Statistical process control and Six Sigma

No two products are ever exactly alike. Two bottles filled by the same machine a second apart contain slightly different volumes; two calls handled by the same agent take different times; two loaves from the same oven differ in weight. Variation is a fact of every process. The question operations managers face is not whether a process varies, but whether the variation is the normal, stable variation the process always shows, or a sign that something has changed and needs attention. Statistical process control (SPC) answers that question with simple charts, and process capability analysis and Six Sigma extend it to ask whether the process is good enough to meet customer requirements.

This lesson explains common and special causes of variation, control charts for measured variables and for attributes, how to interpret them, process capability indices, the Six Sigma approach and its DMAIC method, and acceptance sampling. It assumes the basic statistics of means, standard deviations and the normal distribution covered in BUS-110 and STAT-141.

Common and special causes of variation

Shewhart distinguished two kinds of variation. Common (or random, natural) causes are the many small, unavoidable sources of variation built into a process: tiny differences in raw materials, temperature, machine vibration and human motion. Together they produce a stable pattern that is predictable within limits. Special (or assignable) causes are specific, identifiable events that are not part of the process as designed: a worn tool, a new batch of material from a different supplier, an untrained operator, a sensor that has drifted. A process with only common-cause variation is said to be in statistical control.

The distinction matters because the right response differs. Special causes should be found and removed by the people closest to the process, ideally as soon as they appear. Common-cause variation can only be reduced by changing the process itself, which is usually a management responsibility: better equipment, materials, methods or training. Deming warned of two costly mistakes: treating common-cause variation as if it had a special cause (tampering, or adjusting a stable process in response to random noise, which actually increases variation), and failing to react to a real special cause. Control charts help avoid both.

Control charts for variables

A control chart plots a sample statistic over time against a center line and upper and lower control limits. Control limits are calculated from the process's own data, usually set at three standard deviations of the plotted statistic above and below the center line. If the process is in control and the statistic is roughly normally distributed, about 99.7 % of points fall within the limits, so a point outside them is strong evidence of a special cause. Control limits describe what the process does; they are not specification limits, which describe what the customer wants.

For variables measured on a continuous scale, such as weight, length, temperature or time, two charts are normally used together. The x-bar chart tracks the average of each small sample (often 4 or 5 units taken at regular intervals) and detects shifts in the process center. The R chart tracks the range (largest minus smallest value) of each sample and detects changes in process spread. With the grand mean \(\bar{\bar{x}}\) (the average of sample means) and the average range \(\bar{R}\), the limits are:

import random, statistics as st
random.seed(230)
# 25 samples of 5 bottles each; fill volume in mL (target 500)
samples = [[random.gauss(500.4, 1.2) for _ in range(5)] for _ in range(25)]
for s in samples[20:]:                 # a shift in the process after sample 20
    for i in range(5):
        s[i] += 2.0
A2, D3, D4 = 0.577, 0.0, 2.114          # constants for n = 5
base = samples[:20]                     # set limits from the first 20 samples
xbar = [st.mean(s) for s in base]
rng = [max(s) - min(s) for s in base]
xbb, rbar = st.mean(xbar), st.mean(rng)
ucl, lcl = xbb + A2 * rbar, xbb - A2 * rbar
print(f"Grand mean {xbb:.2f} mL, average range {rbar:.2f} mL")
print(f"x-bar chart: LCL {lcl:.2f}  UCL {ucl:.2f};  R chart: LCL {D3*rbar:.2f}  UCL {D4*rbar:.2f}")
for k, s in enumerate(samples, 1):
    m, r = st.mean(s), max(s) - min(s)
    flag = "OUT" if not (lcl <= m <= ucl) or r > D4 * rbar else ""
    if k > 17 or flag:
        print(f"sample {k:>2}: mean {m:7.2f}  range {r:4.2f} {flag}")
# Capability from the in-control data: sigma estimated as R-bar / d2 (d2 = 2.326 for n = 5)
sigma = rbar / 2.326
LSL, USL = 496.0, 504.0
cp = (USL - LSL) / (6 * sigma)
cpk = min(USL - xbb, xbb - LSL) / (3 * sigma)
print(f"sigma {sigma:.2f} mL; Cp {cp:.2f}; Cpk {cpk:.2f}")

Grand mean 500.28 mL, average range 2.98 mL
x-bar chart: LCL 498.56  UCL 502.00;  R chart: LCL 0.00  UCL 6.31
sample 18: mean  499.94  range 4.91 
sample 19: mean  499.72  range 4.22 
sample 20: mean  500.87  range 3.04 
sample 21: mean  502.07  range 1.93 OUT
sample 22: mean  502.23  range 2.47 OUT
sample 23: mean  503.06  range 2.01 OUT
sample 24: mean  504.13  range 2.09 OUT
sample 25: mean  502.18  range 4.70 OUT
sigma 1.28 mL; Cp 1.04; Cpk 0.97

The limits are set from the first 20 samples, when the filler was running normally. From sample 21, a simulated shift of 2 mL pushes every sample mean above the upper control limit, while the ranges stay normal: the filler is overfilling consistently rather than becoming erratic. In practice, an operator would stop and adjust the filler at the first signal. The last line of the output previews capability analysis, discussed below: even in control, this filler is only marginally capable of meeting a 496 to 504 mL specification.

