5. Control Charts for Variables

Introduction and Context

Statistical process control (SPC) provides a disciplined method for distinguishing routine process variation from unusual variation requiring investigation. In quality control, this distinction is essential because every process varies. Measurements such as diameter, length, weight, tensile strength, temperature, pressure, viscosity, and response time never remain perfectly constant. The managerial problem is not to eliminate all variation immediately, but to determine whether the observed variation is stable and predictable and whether the process is capable of meeting requirements.

Control charts for variables are among the most important SPC instruments for this purpose. They are used when quality characteristics can be measured on a continuous numerical scale. Unlike attribute charts, which classify observations as conforming or nonconforming or count defects, variable charts retain the magnitude of each observation. This generally provides more information about process behaviour and may reveal deterioration before an individual measurement exceeds a specification limit.

This chapter develops three major variable-chart systems: the X-bar–R chart, the X-bar–S chart, and the individuals–moving range chart. It explains their statistical foundations, appropriate use, construction, interpretation, assumptions, limitations, and connection with reaction plans. Particular attention is given to the distinction between control limits and specification limits, the interpretation of non-random patterns, rational subgrouping, and the responsible use of digital SPC systems.

Learning Objectives

After studying this chapter, the learner should be able to:

1. Explain the purpose and statistical logic of control charts for variables.

2. Distinguish common-cause variation from special-cause variation.

3. Construct and interpret X-bar–R, X-bar–S, and individuals–moving range charts.

4. Select an appropriate chart according to subgroup size, sampling structure, and process conditions.

5. Evaluate assumptions concerning independence, rational subgrouping, distribution, and measurement quality.

6. Identify non-random patterns, trends, shifts, cycles, and instability.

7. Distinguish statistical control from process capability and product conformance.

8. Develop statistically sound reaction plans for production and service processes.

9. Assess professional, technological, and managerial risks in the use of SPC.

The Statistical Logic of Variable Control Charts

A control chart plots a statistic in time order, together with a central line and statistically derived upper and lower control limits. The central line represents the process average or another relevant process parameter. The limits describe the amount of variation expected when only common causes are operating. They are not customer requirements and do not indicate what the process ought to achieve; rather, they indicate what the process has historically demonstrated under a defined set of conditions.

A process is said to be statistically stable, or in statistical control, when its plotted behaviour is consistent with a stable probability model. This does not necessarily mean that all output meets specifications. A stable process can be consistently incapable if its mean is badly positioned or its natural variation is too large. Conversely, a process may produce many conforming units while still being unstable and unpredictable. Control charts therefore answer the question, “Is the process behaving predictably?” Capability analysis answers a different question, “Can the predictable process meet requirements?”

The fundamental distinction is between common-cause and special-cause variation. Common causes are numerous, individually small sources inherent in the design and operation of the process—for example, normal tool wear, routine environmental fluctuation, or small differences among materials. Special causes are identifiable and unusual influences, such as an incorrectly adjusted machine, a failed sensor, an inexperienced operator, or a contaminated batch of material. A control chart does not prove that a special cause exists; it signals that the observed pattern is unlikely under the assumed stable process model and therefore merits investigation.

Rational Subgrouping

The validity of a control chart depends heavily on how observations are grouped. A rational subgroup should contain items produced under conditions that are as similar as possible, while successive subgroups should represent the process across time. For example, five consecutive parts taken every hour may form a rational subgroup when the purpose is to detect shifts between production periods. If five parts are collected from different machines and mixed together, the subgroup average may conceal machine-to-machine differences.

Subgrouping reflects the question being asked. Within-subgroup variation is intended to estimate short-term, common-cause variation. Between-subgroup variation is examined through the subgroup averages. Poor subgrouping can make limits too wide, hide important changes, or create false signals.

The X-bar–R Chart System

The X-bar–R system is commonly used when measurements are collected in rational subgroups of relatively small size, often between 2 and 10 observations. The X-bar chart monitors changes in the process location, while the R chart monitors short-term within-subgroup dispersion.

Suppose subgroup i contains n observations, denoted by x_i1, x_i2, ..., x_in. The subgroup mean is

x̄_i = (x_i1 + x_i2 + ... + x_in) / n

and the subgroup range is

R_i = maximum observation − minimum observation.

Across k subgroups, the average subgroup mean and average range are

x̄̄ = (Σx̄_i) / k

R̄ = (ΣR_i) / k.

