Unit 8: Statistical Quality Control - Subjective Questions

Statistical Quality Control (SQC) is the application of statistical methods to measure, monitor, control, and improve the quality of a process or product.

Main objectives:

• Maintain process stability: Detect changes in the process over time.
• Reduce variation: Identify and eliminate avoidable sources of variation.
• Ensure conformance: Verify that output satisfies specified quality requirements.
• Prevent defects: Emphasize prevention rather than inspection after production.
• Improve productivity: Reduce scrap, rework, delays, and inspection costs.
• Support decisions: Provide objective, data-based information for process improvement.

SQC commonly includes process control, control charts, acceptance sampling, and process capability analysis. A process is statistically controlled when its variation is caused only by stable, random factors.

Chance causes, also called common causes, are small and unavoidable sources of variation naturally present in a stable process. Examples include minor temperature changes, normal machine vibration, and small differences in raw materials.

Assignable causes, also called special causes, arise from identifiable and correctable factors. Examples include an improperly adjusted machine, a defective tool, an untrained operator, or a poor batch of material.

Key differences:

• Chance causes produce random variation within the control limits, whereas assignable causes may produce points or patterns outside the limits.
• Chance causes generally require improvement of the overall process system.
• Assignable causes require investigation and corrective action at their specific source.
• A process containing only chance-cause variation is said to be in statistical control.

Control charts help distinguish between these two types of variation.

Process control is the systematic monitoring of process performance so that variations caused by assignable factors can be detected and corrected before they produce substantial nonconforming output.

Basic steps:

1. Select an important quality characteristic.
2. Choose an appropriate variables or attributes control chart.
3. Decide the rational subgroup size, sampling frequency, and sampling method.
4. Collect preliminary process data under normal operating conditions.
5. Calculate the center line and trial control limits.
6. Plot the sample statistics in time order.
7. Investigate out-of-control points and nonrandom patterns.
8. Remove verified assignable causes and revise the limits if necessary.
9. Continue monitoring using the established limits.
10. Use the results for continuous process improvement.

Statistical control does not necessarily mean that the process meets specifications; it means that its behavior is stable and predictable.

Let a plotted sample statistic have mean and standard deviation . The general control-chart lines are

If is approximately normally distributed, about of its values lie within three standard deviations of its mean:

Therefore, the probability that an in-control statistic falls outside the limits is approximately

This small probability provides a practical balance between detecting genuine process shifts and avoiding excessive false alarms. A point outside a limit suggests an assignable cause, but it does not prove one. Nonrandom patterns within the limits can also indicate process changes.

The chart monitors the process mean, while the chart monitors within-subgroup dispersion.

For subgroups, each of size , calculate

Then calculate

Because the process standard deviation can be estimated by , the three-sigma limits can be expressed using tabulated constants.

chart:

chart:

The constants , , and depend on subgroup size . The chart should be interpreted first because unstable dispersion affects the reliability of the chart.

For the chart:

Thus,

For the chart:

Therefore, the -chart limits are and approximately , with center line .

The charts provide complementary information:

• The chart shows whether within-subgroup variability is stable.
• The chart shows whether the process average is stable.

The chart is examined first because the control limits of the chart are based on the estimate of process variability obtained from . If variability is unstable, the limits may be unreliable.

Interpretation procedure:

1. Examine the chart for points outside the limits and nonrandom patterns.
2. Investigate and remove assignable causes affecting variability.
3. Recalculate limits if preliminary out-of-control observations are legitimately removed.
4. Examine the chart for shifts in process location.

A process is statistically controlled only when both charts show stable behavior. An in-control chart cannot compensate for an out-of-control chart.

The and charts monitor the process mean and standard deviation. They are often preferred to and charts for moderate or large subgroup sizes because uses all observations and is a more efficient measure of dispersion.

For subgroup ,

Calculate

chart:

chart:

The constants , , and depend on . As with an chart, the chart should be checked before interpreting the chart.

Similarities:

• Both combinations monitor process location and dispersion.
• Both require quantitative measurements and rational subgroups.
• Both use control-chart constants determined by subgroup size.
• In both cases, the dispersion chart should be examined first.

Differences:

• The chart uses , whereas the chart uses the sample standard deviation.
• The range is simple to calculate but uses only the smallest and largest observations.
• The standard deviation uses every observation and estimates variability more efficiently.
• - charts are commonly used for small subgroup sizes, often about to .
• - charts are preferred for larger subgroups or when calculations are performed by software.

The selection depends on subgroup size, computational facilities, established practice, and the required precision in monitoring dispersion.

A chart monitors the fraction nonconforming. If sample contains nonconforming units out of inspected units, then

The pooled estimate of the process fraction nonconforming is

Under the binomial model,

Replacing by gives the limits for sample :

A negative lower limit is set to , and an upper limit above is set to . For unequal sample sizes, separate limits should be calculated for each , producing varying control limits.

The overall fraction nonconforming is

The standard error is

Therefore,

Since a fraction cannot be negative, set .

For the specified sample,

Because

the sample is within the control limits. It does not generate a point-based out-of-control signal, although surrounding points should still be checked for nonrandom patterns.

