91Ó°ÊÓ

Control limits are calculated using the interquartile range\((IQR)\)

\(IQR = \left\{ {\begin{aligned}{*{20}{l}}{(2nd{\rm{ largest }}}&{\left. {{x_i}} \right) - \left( {2nd{\rm{ smallest }}{x_i}} \right)}&{,n = 4,5,6,7}\\{(3rd{\rm{ largest }}}&{\left. {{x_i}} \right) - \left( {3rd{\rm{ smallest }}{x_i}} \right)}&{,n = 8,9,10,11}\end{aligned}} \right.\)

For a random sample from a normal distribution, \(E(IQR) = {k_n}\sigma \); where the values\({k_n}\)are given by the table.

Control Limits for Robust Control Chart

\(\begin{aligned}{l}LCL = \bar \bar x - 3 \cdot \frac{{\overline {IQR} }}{{{k_n}\sqrt n }}\\UCL = \bar \bar x + 3 \cdot \frac{{\overline {IQR} }}{{{k_n}\sqrt n }}\end{aligned}\)

Now we can calculate the values. There are\(22\)values of IQR and we average them (for each sample), for example, for the first sample we have second largest is\(13\)and second smallest is\(12.8\), therefore the\(IQR\)is\(0.8\). We do that for all and average them (sum all IQR we get and divine with\(22\)) So

\(\overline {IQR} = 0.4318.\)

From the table in the book,

\({k_5} = 0.99\)

and we calculate

\(\bar \bar x = \frac{1}{{22}}\sum\limits_{i = 1}^{22} {{{\bar x}_i}} = \frac{1}{{22}}(12.72 + 12.80 + \ldots + 12.94) = 12.95\)

Therefore, Control Limits for Robust Control Chart

\(\begin{aligned}{l}LCL = \bar \bar x - 3 \cdot \frac{{\overline {IQR} }}{{{k_n}\sqrt n }} = 12.95 - 3 \cdot \frac{{0.4318}}{{0.99\sqrt 5 }} = 12.95 - 0.5852 = 12.3648\\UCL = \bar \bar x + 3 \cdot \frac{{\overline {IQR} }}{{{k_n}\sqrt n }} = 12.95 + 3 \cdot \frac{{0.4318}}{{0.99\sqrt 5 }} = 12.95 + 0.5852 = 13.5352\end{aligned}\)

Finally, the Control Limits Based on the Sample Standard Deviations

\(\begin{aligned}{l}LCL = \bar \bar x - 3 \cdot \frac{{\bar s}}{{{a_n}\sqrt n }} = 59.6683 - 3 \cdot \frac{{0.0108}}{{0.7979\sqrt 2 }} = 59.6395\\UCL = \bar \bar x + 3 \cdot \frac{{\bar s}}{{{a_n}\sqrt n }} = 59.6683 + 3 \cdot \frac{{0.0108}}{{0.7979\sqrt 2 }} = 59.6971\end{aligned}\)

\(2\)-sigma limits are

\(\begin{aligned}{l}\bar \bar x - 2 \cdot \frac{{\bar s}}{{{a_n}\sqrt n }} = 59.6683 - 2 \cdot \frac{{0.0108}}{{0.7979\sqrt 2 }} = 59.6491;\\\bar \bar x + 2 \cdot \frac{{\bar s}}{{{a_n}\sqrt n }} = 59.6683 + 2 \cdot \frac{{0.0108}}{{0.7979\sqrt 2 }} = 59.6875\end{aligned}\)

and \(1\) -sigma limits are

\(\begin{aligned}{l}\bar \bar x - \frac{{\bar s}}{{{a_n}\sqrt n }} = 59.6683 - \frac{{0.0108}}{{0.7979\sqrt 2 }} = 59.6587;\\\bar \bar x + \frac{{\bar s}}{{{a_n}\sqrt n }} = 59.6683 + \frac{{0.0108}}{{0.7979\sqrt 2 }} = 59.6779\end{aligned}\)

Look at the first rule: "On the same side of the centre line, two out of three consecutive points fall outside 2-sigma limits." There are no points that are outside of the 2-sigma range.

Consider the second rule: "On the same side of the centre line, four out of five consecutive points fall outside l-sigma limitations." We can see from the graph that this does not occur.

"Eight consecutive points fall on the same side of the centre line," says the third rule. The following points are all on the same side of the centre line. Processes appear to be under control.