Statistical Process Control: Monitoring and Improving Quality

Variation is the enemy of quality. Every manufacturing process produces output that varies — no two parts are exactly identical. Statistical process control provides the tools to understand, measure, and control this variation. It distinguishes between the normal variation inherent in any process and the special cause variation that indicates a process has gone out of control.

Walter Shewhart developed the concepts of statistical process control at Bell Laboratories in the 1920s. His insight was that processes exhibit two types of variation. Common cause variation is the natural variation inherent in the process. Special cause variation arises from specific, identifiable events. Control charts distinguish between these two types, enabling operators to know when to take action and when to leave the process alone.

The Concept of Variation

Understanding variation is the foundation of statistical process control.

Common Cause vs. Special Cause

Common cause variation is the random variation that results from the combined effect of many small, uncontrollable factors. Temperature fluctuations within a controlled range, slight variations in raw material properties, and minor operator inconsistencies all contribute to common cause variation. A process operating with only common cause variation is said to be in statistical control.

Special cause variation arises from specific events that are not part of the normal process. A tool breaking, a batch of defective raw material arriving, or an operator making a setup error all create special cause variation. These causes can and should be identified and eliminated.

Tampering — making adjustments to a process that is already in statistical control — actually increases variation. Deming illustrated this with the funnel experiment. Moving a funnel to correct each random deviation amplifies the variation instead of reducing it. The operator must resist the urge to adjust a process that is running within normal limits.

The Normal Distribution

Many process characteristics follow approximately normal distributions. The mean of the distribution represents the process center, and the standard deviation represents the process spread. In a normal distribution, 68.3 percent of values fall within one standard deviation of the mean, 95.5 percent within two standard deviations, and 99.7 percent within three standard deviations.

These percentages form the basis for control chart limits. The quality control and Six Sigma article discusses how Six Sigma relates to the normal distribution and defect rates.

Control Charts

Control charts are the primary tool of statistical process control.

X-bar and R Charts

The X-bar chart monitors the process mean. Samples of n units are taken at regular intervals, and the sample mean is plotted on the chart. The center line is the grand mean of all samples. Upper and lower control limits are set at three standard deviations of the sample means above and below the center line.

The R chart monitors the process dispersion. The range — highest minus lowest value — of each sample is plotted. The R chart detects changes in process variation. If the R chart shows an out-of-control condition, the X-bar chart cannot be interpreted until the variation is brought under control.

Individuals and Moving Range Charts

When sample sizes are one — as in many chemical processes or automated measurements — the I-MR chart is used. The individuals chart plots each individual measurement. The moving range chart plots the absolute difference between consecutive measurements.

I-MR charts are less sensitive to process shifts than X-bar charts because they are based on single measurements rather than averages. They are appropriate when sampling is expensive or when the process produces output continuously rather than in discrete batches.

Attribute Control Charts

Attribute charts monitor counts or proportions rather than continuous measurements. The p chart monitors the proportion of defective units in samples of varying size. The np chart monitors the number of defective units when sample size is constant. The c chart monitors the count of defects per unit. The u chart monitors defects per unit when the sample size varies.

Attribute charts require larger sample sizes than variable charts to achieve equivalent statistical power. A p chart typically needs a sample size that gives an expected defect count of at least 5.

Interpreting Control Charts

Control charts are interpreted using rules that detect patterns unlikely to occur by chance. A single point outside the control limits indicates a likely special cause. Seven consecutive points on one side of the center line indicates a process shift. Seven consecutive points trending upward or downward indicates a trend. Cycles and hugging the center line also indicate non-random patterns.

The reliability engineering article discusses how control charts apply to monitoring product reliability in the field.

Process Capability

Process capability analysis determines whether a process can consistently produce output within specification limits.

Capability Indices

Cp compares the specification width to the process spread. It is calculated as the specification width divided by six times the standard deviation. A Cp of 1.0 means the process spread equals the specification width — a minimally capable process. A Cp of 1.33 indicates a good process. A Cp of 2.0 corresponds to Six Sigma capability.

Cpk accounts for process centering. If the process mean is not centered between the specification limits, Cpk will be lower than Cp. Cpk measures the distance from the process mean to the nearest specification limit, divided by three standard deviations.

