Unit 8: Statistical Quality Control

I. Foundations of Statistical Quality Control

A. Introduction to statistical quality control

Statistical quality control (SQC) uses data and statistical methods to distinguish natural process variation from variation caused by identifiable disturbances. Developed notably through Walter A. Shewhart’s work in the 1920s, it emphasizes preventing defects by controlling the process rather than relying only on final inspection.

• Quality: Quality is conformance to stated requirements, such as a shaft diameter of (20.00\pm0.05) mm or a maximum defect rate of 2%.
• Variation: No process produces perfectly identical outputs; measured values vary because of materials, machines, methods, personnel, measurement, and environment.
  • Common-cause variation: Inherent, random variation generated by the existing process system.
  • Special-cause variation: Non-random variation arising from an identifiable event, such as tool breakage, an incorrect machine setting, or contaminated material.
• Statistical control: A process is statistically stable when only common causes operate and its probability distribution remains reasonably constant over time.
• Control chart: A time-ordered graph containing a center line and statistically determined upper and lower control limits.
• Three-sigma convention: Shewhart limits are commonly placed three standard errors from the center line; under normality, about 99.73% of values lie within (\mu\pm3\sigma).
• Control limits versus specification limits:
  • Control limits: Calculated from process data and used to assess stability.
  • Specification limits: Set by design or customer requirements and used to assess acceptability.
• Basic assumptions: Observations should be collected consistently, subgroups should be rationally formed, and the plotted statistic should approximately satisfy the chart’s distributional model.
• Prevention principle: Inspection separates conforming from nonconforming output, whereas SQC seeks timely correction of the process producing that output.

II. Process Stability and Corrective Action

A. Process control

Process control is the systematic measurement, evaluation, and adjustment of a production or service process so that its performance remains stable and predictable.

• Rational subgrouping: Units within a subgroup are selected under nearly identical conditions, while differences between subgroups can reveal changes over time; five consecutive items every hour form a typical subgrouping plan.
• Chart construction: A chosen statistic is plotted in time order against three reference lines.

UCL = center line + 3(standard error)
CL  = expected value of the plotted statistic
LCL = center line - 3(standard error)

Here, UCL is the upper control limit, CL is the center line, and LCL is the lower control limit.

• In-control interpretation: Randomly scattered points within the limits indicate statistical stability, not necessarily compliance with specifications.
• Out-of-control interpretation: Evidence of a special cause includes a point beyond a limit or a non-random sequence such as a sustained shift, trend, cycle, or clustering near a limit.
• Operating sequence:
  1. Select the quality characteristic and sampling plan.
  2. Collect time-ordered preliminary data.
  3. Estimate the center line and control limits.
  4. Investigate special-cause signals and remove confirmed causes.
  5. Recalculate trial limits when justified, then monitor future production.
• Phase I analysis: Historical data are examined to establish stability and estimate reliable chart parameters.
• Phase II monitoring: Established limits are used to detect changes in subsequent process output.
• Reaction plan: A signal should trigger containment of affected output, investigation of machine and material records, correction of the assignable cause, and documentation—not arbitrary adjustment after every fluctuation.
• Tampering risk: Adjusting a stable process in response to common-cause variation can increase overall variability.

III. Control Charts for Measured Characteristics

A. Control charts for variables: X and R charts

The (\bar X)-and-(R) system jointly monitors subgroup location and within-subgroup spread for continuous measurements, especially when subgroup sizes are small.

• Statistics: For subgroup (i) containing (n) observations, calculate its mean (\bar X_i) and range (R_i).

X̄ᵢ = (Σⱼ Xᵢⱼ)/n
Rᵢ = Xmax,i − Xmin,i
X̿ = (Σᵢ X̄ᵢ)/k
R̄ = (Σᵢ Rᵢ)/k

(X_{ij}) is observation (j) in subgroup (i); (n) is subgroup size; (k) is the number of subgroups; (\bar{\bar X}) is the grand mean; and (\bar R) is the average range.

• (\bar X)-chart limits:

UCLX̄ = X̿ + A₂R̄
CLX̄  = X̿
LCLX̄ = X̿ − A₂R̄

• (R)-chart limits:

UCLR = D₄R̄
CLR  = R̄
LCLR = D₃R̄

(A_2), (D_3), and (D_4) are tabulated constants determined by (n).

• Interpretation order: Examine the (R) chart first because unstable dispersion weakens the meaning of the (\bar X) chart; then examine the process mean.
• Suitability: The method is efficient for measurements such as length, mass, temperature, or processing time, commonly with (2\le n\le10).
• Limitation: The range uses only the largest and smallest observations, so it becomes inefficient for larger subgroups.

B. Control charts for variables: X and S charts

The (\bar X)-and-(S) system monitors process location and standard deviation, providing a stronger measure of dispersion when subgroups are moderately large.

• Standard deviation statistic:

Sᵢ = √[Σⱼ(Xᵢⱼ − X̄ᵢ)²/(n − 1)]
S̄ = (Σᵢ Sᵢ)/k

(S_i) is subgroup (i)’s sample standard deviation, and (\bar S) is the average of the (k) subgroup standard deviations; the remaining symbols retain their earlier meanings.

