Construct a 3-sigma R-chart for the length in centimeters of a part from the following table. What is the upper control limit of your R-chart?

Sample # Observation 1 Observation 2 Observation 3 Observation 4
1 0.486 0.499 0.493 0.511
2 0.499 0.506 0.516 0.494
3 0.496 0.5 0.515 0.488
4 0.495 0.506 0.483 0.487
5 0.472 0.502 0.526 0.469
6 0.473 0.495 0.507 0.493
7 0.495 0.512 0.49 0.471
8 0.525 0.501 0.498 0.474
9 0.497 0.501 0.517 0.506
10 0.495 0.505 0.516 0.511

Question

Construct a 3-sigma R-chart for the length in centimeters of a part from the following table. What is the upper control limit of your R-chart?

Sample #	Observation 1	Observation 2	Observation 3	Observation 4
1	0.486	0.499	0.493	0.511
2	0.499	0.506	0.516	0.494
3	0.496	0.5	0.515	0.488
4	0.495	0.506	0.483	0.487
5	0.472	0.502	0.526	0.469
6	0.473	0.495	0.507	0.493
7	0.495	0.512	0.49	0.471
8	0.525	0.501	0.498	0.474
9	0.497	0.501	0.517	0.506
10	0.495	0.505	0.516	0.511

BlackTom AI · Accepted Answer

To determine the upper control limit (UCL) for an R-chart with samples of size n = 4, we first compute the moving ranges within each sample and then average them to get R-bar.

Option-by-option analysis:

Option 1: 2.28
- This value is the common D4 constant for an R-chart with n = 4, not the UCL itself. The UCL is computed as D4 times R-bar, so unless R-bar happened to be 1.0 (which is not the case here), the UCL will not equal D4 itself. In practice, you multiply 2.282 by R-bar, which will yield a far smaller number than 2.28 for typical small R-bar values. Therefore, 2.28 is not the UCL for this data.

Option 2: .032
- This value is numerically close to R-bar, the average range, not the UCL. The R-chart’s UCL is proportional to R-bar via D4, not equal to R-bar. Since R-bar is about 0.032, this option would correspond to R-bar, not the upper control limit. Thus this is not the UCL.

Option 3: 0.73
- If you calculate the UCL as D4 × R-bar with D4 = 2.282 and R-bar ≈ 0.0321, you get UCL ≈ 2.282 × 0.0321 ≈ 0.0733. That is roughly 0.073, not 0.73. The given value 0.73 would imply a tenfold overestimate of the true UCL, which would arise only if R-bar were about 0.32 or if a decimal point was misplaced. Based on the computed R-bar, 0.73 is not the correct UCL.

Option 4: 0.51
- This value is substantially larger than the calculated UCL (≈0.073). There is no reasonable route from the data and standard R-chart constants to yield a UCL around 0.51 for n = 4. Therefore, this option does not match the computed UCL.

Additional context and steps that lead to the correct conclusion:
- Compute each sample’s range: for each of the 10 samples (with 4 observations), the range is max minus min within that sample. The ten ranges are approximately: 0.025, 0.022, 0.027, 0.023, 0.057, 0.034, 0.041, 0.051, 0.020, 0.021.
- Average these ranges to obtain R-bar: (sum of ranges)/10 ≈ 0.321/10 = 0.0321 cm.
- Use the R-chart control constant for n = 4, D4 ≈ 2.282.
- Compute UCL = D4 × R-bar ≈ 2.282 × 0.0321 ≈ 0.0733 cm.

Thus, the technically correct UCL is about 0.073 cm, which does not exactly match any of the provided options. The closest option numerically is 0.73 if a decimal misplacement occurred, but that would be incorrect under standard calculation. The option 0.032 corresponds to R-bar, not the UCL. The other values (2.28 and 0.51) are not correct UCLs for these data under the usual R-chart formula.

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