Question:
The relation between the expected value of R and standard deviation \(\sigma\) is given by:
The relation between the expected value of R and standard deviation \(\sigma\) is given by:
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Small-case \(d_2\) links E(R) to \(\sigma\); capital \(D_1, D_2\) are for the R-chart's own control limits.
Updated On: Jul 4, 2026
- \(E(R) = d_1 \sigma\)
- \(E(R) = d_2 \sigma\)
- \(E(R) = D_1 \sigma\)
- \(E(R) = D_2 \sigma\)
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The Correct Option is B
Solution and Explanation
Step 1: In control chart theory, for a sample of size \(n\) drawn from a normal population with standard deviation \(\sigma\), the sample range \(R\) is a random variable whose distribution depends on \(n\) and \(\sigma\).
Step 2: The relative range \(W = \frac{R}{\sigma}\) has a sampling distribution that depends only on the sample size \(n\), and its expected value is a tabulated constant denoted \(d_2\), so \[ E(W) = E\left(\frac{R}{\sigma}\right) = d_2 \]
Step 3: Rearranging this gives the working relation \[ E(R) = d_2 \, \sigma \] which is why the estimate of \(\sigma\) from the range is taken as \( \hat{\sigma} = \frac{\bar{R}}{d_2} \) in constructing R-charts and X-bar charts.
Step 4: The constants \(D_1, D_2\) (with capital D) are instead used for the lower and upper control limits of the R-chart itself, \(D_1 \sigma\) and \(D_2 \sigma\) being the lower and upper 3-sigma limits for R, not the expected value of R. So the correct relation for the expected value is \(E(R) = d_2\sigma\).
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