For the traditional Shewhart X-bar–R chart, the limits are estimated as:

X-bar chart:

UCL_x̄ = x̄̄ + A2R̄

CL_x̄ = x̄̄

LCL_x̄ = x̄̄ − A2R̄

R chart:

UCL_R = D4R̄

CL_R = R̄

LCL_R = D3R̄

The constants A2, D3, and D4 depend on subgroup size and are derived from the distribution of subgroup means and ranges. For some subgroup sizes, D3 is zero, so the lower range limit is zero. The constants must be selected for the actual subgroup size; using an incorrect table value produces incorrect limits.

The R chart should generally be examined before the X-bar chart. If within-subgroup variation is unstable, the estimated limits for the X-bar chart are unreliable. A process shift in dispersion can also make points on the X-bar chart appear unusual even when the process centre has not changed.

Illustrative Calculation

Consider a filling process in which five bottles are sampled every hour. Assume that, after preliminary analysis, the average subgroup mean is 500.4 ml and the average range is 3.2 ml. For subgroup size n = 5, commonly used constants are A2 = 0.577, D3 = 0, and D4 = 2.114.

The X-bar limits are:

UCL_x̄ = 500.4 + (0.577)(3.2) = 502.25 ml approximately

LCL_x̄ = 500.4 − (0.577)(3.2) = 498.55 ml approximately.

The range limits are:

UCL_R = (2.114)(3.2) = 6.76 ml

LCL_R = 0.

A subgroup mean outside 498.55–502.25 ml is evidence of a possible shift in process location. A subgroup range above 6.76 ml indicates unusually large within-subgroup variation. A range of zero is not automatically a problem; it may occur with a coarse measurement system or a process whose readings are highly rounded. Such a result should prompt examination of measurement resolution rather than immediate claims of exceptional control.

The X-bar–S Chart System

The X-bar–S chart system is preferred when subgroup sizes are larger, when estimates of dispersion should be more efficient, or when software naturally calculates the sample standard deviation. The X-bar chart again monitors location, while the S chart monitors within-subgroup standard deviation.

For subgroup i, the sample standard deviation is:

s_i = √[Σ(x_ij − x̄_i)²/(n − 1)].

Across k subgroups, let s̄ be the average subgroup standard deviation. The conventional limits are:

X-bar chart:

UCL_x̄ = x̄̄ + A3s̄

CL_x̄ = x̄̄

LCL_x̄ = x̄̄ − A3s̄

S chart:

UCL_S = B4s̄

CL_S = s̄

LCL_S = B3s̄.

The constants A3, B3, and B4 depend on subgroup size. Because the standard deviation uses all observations in a subgroup, it is generally a more stable estimator of dispersion than the range for larger subgroups. The X-bar–S system is therefore appropriate for processes in which subgroup sizes are sufficiently large or where precision in estimating variation is important.

The same interpretive logic applies as with the X-bar–R system: the S chart should be checked first, and an unstable dispersion estimate undermines interpretation of the location chart. When subgroup sizes vary, control-chart calculations require appropriate subgroup-specific methods or software; simply averaging statistics without considering subgroup size can be misleading.

Individuals–Moving Range Charts

Some processes do not permit rational subgroups larger than one. Measurements may be expensive, destructive, infrequent, or generated one at a time. Examples include laboratory test results, batch-release characteristics, aircraft inspection measurements, and monthly financial or service indicators. In these situations, the individuals–moving range (I–MR) chart is appropriate.

The individuals chart plots each observation x_i. The moving range of order two is usually defined as:

MR_i = |x_i − x_{i−1}|, for i = 2, 3, ...

The average moving range is:

MR̄ = ΣMR_i/(m − 1),

where m is the number of individual observations. An estimate of the process standard deviation is often calculated as:

σ̂ = MR̄/d2,

where d2 = 1.128 for moving ranges of two observations.

The individuals-chart limits are:

UCL_I = x̄ + 3(MR̄/d2)

CL_I = x̄

LCL_I = x̄ − 3(MR̄/d2).

For the moving-range chart:

UCL_MR = D4MR̄

CL_MR = MR̄

LCL_MR = D3MR̄,

with D3 = 0 and D4 = 3.267 for moving ranges of two.

The I–MR chart is convenient but has important limitations. Its moving ranges overlap, so successive range values are not independent. The chart is also highly sensitive to autocorrelation. If observations naturally drift, cycle, or persist because of the process dynamics, the standard limits may generate misleading signals. In time-dependent processes, a time-series model or a specialized control chart may be more appropriate.