Process Capability Studies

A process capability study involves collecting data from the process while it is operating in statistical control, estimating the process mean and standard deviation, and calculating capability indices. The data must be collected over sufficient time to capture the full range of common cause variation.

Capability studies are the basis for process improvement prioritization. Processes with low Cpk values are candidates for process improvement projects. The project management for industrial engineering article discusses how to manage process improvement initiatives.

Non-Normal Distributions

Not all processes follow normal distributions. Some processes produce skewed, multimodal, or bounded distributions. For non-normal processes, percentiles are used instead of standard deviations to calculate capability. The Johnson transformation or Box-Cox transformation can normalize some non-normal data.

Implementation of SPC

Implementing statistical process control requires more than just drawing charts.

Selecting Critical Characteristics

Data collection is expensive. Focus control charting on critical-to-quality characteristics — those that affect product function, customer satisfaction, or downstream processes. The Pareto principle applies — 20 percent of characteristics cause 80 percent of quality problems.

Training Operators

Operators must be trained to collect samples, measure parts, plot data, and interpret control charts. They must understand the difference between common and special causes. They must be empowered to stop the process when they detect special causes. Without this empowerment, SPC becomes a paper exercise.

Reaction Plans

A control chart is useless without a defined response to out-of-control signals. Reaction plans specify who to notify, what adjustments to make, how to document the event, and how to handle product produced while the process was out of control.

Advanced SPC Methods

Beyond basic control charts, advanced methods address special situations.

Multivariate Control Charts

When multiple correlated quality characteristics must be monitored simultaneously, multivariate control charts are needed. Hotelling’s T-squared chart monitors the vector of means for multiple variables. The chart detects shifts in the relationship between variables that univariate charts would miss.

In chemical processes where temperature, pressure, and concentration are interrelated, multivariate SPC detects process shifts sooner than individual charts for each variable. The principal component analysis approach reduces the dimensionality of multivariate data for monitoring.

Short-Run SPC

Traditional control charts require 20 to 30 samples to establish control limits. Short-run SPC methods handle processes with low production volumes. The standardized X-bar and R chart uses deviation from target rather than absolute values. Z-MR charts plot the standardized values, allowing multiple part types on the same chart.

Short-run methods are essential for job shops and high-mix, low-volume manufacturing. Without these methods, low-volume processes cannot benefit from SPC.

Economic Design of Control Charts

Control chart design parameters — sample size, sampling frequency, and control limit width — are traditionally selected based on statistical criteria. Economic design selects these parameters to minimize total quality cost. The economic model considers sampling cost, investigation cost, and the cost of producing defective products.

Economic designs typically use smaller sample sizes and wider control limits than statistical designs. The cost savings from economic design are 10 to 30 percent compared to heuristic parameter selection. Implementation requires estimating the various cost components, which can be difficult in practice.

Frequently Asked Questions

What is the difference between control limits and specification limits? Control limits are calculated from the process data and reflect actual process variation. Specification limits are engineering requirements that define acceptable product. Control limits determine whether the process is stable. Specification limits determine whether the product is acceptable. A process can be in statistical control but still produce defective products if its capability is inadequate.

How often should control chart limits be recalculated? Control limits are recalculated when the process has been changed or improved. Initial limits are calculated from 20 to 30 subgroups. Revised limits extend the baseline as more data accumulates. Limits should not be recalculated reactively every time a special cause is found — that defeats the purpose of the chart.

Can SPC be applied to non-manufacturing processes? Yes. SPC is used in healthcare to monitor surgical outcomes, infection rates, and patient waiting times. It is used in finance to track transaction errors and fraud rates. It is used in software development to monitor defect rates and cycle times. Any process with measurable output can benefit from SPC.

What sample size should I use for control charts? For X-bar and R charts, sample sizes of 4 to 6 are standard. Larger samples provide greater sensitivity to process shifts but increase inspection cost. The sample size should be large enough to detect process shifts of practical importance within a reasonable number of samples.

Quality Control and Six Sigma — Production Systems Design — Reliability Engineering