Question

Should the value 216 be rejected from the set of results 192,216,202,195, and 204?

Short answer

The value 216 should not be rejected but has to be retained in the given results.

Step-by-step solution

Grubbs Test for an Outlier

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:

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

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.

Determine whether the value 216 can be rejected

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:

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

The Standard deviation is calculated as follows:

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

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.