A quality engineer sets up an average and range chart for a machining process. Given grand average (xbb) = 70.1000; control chart factor A2 (A2) = 0.6500; average range (Rbar) = 3.6000, determine the upper control limit (UCL).

Start with the thinking

• The governing relation printed in this handbook section is Average and range control chart limits.
• Everything except UCL is given, so isolate UCL symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Average and range charts set control limits on the process mean using the grand average and average range.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

2. Step 2 — Rearrange symbolically for UCL:

   $$UCL = \bar{\bar{x}} + A_2\bar{R}$$

3. Step 3 — List the givens: grand average (xbb) = 70.1000, control chart factor A2 (A2) = 0.6500, average range (Rbar) = 3.6000.

4. Step 4 — Substitute the given values:

   $$UCL = \bar{\bar{x}} + A_2\bar{R}$$

5. Step 5 — Evaluate:

   $$UCL = 72.4400$$

6. Step 6 — Check: returning UCL = 72.4400 to

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 144.9 — kept a factor of two that cancels in the correct rearrangement.
• 36.2200 — dropped that same factor in the other direction.
• 79.6840 — rounded an intermediate value before the final step.

Reference: FE Handbook — Average and Range Charts

An analyst computes the upper control limit for an average and range chart. Given control chart factor A2 (A2) = 0.7300; average range (Rbar) = 15.5000; upper control limit (UCL) = 62.2000, determine the grand average (xbb).

Start with the thinking

• The governing relation printed in this handbook section is Average and range control chart limits.
• Everything except xbb is given, so isolate xbb symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Average and range charts set control limits on the process mean using the grand average and average range.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

2. Step 2 — Rearrange symbolically for xbb:

   $$xbb = UCL - A_2\bar{R}$$

3. Step 3 — List the givens: control chart factor A2 (A2) = 0.7300, average range (Rbar) = 15.5000, upper control limit (UCL) = 62.2000.

4. Step 4 — Substitute the given values:

   $$xbb = 62.2000 - A_2\bar{R}$$

5. Step 5 — Evaluate:

   $$xbb = 50.8850$$

6. Step 6 — Check: returning xbb = 50.8850 to

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 101.8 — kept a factor of two that cancels in the correct rearrangement.
• 25.4425 — dropped that same factor in the other direction.
• 55.9735 — rounded an intermediate value before the final step.

Reference: FE Handbook — Average and Range Charts

The average and range chart signals a process shift beyond the control limit. Given grand average (xbb) = 85.7000; control chart factor A2 (A2) = 0.6600; upper control limit (UCL) = 92.0000, determine the average range (Rbar).

Start with the thinking

• The governing relation printed in this handbook section is Average and range control chart limits.
• Everything except Rbar is given, so isolate Rbar symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Average and range charts set control limits on the process mean using the grand average and average range.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

2. Step 2 — Rearrange symbolically for Rbar:

   $$Rbar = \dfrac{UCL - \bar{\bar{x}}}{A_2}$$

3. Step 3 — List the givens: grand average (xbb) = 85.7000, control chart factor A2 (A2) = 0.6600, upper control limit (UCL) = 92.0000.

4. Step 4 — Substitute the given values:

   $$Rbar = \dfrac{92.0000 - \bar{\bar{x}}}{A_2}$$

5. Step 5 — Evaluate:

   $$Rbar = 9.5455$$

6. Step 6 — Check: returning Rbar = 9.5455 to

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 19.0909 — kept a factor of two that cancels in the correct rearrangement.
• 4.7727 — dropped that same factor in the other direction.
• 10.5000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Average and Range Charts

A quality engineer sets up an average and range chart for a machining process. Given grand average (xbb) = 56.5000; control chart factor A2 (A2) = 0.7000; average range (Rbar) = 5.9000, determine the upper control limit (UCL).

