Question 2 of 7: Control Limits for the Tabulated Data

Question 2: Control Limits for the Tabulated Data (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Six subgroups of four measurements each, taken in production order on one quality characteristic. The paper prints the readings only, without naming the characteristic or its units, so they are carried through as recorded.

Find. The upper and lower control limits for the subgroup-average chart and for the subgroup-range chart, and a statement of whether the process is in statistical control.

Approach. Because the data arrive as small rational subgroups of a variable measurement, the appropriate pair is the Shewhart x̄–R chart: estimate the process centre from the grand average, estimate the within-subgroup spread from the average range, and convert that spread into three-sigma limits using the tabulated constants for n = 4.

1. Compute each subgroup average and range. For subgroup i, $$\bar{x}_i=\frac{1}{n}\sum_{j=1}^{n}x_{ij},\qquad R_i=\max_j x_{ij}-\min_j x_{ij}$$ Applying these to the six rows, for example subgroup 1 gives $\bar{x}_1=(0.55+0.60+0.57+0.55)/4=0.5675$ and $R_1=0.60-0.55=0.05$:

   Subgroup	1	2	3	4	5	6
   Average x̄i	0.5675	0.5800	0.5275	0.5275	0.5800	0.5925
   Range Ri	0.05	0.05	0.05	0.07	0.04	0.06

2. Average the subgroup statistics to locate the two centre lines. With k = 6 subgroups, $$\bar{\bar{x}}=\frac{1}{k}\sum_{i=1}^{k}\bar{x}_i=\frac{3.3750}{6}=0.5625, \qquad \bar{R}=\frac{1}{k}\sum_{i=1}^{k}R_i=\frac{0.32}{6}=0.05333$$ These are the centre lines of the average chart and the range chart respectively.
3. Read the Shewhart constants for the subgroup size. For n = 4 the standard tables (Montgomery, Appendix VI; identical values appear in ISO 7870-2) give $$A_2=0.729,\qquad D_3=0,\qquad D_4=2.282,\qquad d_2=2.059$$ $A_2$ converts an average range into the three-sigma half-width of the mean, $D_3$ and $D_4$ do the same job for the range, and $d_2$ converts an average range into an estimate of the process standard deviation.
4. Construct the limits for the subgroup-average chart. Substituting into $\mathrm{UCL}_{\bar{x}},\ \mathrm{LCL}_{\bar{x}}=\bar{\bar{x}}\pm A_2\bar{R}$, the half-width is $A_2\bar{R}=0.729\times 0.05333=0.03888$, so $$\boxed{\ \mathrm{UCL}_{\bar{x}}=0.5625+0.03888=0.6014,\qquad \mathrm{CL}_{\bar{x}}=0.5625,\qquad \mathrm{LCL}_{\bar{x}}=0.5625-0.03888=0.5236\ }$$
5. Construct the limits for the subgroup-range chart. The range limits are pure multiples of the average range, $\mathrm{UCL}_R=D_4\bar{R}$ and $\mathrm{LCL}_R=D_3\bar{R}$: $$\boxed{\ \mathrm{UCL}_{R}=2.282\times 0.05333=0.1217,\qquad \mathrm{CL}_{R}=0.05333,\qquad \mathrm{LCL}_{R}=0\times 0.05333=0\ }$$ The lower range limit is zero for every subgroup size up to six, which is why a range chart on small subgroups can never signal an improvement in spread; only the upper limit is informative.
6. Test the data against the limits. The range chart is read first, because the average chart borrows its limits from $\bar{R}$ and is meaningless if the spread itself is unstable. The largest range is $R_4=0.07$, comfortably below 0.1217, so the within-subgroup spread is stable. The subgroup averages run from 0.5275 to 0.5925, and both extremes lie inside the band $0.5236$ to $0.6014$. There is no run of seven on one side of the centre line, no trend, and no pattern of points hugging the limits, so the process is in statistical control and $\bar{R}$ is a valid basis for the limits just computed.
7. Convert the average range into a process standard deviation and natural tolerance. Now that control is established, the spread of individual pieces follows from $\hat{\sigma}=\bar{R}/d_2$: $$\hat{\sigma}=\frac{0.05333}{2.059}=0.02590$$ so the natural tolerance of the process, the band inside which essentially all individual items fall, is $$\bar{\bar{x}}\pm 3\hat{\sigma}=0.5625\pm 0.0777\ \Rightarrow\ 0.4848\ \text{to}\ 0.6402$$ This is the number to compare against a drawing tolerance when the specification is eventually supplied; the control limits themselves say nothing about conformance.