Statistical Process Control (SPC) Charts: X-bar, R, p, c

These charts are graphical tools used in Statistical Process Control (SPC) to monitor processes over time, ensuring they are stable and operating within desired limits. They help distinguish between normal process variation (common cause) and variation due to specific events (special cause).

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1. X-bar (Average) Chart

• Purpose: To monitor the central tendency or average of a process based on measurable data (variables). It helps detect shifts in the process mean.
• Data Type: Variable data (measurements like weight, length, volume, time).
• Used in Conjunction: Often used together with an R chart or S chart, which monitor process variability.

Formulas

You typically collect data in small subgroups (samples) of size 'n' over time. Let 'k' be the number of subgroups.

1. Calculate the average (X̄ᵢ) for each subgroup i: X̄ᵢ = (Sum of measurements in subgroup i) / n
2. Calculate the overall average (Grand Average, X̿), which is the Center Line (CL) for the X-bar chart: X̿ = (Sum of all subgroup averages) / k = (X̄₁ + X̄₂ + ... + X̄<0xE2><0x82><0x96>) / k
3. Calculate the range (Rᵢ) for each subgroup i: Rᵢ = (Maximum measurement in subgroup i) - (Minimum measurement in subgroup i)
4. Calculate the average range (R̄): R̄ = (Sum of all subgroup ranges) / k = (R₁ + R₂ + ... + R<0xE2><0x82><0x96>) / k
5. Calculate the Control Limits (UCL and LCL) for the X-bar chart:
   • UCL = X̿ + A₂ * R̄
   • LCL = X̿ - A₂ * R̄ (Note: The factor A₂ depends on the subgroup size 'n' and is found in standard SPC constant tables).

Example Problem: X-bar Chart

A bottling plant wants to monitor the fill volume of 1-liter juice bottles. They take samples of 5 bottles (n=5) every hour for 4 hours (k=4).

Data:

Subgroup	Bottle 1	Bottle 2	Bottle 3	Bottle 4	Bottle 5	Subgroup Avg (X̄ᵢ)	Subgroup Range (Rᵢ)
1	1.01	0.99	1.03	1.00	1.02	1.010	0.04
2	0.98	1.00	1.01	1.02	0.99	1.000	0.04
3	1.02	1.04	1.00	1.01	1.03	1.020	0.04
4	0.99	1.01	0.98	1.00	1.02	1.000	0.04
Sums						4.030	0.16

Calculations:

• Subgroup Averages (X̄ᵢ) and Ranges (Rᵢ): Calculated in the table.
• Grand Average (Center Line for X-bar Chart): X̿ = 4.030 / 4 = 1.0075 L
• Average Range: R̄ = 0.16 / 4 = 0.04 L
• Control Limits for X-bar Chart:
  • From SPC tables, for n=5, the factor A₂ = 0.577.
  • UCL = 1.0075 + (0.577 * 0.04) = 1.0075 + 0.02308 = 1.0306 L
  • LCL = 1.0075 - (0.577 * 0.04) = 1.0075 - 0.02308 = 0.9844 L

Plotting: Plot the subgroup averages (1.010, 1.000, 1.020, 1.000) on a chart with CL=1.0075, UCL=1.0306, and LCL=0.9844. All points are within the limits.

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2. R (Range) Chart

• Purpose: To monitor the process variability or dispersion within subgroups based on measurable data (variables). It helps detect changes in process consistency.
• Data Type: Variable data (measurements like weight, length, volume, time).
• Used in Conjunction: Almost always used with an X-bar chart.

Formulas

Using the same subgroup data and calculations (Rᵢ and R̄) from the X-bar chart example:

1. Calculate the Center Line (CL) for the R chart: CL = R̄
2. Calculate the Control Limits (UCL and LCL) for the R chart:
   • UCL = D₄ * R̄
   • LCL = D₃ * R̄ (Note: Factors D₃ and D₄ depend on the subgroup size 'n' and are found in standard SPC constant tables. For small n (≤6), D₃ is often 0, meaning the LCL is 0).

Example Problem: R Chart (Continuing from X-bar)

Using the bottling plant data (n=5, k=4, R̄=0.04 L):

Calculations:

• Center Line for R Chart: CL = R̄ = 0.04 L
• Control Limits for R Chart:
  • From SPC tables, for n=5, D₄ = 2.114 and D₃ = 0.
  • UCL = 2.114 * 0.04 = 0.08456 L
  • LCL = 0 * 0.04 = 0 L

Plotting: Plot the subgroup ranges (0.04, 0.04, 0.04, 0.04) on a chart with CL=0.04, UCL=0.08456, and LCL=0. In this example, all points are on the center line, indicating consistent variability.

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3. p-Chart (Proportion Defective)

• Purpose: To monitor the proportion or fraction of nonconforming (defective) items in a sample or subgroup.
• Data Type: Attribute data (Go/No-Go, Pass/Fail, Defective/Non-defective).
• Sample Size: The subgroup size (n) can vary.