• (\bar X)-chart limits:

UCLX̄ = X̿ + A₃S̄
CLX̄  = X̿
LCLX̄ = X̿ − A₃S̄

• (S)-chart limits:

UCLS = B₄S̄
CLS  = S̄
LCLS = B₃S̄

(A_3), (B_3), and (B_4) are constants based on subgroup size (n).

• Advantage over the range: (S_i) incorporates every observation, making it more sensitive and statistically efficient than (R_i), particularly when (n>10).
• Interpretation: A signal on the (S) chart indicates changed variability; a signal only on the (\bar X) chart indicates a likely shift in location.
• Condition: These conventional constants rely on approximate normality and consistent subgroup size; severe skewness may require transformation or another charting method.

IV. Control Charts for Counted Characteristics

A. Control charts for attributes: p chart

A (p) chart monitors the proportion of inspected units classified as nonconforming and permits subgroup sample sizes to vary.

• Statistic:

pᵢ = dᵢ/nᵢ
p̄ = (Σᵢ dᵢ)/(Σᵢ nᵢ)

(d_i) is the number of nonconforming units in subgroup (i), (n_i) is the number inspected, (p_i) is its nonconforming proportion, and (\bar p) is the overall proportion.

• Limits for subgroup (i):

UCLᵢ = p̄ + 3√[p̄(1 − p̄)/nᵢ]
CL   = p̄
LCLᵢ = p̄ − 3√[p̄(1 − p̄)/nᵢ]

A negative LCL is set to zero; a UCL above one is set to one.

• Model: The chart uses the binomial model: each unit is conforming or nonconforming, classifications are independent, and the underlying probability is constant while the process is stable.
• Variable limits: Larger (n_i) produces narrower limits because the standard error decreases as (1/\sqrt{n_i}).
• Application: The chart can track the weekly proportion of rejected invoices, leaking containers, or incorrectly assembled products.

B. Control charts for attributes: np chart

An (np) chart monitors the number of nonconforming units per subgroup and is appropriate only when the inspected sample size remains constant.

• Statistic and center: The plotted value is (d_i), while its expected value is (n\bar p).
• Control limits:

UCL = n p̄ + 3√[n p̄(1 − p̄)]
CL  = n p̄
LCL = n p̄ − 3√[n p̄(1 − p̄)]

(n) is the fixed sample size, (\bar p) is the estimated nonconforming probability, and negative lower limits are replaced by zero.

• Relationship to the (p) chart: Both use the binomial distribution, but the (np) chart displays counts rather than fractions.
• Practical advantage: Counts such as “7 defective bulbs out of 200” are often easier for operators to understand than a plotted proportion of (0.035).
• Limitation: If subgroup sizes change, the expected count and limits also change; a (p) chart should normally be used instead.

C. Control charts for attributes: c chart

A (c) chart monitors the number of nonconformities found in a constant-sized inspection unit, where one unit may contain multiple defects.

• Distinction: A nonconforming unit fails requirements, whereas a nonconformity is an individual defect; one fabric roll may contain several stains or tears.
• Statistic and limits:

c̄   = (Σᵢ cᵢ)/k
UCL = c̄ + 3√c̄
CL  = c̄
LCL = c̄ − 3√c̄

(c_i) is the defect count in inspection unit (i), (k) is the number of units, and (\bar c) is the average count; a negative LCL becomes zero.

• Model: The Poisson model assumes defects occur independently over a constant area, time, length, or opportunity and have a stable mean rate.
• Constant opportunity: Each inspected unit must be comparable—for example, every printed sheet must have the same area.
• Limitation: If inspection-unit size varies, raw counts are not comparable; defects per unit should instead be monitored with a rate-based chart.

V. Operational Use of Control Charts

A. Applications of control charts in process control

Control charts convert repeated process data into signals that support timely, evidence-based operational decisions.

• Manufacturing: (\bar X)-and-(R) charts monitor dimensions or fill weights, while (p), (np), and (c) charts track rejected items or surface defects.
• Services: Charts can monitor transaction errors, delayed deliveries, complaint proportions, or documentation defects, provided observations have consistent definitions.
• Chart selection:
  • Continuous measurement: Use (\bar X)-(R) for small subgroups or (\bar X)-(S) for larger ones.
  • Nonconforming units: Use (p) for variable sample sizes and (np) for fixed sample sizes.
  • Defect counts: Use (c) when the inspection opportunity is constant.
• Improvement cycle: Establish stability, identify and remove special causes, revise the process to reduce common-cause variation, and then recalculate limits from the improved stable process.
• Capability assessment: Specification-based indices such as (Cp) and (C{pk}) are meaningful only after control charts indicate a stable process.
• Decision discipline: A control signal prompts investigation rather than automatic rejection; a point inside the limits indicates stability rather than guaranteed product conformity.
• Effective implementation: Reliable measurement systems, trained operators, rational sampling, prompt reaction plans, and documented cause analysis determine whether charts lead to genuine improvement.