Start with the thinking

• The governing relation printed in this handbook section is Average and range control chart limits.
• Everything except UCL is given, so isolate UCL symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Average and range charts set control limits on the process mean using the grand average and average range.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

2. Step 2 — Rearrange symbolically for UCL:

   $$UCL = \bar{\bar{x}} + A_2\bar{R}$$

3. Step 3 — List the givens: grand average (xbb) = 56.5000, control chart factor A2 (A2) = 0.7000, average range (Rbar) = 5.9000.

4. Step 4 — Substitute the given values:

   $$UCL = \bar{\bar{x}} + A_2\bar{R}$$

5. Step 5 — Evaluate:

   $$UCL = 60.6300$$

6. Step 6 — Check: returning UCL = 60.6300 to

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 121.3 — kept a factor of two that cancels in the correct rearrangement.
• 30.3150 — dropped that same factor in the other direction.
• 66.6930 — rounded an intermediate value before the final step.

Reference: FE Handbook — Average and Range Charts

An analyst computes the upper control limit for an average and range chart. Given control chart factor A2 (A2) = 0.4100; average range (Rbar) = 8.6000; upper control limit (UCL) = 72.5000, determine the grand average (xbb).

Start with the thinking

• The governing relation printed in this handbook section is Average and range control chart limits.
• Everything except xbb is given, so isolate xbb symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Average and range charts set control limits on the process mean using the grand average and average range.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

2. Step 2 — Rearrange symbolically for xbb:

   $$xbb = UCL - A_2\bar{R}$$

3. Step 3 — List the givens: control chart factor A2 (A2) = 0.4100, average range (Rbar) = 8.6000, upper control limit (UCL) = 72.5000.

4. Step 4 — Substitute the given values:

   $$xbb = 72.5000 - A_2\bar{R}$$

5. Step 5 — Evaluate:

   $$xbb = 68.9740$$

6. Step 6 — Check: returning xbb = 68.9740 to

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 137.9 — kept a factor of two that cancels in the correct rearrangement.
• 34.4870 — dropped that same factor in the other direction.
• 75.8714 — rounded an intermediate value before the final step.

Reference: FE Handbook — Average and Range Charts

The average and range chart signals a process shift beyond the control limit. Given grand average (xbb) = 53.9000; control chart factor A2 (A2) = 0.3200; upper control limit (UCL) = 82.7300, determine the average range (Rbar).

Start with the thinking

• The governing relation printed in this handbook section is Average and range control chart limits.
• Everything except Rbar is given, so isolate Rbar symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Average and range charts set control limits on the process mean using the grand average and average range.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

2. Step 2 — Rearrange symbolically for Rbar:

   $$Rbar = \dfrac{UCL - \bar{\bar{x}}}{A_2}$$

3. Step 3 — List the givens: grand average (xbb) = 53.9000, control chart factor A2 (A2) = 0.3200, upper control limit (UCL) = 82.7300.

4. Step 4 — Substitute the given values:

   $$Rbar = \dfrac{82.7300 - \bar{\bar{x}}}{A_2}$$

5. Step 5 — Evaluate:

   $$Rbar = 90.0938$$

6. Step 6 — Check: returning Rbar = 90.0938 to

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 180.2 — kept a factor of two that cancels in the correct rearrangement.
• 45.0469 — dropped that same factor in the other direction.
• 99.1031 — rounded an intermediate value before the final step.

Reference: FE Handbook — Average and Range Charts

A quality engineer sets up an average and range chart for a machining process. Given grand average (xbb) = 78.5000; control chart factor A2 (A2) = 0.3400; average range (Rbar) = 4.0000, determine the upper control limit (UCL).

Start with the thinking

• The governing relation printed in this handbook section is Average and range control chart limits.
• Everything except UCL is given, so isolate UCL symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Average and range charts set control limits on the process mean using the grand average and average range.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

2. Step 2 — Rearrange symbolically for UCL:

   $$UCL = \bar{\bar{x}} + A_2\bar{R}$$

3. Step 3 — List the givens: grand average (xbb) = 78.5000, control chart factor A2 (A2) = 0.3400, average range (Rbar) = 4.0000.