Control charts for attributes

Attributes are characteristics that are counted rather than measured: a unit is defective or not, an invoice has an error or not, a call is resolved or not. For the proportion defective in samples of size \(n\), the p-chart is used. Its center line is the average proportion defective \(\bar{p}\), and its limits are:

When the count of defects per unit is of interest, such as the number of scratches on a car door or the number of complaints per day, the c-chart is used, with center line \(\bar{c}\) and limits \(\bar{c} \pm 3\sqrt{\bar{c}}\). Attribute charts need larger samples than variable charts because a yes-or-no observation carries less information than a measurement.

Reading control charts

A point beyond the control limits is the clearest signal, but non-random patterns inside the limits also indicate special causes. Common rules, such as the Western Electric rules, flag a run of seven or eight consecutive points on one side of the center line, six points in a row steadily increasing or decreasing, two of three consecutive points near a control limit, and cycles or other systematic patterns. Using too many rules at once raises the chance of false alarms, so organizations usually choose a few.

Control charts are most useful when they are kept by the people who run the process and acted on immediately. A control chart posted on a wall and updated weekly by a quality technician catches problems long after the defective product has been made. Modern sensors and software can chart thousands of variables automatically and alert operators in real time. Before charts are used for monitoring, the process must be brought into control, with special causes removed, and the limits calculated from that stable data. Limits should be recalculated after a deliberate process change.

Process capability

A process can be in statistical control and still produce many defects if its natural variation is too wide for the specification. Process capability compares the spread of the process with the specification limits set by the customer or designer. The capability ratio \(C_p\) compares the width of the specification with the natural spread of the process (six standard deviations). The capability index \(C_{pk}\) also accounts for whether the process is centered, using the distance from the mean to the nearer specification limit:

A \(C_p\) of 1.0 means the process spread exactly fills the specification, so even a perfectly centered process produces about 0.27 % defects (2,700 per million). Many firms require \(C_{pk}\) of at least 1.33 from their suppliers, and automotive and aerospace customers often require 1.67 for critical characteristics. If \(C_{pk}\) is much lower than \(C_p\), the process could be improved simply by centering it; if \(C_p\) itself is low, the process variation must be reduced.

Six Sigma

Six Sigma is a disciplined, data-driven approach to reducing variation and defects, developed at Motorola in the 1980s and popularized by General Electric in the 1990s. The name refers to a process whose specification limits lie six standard deviations from the mean. Allowing for a conventional long-term drift of 1.5 standard deviations in the process mean, a six sigma process produces about 3.4 defects per million opportunities (DPMO). A typical three sigma process, by the same convention, produces about 66,800 DPMO. Defects per million opportunities is calculated as:

Six Sigma projects follow the DMAIC method. Define the problem, the customer requirements and the project scope, often in a project charter. Measure the current process performance and validate the measurement system. Analyze the data to find and verify root causes. Improve the process by developing, testing and implementing solutions. Control the improved process with control charts, standard work and monitoring so the gains last. For new processes or products, a related method, DMADV (define, measure, analyze, design, verify), is used.

Six Sigma also defines roles. Executive champions select projects aligned with strategy; Master Black Belts coach and train; Black Belts lead projects full time; Green Belts lead smaller projects alongside their regular jobs. Many organizations combine Six Sigma with lean (lesson 11) as Lean Six Sigma, using lean to remove waste and speed flow and Six Sigma to reduce variation. Critics note that Six Sigma can become bureaucratic and that the 3.4 DPMO target means little for processes where 'opportunities' are defined loosely; the underlying discipline of defining problems with data and verifying causes before acting is its lasting value.

Acceptance sampling

Acceptance sampling inspects a random sample from an incoming or outgoing lot and decides whether to accept or reject the whole lot. A sampling plan specifies the sample size and the maximum number of defectives allowed. An operating characteristic (OC) curve shows the probability of accepting lots of different quality levels under the plan. The producer's risk is the chance of rejecting a good lot (one at the acceptable quality level, or AQL); the consumer's risk is the chance of accepting a bad lot. Acceptance sampling is useful when inspecting every item is too costly or destructive, but it does nothing to improve quality, and modern practice shifts effort from inspecting lots to ensuring suppliers' processes are capable and in control.