Selecting the Appropriate Chart

Chart selection should be based on the measurement structure and the process question rather than personal preference.

• Use X-bar–R when rational subgroups are available and subgroup sizes are small.
• Use X-bar–S when rational subgroups are available and subgroup sizes are moderate or large, or when standard deviation is the preferred dispersion measure.
• Use I–MR when observations occur individually and meaningful subgroups of size greater than one cannot be formed.

The decision must also consider sampling frequency, cost, destructive testing, process speed, expected distribution, autocorrelation, and the consequences of delayed detection. A chart with a large subgroup may estimate the mean precisely but could delay recognition of a short-lived shift. A chart with very small subgroups may provide rapid information but less reliable estimates of variation.

Reading Control Charts: Signals and Patterns

A single point beyond a three-sigma control limit is a strong signal of possible special-cause variation, provided the chart assumptions and calculations are sound. However, points within the limits do not automatically establish control. Non-random patterns can be equally important.

Common diagnostic signals include:

• A point beyond a control limit, suggesting a sudden shift or unusual event.
• A sustained run of points on one side of the centre line, suggesting a change in process level.
• A trend of successive increases or decreases, suggesting tool wear, temperature drift, chemical depletion, or another systematic influence.
• A cycle or repeating pattern, suggesting shifts, maintenance schedules, ambient conditions, or batch effects.
• Stratification, in which points cluster unusually close to the centre line, possibly indicating mixed data sources, excessive rounding, or limits calculated from inappropriate variation.
• An alternating pattern, which may result from overadjustment or systematic differences between operators, machines, or materials.
• Instability in the R or S chart, indicating that location-chart signals should not be interpreted in isolation.

Western Electric and related run rules are often used to supplement the basic beyond-limit rule. Examples include two of three successive points beyond a two-sigma zone on the same side, four of five beyond a one-sigma zone on the same side, or eight successive points on one side of the centre line. Such rules increase sensitivity but also increase false alarms. They should therefore be selected deliberately, documented, and validated against the cost of investigation and the consequences of missed signals.

Control charts should always be interpreted in process context. A point may be statistically unusual because of a recorded setup change, a new supplier, or a planned maintenance event. The correct response is not to delete the point automatically. The event and its technical justification should be documented, and limits should be recalculated only when a stable, homogeneous process has been established and the reason for doing so is defensible.

Assumptions and Data Quality

Variable control charts rely on several assumptions. First, the measurement system must be capable of distinguishing relevant process differences. Bias, poor repeatability, poor reproducibility, inadequate resolution, and transcription errors can create false signals or hide real ones. Measurement-system analysis should precede serious interpretation.

Second, observations within a rational subgroup should represent comparable operating conditions. Mixing different products, machines, shifts, or recipes can inflate variation and obscure meaningful structure. Conversely, selecting only convenient observations can produce falsely narrow limits.

Third, the process should be sampled in time order. Randomly reordering measurements destroys evidence of trends, shifts, and cycles. Missing observations, changes in sampling frequency, and retrospective data selection should be recorded.

Fourth, the usual three-sigma limits assume an approximately stable and reasonably well-behaved distribution of the chart statistic. Moderate non-normality may not seriously compromise subgroup-mean charts, particularly with larger subgroups, but extreme skewness, heavy tails, mixtures, or dependence require investigation. Transformations, alternative charts, robust methods, or distribution-specific approaches may be appropriate.

Finally, control limits should be based on a process that is representative of the intended operating conditions. Limits calculated from a period containing known special causes may be too wide and fail to detect future problems. On the other hand, removing every unusual point merely to obtain attractive limits is a form of statistical manipulation.

From Signals to Reaction Plans

A control chart has value only when it leads to appropriate action. A reaction plan should specify what constitutes a signal, who is responsible, what immediate containment is required, how the cause will be investigated, and what conditions permit restart or release.

A sound reaction plan commonly follows this sequence:

1. Verify the signal and the measurement record, including instrument status, units, data entry, and chart configuration.

2. Protect the customer and the process by segregating potentially affected output or pausing the operation when risk warrants it.

3. Identify the time window and conditions associated with the signal: machine, operator, material, batch, software version, environment, and recent adjustments.