An chart monitors the number of nonconforming units in each sample. It is appropriate when every inspected unit is classified as conforming or nonconforming and the sample size remains constant.

If the process fraction nonconforming is estimated by , then the expected number nonconforming is

and its estimated standard deviation is

Thus, the three-sigma limits are

A negative lower limit is replaced by , and the upper limit cannot exceed . The chart is easy to interpret because its plotted values are counts rather than proportions.

chart:

• Plots the fraction or proportion nonconforming, .
• Can accommodate equal or unequal sample sizes.
• Uses the center line .
• Has varying control limits when sample sizes differ.

chart:

• Plots the number of nonconforming units, .
• Requires a constant sample size.
• Uses the center line .
• Is often easier for operators because it displays actual counts.

Both charts are based on the binomial model and require independent classifications with an approximately constant probability of nonconformance. If sample sizes vary, the chart is normally selected. If sample sizes are constant and counts are operationally clearer, the chart may be preferred.

A chart monitors the number of nonconformities or defects per inspection unit when the area of opportunity remains constant. One unit may contain more than one defect.

Typical applications:

• Number of scratches on a sheet
• Number of errors on a standard form
• Number of flaws in a fixed length of fabric
• Number of soldering defects on a circuit board

The chart is based on the Poisson model, for which the mean and variance are both approximately . If

then the limits are

If the lower limit is negative, it is set to . The inspected unit and opportunity for defects must remain reasonably constant.

The average number of defects is

The control limits are

Because a defect count cannot be negative, set .

Thus, the chart has:

An inspection unit containing defects lies above the upper control limit. It indicates a possible assignable cause and should be investigated. Possible causes include a material problem, equipment malfunction, inspection change, or unusual operating condition.

Variables control charts:

• Use measured numerical characteristics such as length, weight, pressure, or temperature.
• Examples include , , and charts.
• Provide information about both process location and dispersion.
• Usually detect process changes with smaller sample sizes.
• Require measurement equipment and may involve higher data-collection costs.

Attributes control charts:

• Use classifications or counts, such as conforming/nonconforming or number of defects.
• Examples include , , and charts.
• Are simpler when precise measurement is impractical.
• Often require larger sample sizes to detect small changes.
• May combine different defect types into one count and therefore provide less diagnostic detail.

The choice depends on the form of available data, sample-size conditions, cost of measurement, and purpose of process monitoring.

Important nonrandom patterns include:

• Point outside a control limit: Suggests a sudden assignable cause.
• Run on one side of the center line: May indicate a sustained shift in the process mean.
• Upward or downward trend: May result from tool wear, temperature drift, or gradual deterioration.
• Cycle: May be associated with shifts, seasons, maintenance schedules, or periodic environmental changes.
• Stratification: Points cluster near the center line, possibly because subgroups mix observations from different sources.
• Mixture: Points avoid the center line and cluster toward both limits, possibly due to alternating process populations.
• Repeated proximity to a control limit: May indicate a smaller shift not yet producing an outside point.

Such patterns should be evaluated using predefined run rules. A signal initiates investigation; it does not by itself identify the physical cause.

Statistical process control concerns process stability over time. A process is in control when only common-cause variation is present and its statistical behavior is predictable.

Process capability concerns whether the stable process distribution fits within engineering specification limits. Capability compares the process mean and spread with the lower and upper specification limits.

Therefore, an in-control process can still produce nonconforming items if:

• Its natural variation is wider than the specification interval.
• Its stable mean is too close to one specification limit.
• The process is stably centered at an unsuitable target.

Control limits are calculated from process data, whereas specification limits are established by design, customers, or engineering requirements. Control limits must not be replaced by specification limits. Capability should normally be assessed only after statistical stability has been established.

A practical implementation includes the following activities:

1. Select the process characteristic: Choose a measure linked to customer requirements or process risk.
2. Select the chart: Use - or - for measurements, or for nonconforming units, and for defects on a constant inspection unit.
3. Create rational subgroups: Form subgroups so that within-group variation represents short-term common causes.
4. Collect baseline data: Gather observations in time order under representative conditions.
5. Establish trial limits: Calculate limits and investigate preliminary signals.
6. Prepare a reaction plan: Specify responsibilities and actions for each signal.
7. Monitor routinely: Plot data promptly and apply consistent decision rules.
8. Investigate causes: Record machine, material, method, operator, measurement, and environmental factors.
9. Standardize improvements: Retain verified corrective actions and prevent recurrence.
10. Recalculate limits appropriately: Revise limits after a sustained, verified process improvement rather than after every short-term fluctuation.

1. Shaft diameter measurements:
Use an - chart for small subgroups or an - chart for larger subgroups. Diameter is continuous variables data. The chart monitors the average diameter, while the or chart monitors variability.

2. Number of rejected shafts:

• Use an chart if the number inspected in every sample is constant.
• Use a chart if sample sizes vary or if the fraction rejected is the preferred measure.

Each shaft is classified as conforming or nonconforming, so a binomial attributes chart is appropriate.

3. Number of surface flaws per standard sheet:
Use a chart because multiple flaws can occur on one sheet and every inspected sheet has the same area of opportunity.

The selections depend not only on whether the data are variables or attributes but also on whether sample size or inspection-unit size remains constant.