Formulas

1. For each subgroup i, calculate the proportion defective (pᵢ): pᵢ = (Number of defective items in subgroup i) / (Total number of items inspected in subgroup i (nᵢ))
2. Calculate the average proportion defective (p̄), which is the Center Line (CL) for the p-chart: p̄ = (Total number of defective items across all subgroups) / (Total number of items inspected across all subgroups)
3. Calculate the Control Limits (UCL and LCL) for each subgroup i (since n can vary):
   • UCL<0xE1><0xB5><0x96>ᵢ = p̄ + 3 * sqrt( p̄ * (1 - p̄) / nᵢ )
   • LCL<0xE1><0xB5><0x96>ᵢ = p̄ - 3 * sqrt( p̄ * (1 - p̄) / nᵢ ) (Note: If LCL calculates to a negative value, it is set to 0. If the sample size 'n' is constant for all subgroups, use a single average 'n' for fixed control limits).

Example Problem: p-Chart

A call center monitors the proportion of calls per day where the issue wasn't resolved on the first call ('defective').

Data:

Day (Subgroup i)	Total Calls (nᵢ)	Unresolved Calls	Proportion Defective (pᵢ)
1	200	10	10 / 200 = 0.050
2	250	15	15 / 250 = 0.060
3	180	9	9 / 180 = 0.050
4	220	18	18 / 220 = 0.082
5	210	8	8 / 210 = 0.038
Totals	1060	60

Calculations:

• Proportions (pᵢ): Calculated in the table.
• Average Proportion (Center Line): p̄ = 60 / 1060 = 0.0566
• Control Limits (calculated for each day due to varying nᵢ):
  • Day 1 (n₁=200):
    • sqrt(0.0566 * (1 - 0.0566) / 200) ≈ 0.0163
    • UCL₁ = 0.0566 + 3 * 0.0163 = 0.0566 + 0.0489 = 0.1055
    • LCL₁ = 0.0566 - 3 * 0.0163 = 0.0566 - 0.0489 = 0.0077
  • Day 2 (n₂=250):
    • sqrt(0.0566 * 0.9434 / 250) ≈ 0.0146
    • UCL₂ = 0.0566 + 3 * 0.0146 = 0.0566 + 0.0438 = 0.1004
    • LCL₂ = 0.0566 - 3 * 0.0146 = 0.0566 - 0.0438 = 0.0128
  • (Calculate similarly for Day 3, 4, 5)
  • Day 4 (n₄=220):
    • sqrt(0.0566 * 0.9434 / 220) ≈ 0.0156
    • UCL₄ = 0.0566 + 3 * 0.0156 = 0.0566 + 0.0468 = 0.1034
    • LCL₄ = 0.0566 - 3 * 0.0156 = 0.0566 - 0.0468 = 0.0098

Plotting: Plot the daily proportions (0.050, 0.060, 0.050, 0.082, 0.038) on a chart. The CL is constant at 0.0566, but the UCL and LCL change for each point based on nᵢ. All calculated pᵢ values are within their respective control limits.

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4. c-Chart (Count of Defects)

• Purpose: To monitor the number of defects or nonconformities found in a single unit or a standard sample size (area of opportunity).
• Data Type: Attribute data (counts of defects, e.g., scratches on a table, errors on a page).
• Sample Size: The size of the inspection unit (e.g., one table, one page, one batch of fixed size) must be constant.

Formulas

1. Count the number of defects (cᵢ) for each inspection unit i.
2. Calculate the average number of defects per unit (c̄), which is the Center Line (CL) for the c-chart: c̄ = (Total number of defects found across all units) / (Total number of units inspected)
3. Calculate the Control Limits (UCL and LCL) for the c-chart:
   • UCL = c̄ + 3 * sqrt(c̄)
   • LCL = c̄ - 3 * sqrt(c̄) (Note: If LCL calculates to a negative value, it is set to 0).

Example Problem: c-Chart

A textile factory inspects rolls of fabric (constant unit size) and counts weaving defects.

Data:

Roll (Unit i)	Number of Defects (cᵢ)
1	3
2	2
3	5
4	1
5	4
6	2
7	3
8	6
9	2
10	4
Total	32

Calculations:

• Defects per unit (cᵢ): Listed in the table.
• Average Defects per unit (Center Line): c̄ = 32 / 10 = 3.2
• Control Limits:
  • sqrt(c̄) = sqrt(3.2) ≈ 1.789
  • UCL = 3.2 + 3 * 1.789 = 3.2 + 5.367 = 8.567
  • LCL = 3.2 - 3 * 1.789 = 3.2 - 5.367 = -2.167, which is set to 0.

Plotting: Plot the number of defects for each roll (3, 2, 5, 1, 4, 2, 3, 6, 2, 4) on a chart with CL=3.2, UCL=8.567, and LCL=0. All points fall within the control limits.

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Summary

• X-bar and R charts: Used together for variable data (measurements) to monitor process average and variation. Require constant subgroup size.
• p-chart: Used for attribute data to monitor the proportion of defective items. Allows for varying sample sizes.
• c-chart: Used for attribute data to monitor the number of defects per constant unit. Requires a constant inspection unit size.

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