4. Investigate assignable causes using evidence rather than speculation.

5. Correct the cause and verify that the process has returned to a stable condition.

6. Document the event, disposition of affected product, corrective action, and follow-up evidence.

7. Recalculate limits only when the process has changed permanently or the original limits were based on an invalid or nonrepresentative period.

A reaction plan must avoid two opposite errors. Underreaction allows special causes to continue and may release defective output. Overreaction leads operators to adjust a stable process whenever a point differs slightly from the target. This “tampering” can increase variation. Operators should adjust a process only when there is a justified signal and a known or strongly supported cause.

Control Limits, Specification Limits, and Capability

Specification limits are established by design, customer need, regulation, engineering requirements, or contractual agreement. Control limits are calculated from process data. A specification limit may remain unchanged while control limits change as the process improves or deteriorates.

Once a process is demonstrably stable, capability indices can be used cautiously. For a two-sided specification with lower specification limit LSL and upper specification limit USL:

Cp = (USL − LSL)/(6σ̂)

Cpk = min[(USL − μ)/(3σ̂), (μ − LSL)/(3σ̂)].

Cp reflects potential capability if the process is centred; Cpk also reflects displacement of the mean. These indices are not meaningful when the process is unstable, because the estimates of μ and σ̂ do not represent a consistent process. Capability analysis should therefore follow, not replace, control-chart analysis.

Applied Case: Stabilising a Precision Machining Process

Background

The following is an illustrative case. A manufacturer of hydraulic valve bodies measures the diameter of a critical bore after machining. The target is 25.000 mm, and the engineering specification is 24.960 to 25.040 mm. Five consecutive parts are sampled every two hours from the same machine, creating rational subgroups of size five.

Situation and Problem

The quality team initially uses an X-bar–R chart. During the first week, the R chart shows a gradual increase in ranges, followed by two points above the upper range limit. The X-bar chart then displays a downward trend, although no individual measurement has yet violated the specification. Production personnel propose lowering the machine offset after each subgroup to keep the average close to target.

Analysis

The rising ranges indicate that within-subgroup variation is changing. Investigation finds that the coolant concentration is falling during long production runs and that the cutting tool is approaching the end of its useful life. The downward trend in the X-bar chart is therefore not simply random fluctuation. Repeated offset adjustments would mask the tool-wear mechanism and might produce overcorrection.

The team also checks the measurement system. A gauge repeatability and reproducibility study indicates that measurement variation is small relative to the engineering tolerance and process variation. The subgroup structure is retained because the five consecutive parts represent short-term machine behaviour, while the two-hour interval allows changes across the production cycle to be observed.

Approach and Solution

The immediate response is to quarantine parts produced after the first confirmed range signal until the last verified stable subgroup. The coolant concentration is restored, the tool is replaced according to a defined wear criterion, and the machine is restarted under controlled conditions. The team records tool age, coolant concentration, operator, and bore measurements for each subgroup.

After a stable period is established, revised limits are calculated using only representative stable data. The X-bar–R chart is retained because the sampling design remains appropriate. A documented reaction plan is introduced: an R-chart signal requires checking coolant, tool condition, and measurement setup before any offset adjustment; an X-bar signal requires checking machine centring, fixture condition, and recent changes.

Outcome

The process becomes more predictable, and the average range decreases. Capability is evaluated only after stability is demonstrated. The organization learns that the absence of specification violations did not prove process control; the chart provided earlier evidence of deterioration.

Learning Points

The case illustrates five principles. First, the dispersion chart is essential because changing variation can precede a location shift. Second, rational subgrouping determines what the chart can reveal. Third, operator adjustment without diagnosis may increase variation. Fourth, measurement-system adequacy is a prerequisite for interpretation. Fifth, a reaction plan must connect statistical signals with physical process checks and product protection.

Managerial, Professional, and Digital Considerations

Managers should treat control charts as decision-support systems rather than surveillance devices for blaming operators. Signals may reflect inadequate process design, unstable materials, poor maintenance, or unrealistic production schedules. A mature quality culture asks what system condition produced the signal and whether the organization has provided the resources needed for prevention.

Digital SPC platforms can automate data collection, charting, alarm notification, traceability, and linkage with maintenance or manufacturing-execution systems. Advanced analytics may classify patterns, associate signals with contextual variables, and prioritize investigations. Nevertheless, automation does not correct poor sampling, unsuitable limits, measurement error, or inappropriate reaction rules. Algorithms should be validated, version-controlled, and monitored for false alarms and missed signals.

