.8\right|}{9.34} \\ =1.52 \end{gathered}$$

For five observations, the $G_{table}$value is 1.672.

Compare the result and see that ${\mathrm{G}}_{\mathrm{calculated}}(1.52)<{\mathrm{G}}_{\mathrm{table}}(1.672).$

Hence, the questionable value should be retained.