4. Step 4 — Substitute the given values:

   $$UCL = \bar{\bar{x}} + A_2\bar{R}$$

5. Step 5 — Evaluate:

   $$UCL = 79.8600$$

6. Step 6 — Check: returning UCL = 79.8600 to

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 159.7 — kept a factor of two that cancels in the correct rearrangement.
• 39.9300 — dropped that same factor in the other direction.
• 87.8460 — rounded an intermediate value before the final step.

Reference: FE Handbook — Average and Range Charts

An analyst computes the upper control limit for an average and range chart. Given control chart factor A2 (A2) = 0.4300; average range (Rbar) = 16.0000; upper control limit (UCL) = 60.4000, determine the grand average (xbb).

Start with the thinking

• The governing relation printed in this handbook section is Average and range control chart limits.
• Everything except xbb is given, so isolate xbb symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Average and range charts set control limits on the process mean using the grand average and average range.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

2. Step 2 — Rearrange symbolically for xbb:

   $$xbb = UCL - A_2\bar{R}$$

3. Step 3 — List the givens: control chart factor A2 (A2) = 0.4300, average range (Rbar) = 16.0000, upper control limit (UCL) = 60.4000.

4. Step 4 — Substitute the given values:

   $$xbb = 60.4000 - A_2\bar{R}$$

5. Step 5 — Evaluate:

   $$xbb = 53.5200$$

6. Step 6 — Check: returning xbb = 53.5200 to

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 107.0 — kept a factor of two that cancels in the correct rearrangement.
• 26.7600 — dropped that same factor in the other direction.
• 58.8720 — rounded an intermediate value before the final step.

Reference: FE Handbook — Average and Range Charts

The average and range chart signals a process shift beyond the control limit. Given grand average (xbb) = 67.3000; control chart factor A2 (A2) = 0.7400; upper control limit (UCL) = 98.5200, determine the average range (Rbar).

Start with the thinking

• The governing relation printed in this handbook section is Average and range control chart limits.
• Everything except Rbar is given, so isolate Rbar symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Average and range charts set control limits on the process mean using the grand average and average range.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

2. Step 2 — Rearrange symbolically for Rbar:

   $$Rbar = \dfrac{UCL - \bar{\bar{x}}}{A_2}$$

3. Step 3 — List the givens: grand average (xbb) = 67.3000, control chart factor A2 (A2) = 0.7400, upper control limit (UCL) = 98.5200.

4. Step 4 — Substitute the given values:

   $$Rbar = \dfrac{98.5200 - \bar{\bar{x}}}{A_2}$$

5. Step 5 — Evaluate:

   $$Rbar = 42.1892$$

6. Step 6 — Check: returning Rbar = 42.1892 to

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 84.3784 — kept a factor of two that cancels in the correct rearrangement.
• 21.0946 — dropped that same factor in the other direction.
• 46.4081 — rounded an intermediate value before the final step.

Reference: FE Handbook — Average and Range Charts

A quality engineer sets up an average and range chart for a machining process. Given grand average (xbb) = 53.5000; control chart factor A2 (A2) = 0.7100; average range (Rbar) = 4.1000, determine the upper control limit (UCL).

Start with the thinking

• The governing relation printed in this handbook section is Average and range control chart limits.
• Everything except UCL is given, so isolate UCL symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Average and range charts set control limits on the process mean using the grand average and average range.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

2. Step 2 — Rearrange symbolically for UCL:

   $$UCL = \bar{\bar{x}} + A_2\bar{R}$$

3. Step 3 — List the givens: grand average (xbb) = 53.5000, control chart factor A2 (A2) = 0.7100, average range (Rbar) = 4.1000.

4. Step 4 — Substitute the given values:

   $$UCL = \bar{\bar{x}} + A_2\bar{R}$$

5. Step 5 — Evaluate:

   $$UCL = 56.4110$$

6. Step 6 — Check: returning UCL = 56.4110 to

   $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 112.8 — kept a factor of two that cancels in the correct rearrangement.
• 28.2055 — dropped that same factor in the other direction.
• 62.0521 — rounded an intermediate value before the final step.

Reference: FE Handbook — Average and Range Charts