Data governance is also important. Records should preserve time stamps, product and machine identity, operator or shift information, revisions to limits, alarm acknowledgements, and disposition decisions. Access controls should prevent unauthorized alteration of historical data. If predictive models are used, their assumptions and performance should be transparent enough for technical review. Regulatory and customer requirements may require retention of chart records and evidence that out-of-control conditions were handled appropriately.

Common Mistakes and Best Practices

Frequent mistakes include calculating limits from specification limits, ignoring the R or S chart, using control charts with unsuitable subgrouping, treating every point near a limit as a failure, deleting inconvenient observations, changing limits after every signal, and interpreting capability indices before establishing stability. Another mistake is using a control chart as a substitute for engineering understanding; statistical evidence identifies unusual behaviour, but process knowledge is needed to locate and remove causes.

Best practice is to define the process and quality characteristic clearly, verify the measurement system, establish rational subgroups, collect data in time order, calculate limits from a representative stable period, interpret dispersion before location, and document a proportionate reaction plan. Limits and decision rules should be reviewed when the process, product, measurement method, or sampling structure changes. The ultimate objective is not merely a chart with no alarms; it is a stable, capable, understandable, and continually improving process.

Current Developments and Emerging Directions

Modern SPC is moving toward integrated, real-time systems in which sensor data, laboratory results, maintenance records, and production context are analysed together. Adaptive and risk-based monitoring can provide greater sensitivity when process conditions change, although such methods require careful governance. Multivariate control charts are increasingly relevant when several correlated measurements jointly describe product quality. Time-series and autocorrelation-aware methods are important for continuous, chemical, energy, and service processes where observations are not independent.

These developments do not displace the classical X-bar–R, X-bar–S, and I–MR charts. Rather, they reinforce the need to understand the assumptions behind simple methods. A sophisticated dashboard cannot compensate for an invalid sampling design or an untrustworthy measurement system.

Chapter Summary

Control charts for variables monitor measurable quality characteristics and distinguish routine variation from unusual process behaviour. The X-bar–R system is suited to small rational subgroups, the X-bar–S system to larger subgroups or situations where standard deviation is preferred, and the I–MR system to individual observations. In each system, the location chart and dispersion chart must be interpreted together, with dispersion generally examined first.

Control limits describe expected process behaviour; specification limits describe requirements. A stable process may still be incapable, while a temporarily conforming process may be unstable. Rational subgrouping, measurement-system adequacy, time order, distributional assumptions, and independence are central to valid interpretation. Signals include points beyond limits, runs, trends, cycles, stratification, and alternating patterns. A statistically sound reaction plan verifies the signal, protects output, investigates causes, corrects the process, documents decisions, and avoids unnecessary tampering. Digital SPC can strengthen monitoring and traceability, but professional judgement and sound statistical foundations remain indispensable.

Key Terms

Control chart: A time-ordered graph of a process statistic with a centre line and statistically calculated control limits.

Variable data: Numerical measurements recorded on a continuous or quantitative scale.

Common-cause variation: Routine variation inherent in the process system.

Special-cause variation: Unusual variation attributable to a specific, potentially identifiable influence.

Statistical control: A condition in which process behaviour is stable and predictable under a defined model.

Rational subgroup: A group of observations selected to represent similar short-term conditions.

X-bar chart: A chart monitoring subgroup averages and changes in process location.

R chart: A chart monitoring subgroup ranges as an estimate of within-subgroup dispersion.

S chart: A chart monitoring subgroup standard deviations.

Individuals chart: A chart plotting individual observations when rational subgroups larger than one are unavailable.

Moving range: The absolute difference between successive individual observations, usually of order two.

Centre line: The reference line representing the estimated process average or dispersion level.

Control limit: A statistically estimated boundary for expected process behaviour.

Specification limit: A customer, engineering, regulatory, or design requirement for output.

Run: A sequence of points with a meaningful directional or one-sided pattern.

Trend: A sustained sequence of increasing or decreasing observations.

Process capability: The ability of a stable process to meet specified requirements.

Cp: A capability index comparing specification width with six process standard deviations.

Cpk: A capability index that accounts for both process variation and centring.

Reaction plan: A documented set of actions to be taken when a control-chart signal occurs.

