Discussion
For a production process, the sample ranges are found to be 1.2, 1.5, 1.1, 1.4 and 1.5. Suppose that the subgroup size is 5. Then, what will be the process standard deviation, given that \(A_{2} = 0.577\) and \(d_{2} = 2.326\) ?
Answer: D
The process standard deviation is 0.576.
The process standard deviation is estimated from the average range as \(\hat{\sigma} = \bar{R}/d_{2}\):
\[ \begin{aligned} \bar{R} &= \frac{1.2 + 1.5 + 1.1 + 1.4 + 1.5}{5} = 1.34 \\ \hat{\sigma} &= \frac{1.34}{2.326} \approx 0.576 \end{aligned} \]
\(A_{2}\) is used for the control limits of the \(\bar{X}\) chart, not for \(\sigma\); for example \(A_{2}\bar{R} = 0.577 \times 1.34 \approx 0.773\). None of 0.511, 2.463 or 2.322 equals \(\bar{R}/d_{2}\).
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