Review Questions

Short Answer Questions

1. What is the principal difference between common-cause and special-cause variation?

2. Why should the R chart or S chart normally be examined before the X-bar chart?

3. Define rational subgrouping and explain its purpose.

4. When is an individuals–moving range chart appropriate?

5. How do control limits differ from specification limits?

Long Answer Questions

1. Explain the construction and interpretation of an X-bar–R chart, including the roles of x-bar, R-bar, and the constants A2, D3, and D4.

2. Compare the X-bar–R, X-bar–S, and individuals–moving range chart systems and identify suitable applications for each.

3. Discuss the assumptions required for valid variable control-chart interpretation.

4. Explain how runs, trends, cycles, stratification, and alternating patterns can indicate non-random process behaviour.

5. Describe the relationship between statistical control, process capability, Cp, and Cpk.

Analytical / Application Questions

1. A process uses subgroups of five observations with x-bar-bar = 80.6 and R-bar = 1.8. Using A2 = 0.577, D3 = 0, and D4 = 2.114, calculate the control limits for both charts and explain how they would be used.

2. An I–MR chart shows a stable individuals chart but two moving ranges above the upper limit. What possible process and measurement causes should be investigated before changing the control limits?

3. Operators repeatedly adjust a machine whenever an X-bar point moves slightly from the centre line, and process variation subsequently increases. Diagnose the problem and propose a statistically sound reaction plan.

22. Multiple Choice Questions with Answers

Question 1

Which chart system is generally most appropriate when rational subgroups of five consecutive observations are collected from a process?

A. X-bar–R chart

B. Individuals–moving range chart

C. p chart

D. u chart

Correct Answer: A. X-bar–R chart

Question 2

What is the primary purpose of the R chart in an X-bar–R system?

A. To monitor customer specifications

B. To monitor within-subgroup dispersion

C. To estimate the process target

D. To count nonconforming units

Correct Answer: B. To monitor within-subgroup dispersion

Question 3

A process has individual observations that are expensive and cannot be meaningfully grouped. Which chart is most suitable?

A. X-bar–S chart

B. X-bar–R chart

C. Individuals–moving range chart

D. np chart

Correct Answer: C. Individuals–moving range chart

Question 4

Which statement correctly distinguishes control limits from specification limits?

A. Both are always determined by the customer

B. Control limits describe expected process behaviour, whereas specification limits describe requirements

C. Specification limits are recalculated after every process signal

D. Control limits define legal product tolerances

Correct Answer: B. Control limits describe expected process behaviour, whereas specification limits describe requirements

Question 5

Why is the dispersion chart normally examined before interpreting the X-bar chart?

A. The dispersion chart determines the customer target

B. An unstable estimate of dispersion makes location-chart limits unreliable

C. The X-bar chart cannot be calculated until specifications are known

D. Dispersion charts contain no statistical assumptions

Correct Answer: B. An unstable estimate of dispersion makes location-chart limits unreliable

Question 6

A sequence of eight points lies on the same side of the centre line, although none is beyond a control limit. What does this most likely suggest?

A. Possible process shift or non-random behaviour

B. Certain product nonconformance

C. A measurement system with zero variation

D. That specification limits should be widened

Correct Answer: A. Possible process shift or non-random behaviour

Question 7

Which action is most consistent with a sound reaction plan after a confirmed control-chart signal?

A. Delete the point and recalculate limits immediately

B. Adjust the machine repeatedly until the point returns to the centre line

C. Verify the signal, protect potentially affected output, and investigate assignable causes

D. Ignore the signal if all products remain within specifications

Correct Answer: C. Verify the signal, protect potentially affected output, and investigate assignable causes

Question 8

Why can a stable process still have a poor Cpk value?

A. Statistical control guarantees capability

B. The process may be stable but poorly centred or too variable relative to specifications

C. Cpk measures only measurement-system bias

D. Cpk is calculated without using specifications

Correct Answer: B. The process may be stable but poorly centred or too variable relative to specifications

Question 9

Which practice can create false confidence in a control chart?

A. Recording observations in time order

B. Checking measurement-system adequacy

C. Mixing observations from different machines into one subgroup without justification

D. Documenting the sampling method

Correct Answer: C. Mixing observations from different machines into one subgroup without justification

Question 10

What is a major risk of repeatedly adjusting a stable process in response to ordinary variation?

A. It may increase process variation through tampering

B. It always improves Cpk

C. It removes the need for rational subgrouping

D. It converts variable data into attribute data

Correct Answer: A. It may increase process variation through tampering