Statistical process control

The formula

The whole discipline hangs on one decision rule. Estimate the process center and its short-term standard deviation while it runs undisturbed, then draw limits at three sigma:

UCL = μ̂ + 3·σ̂        CL = μ̂        LCL = μ̂ − 3·σ̂

Reading: a point inside the limits is the process being itself; a point outside is worth walking to the machine for. The critical detail is where σ̂ comes from — within-subgroup variation, never from the drawing tolerance ^[4]. And σ̂ must be the standard deviation of whatever statistic you actually plot: if you chart subgroup means from rational subgroups of size n, then R̄/d₂ estimates the sigma of individual pieces, and the limits are μ̂ ± 3·(R̄/d₂)/√n — the tabled shortcut μ̂ ± A₂·R̄ — not μ̂ ± 3·R̄/d₂, which is √n too wide and never signals. On an individuals chart (n = 1) the estimate comes from the moving range instead, σ̂ = M̄R/1.128. The constant d₂ depends on subgroup size: 1.128 for n = 2, 2.326 for n = 5 ^[4].

How often a chart signals is measured by average run length:

ARL = 1/p        p = probability a single point signals

Reading: for a stable, roughly normal process, p = 0.0027 outside 3-sigma limits, so a false alarm arrives about once every 370 points ^[4]. Shift the mean by one sigma and p rises to 0.0228 — the chart now signals in about 44 points on average.

For shifts too small for a Shewhart chart to catch quickly, the CUSUM accumulates evidence ^[5]^[8]:

S⁺_i = max(0, S⁺_{i−1} + x_i − μ₀ − k)        signal when S⁺ > h

Reading: each point deposits its excess over the target (minus an allowance k, usually half the shift you care about, with h around 4 to 5 sigma) into a running sum; a real shift makes the sum climb steadily while noise keeps draining back to zero. The mirror-image S⁻ watches the low side.

The EWMA chart does the same job with a fading memory ^[6]^[11]:

z_i = λ·x_i + (1−λ)·z_{i−1}        limits: μ₀ ± L·σ·√( λ/(2−λ) · (1 − (1−λ)^(2i)) )

Reading: each plotted point is a weighted blend of the newest reading and everything before it; λ between about 0.05 and 0.3 sets how fast old data fades ^[6]^[11].

Where you meet it

The SMT line. Solder paste height off the stencil printer, sampled every panel, charted at the machine. The chart is what decides whether a low reading means "wipe the stencil and requalify" or "keep running" — and it makes that call before X-ray finds the voids two stations later.
The test stand, before first ignition. A load-cell check standard exercised every test morning, one point per day on an individuals chart. When the customer later asks whether the thrust data from March can be trusted, that chart is the answer — the channel was demonstrably in control the whole campaign.
The environmental lab. Chamber temperature uniformity run against a reference profile weekly, charted per zone. A drifting zone shows up as a run below the center line weeks before it would fail the annual survey, which is the difference between a scheduled fix and an impounded month of qual data.
The supplier review board. A PPAP or source-inspection package lands with charts attached, and the working question is whether the process was actually in statistical control when the capability numbers were computed. Cpk calculated on an out-of-control process is a number about the past with no claim on the next lot.

How it works

Shewhart's founding move was to split variation into two kinds and attach a different management action to each ^[1]^[2]. Common causes are the ordinary noise of the process — the sum of a hundred small influences nobody can economically remove one at a time. Assignable (special) causes are specific, findable events: a worn tool, a bad solder paste lot, a loose thermocouple. The chart is a filter that routes each point to the right action. Signal: go find the assignable cause. No signal: leave the process alone.

Both halves of that rule get violated on real floors, and each violation has its own cost.

Violation one is drawing the limits at the tolerance. Spec limits describe what the customer needs; control limits describe what the process does. They are different questions with different math, and a process can be comfortably in control while shipping scrap, or out of control while every part happens to measure in spec. Control answers "did something change?"; a capability study (Cp, Cpk) answers "is the process good enough?" — in that order, because capability computed from an unstable process predicts nothing.

Violation two is tampering: adjusting an in-control process every time a point lands off center. Each adjustment responds to noise, so it adds a fresh offset on top of the existing scatter, and total variation goes up, not down. The chart's real economic content is the permission it gives you to not act. Deming built his management argument on exactly this distinction between the two kinds of causes ^[10].

The practice runs in two phases ^[1]. Phase I is retrospective: collect twenty-plus subgroups, compute trial limits, hunt down and remove assignable causes, recompute, and freeze the limits. Phase II is live monitoring against those frozen limits. Recomputing limits every week quietly converts drift into "the new normal," which defeats the entire point.

Chart choice is a matter of what size shift you need to catch, and the trade is quantifiable. A plain 3-sigma Shewhart chart false-alarms about every 370 points and catches a 2-sigma shift in about 6 points — but takes about 44 points to notice a 1-sigma shift. Adding the Western Electric zone rules (2 of 3 beyond 2σ, 4 of 5 beyond 1σ, 8 in a row one side) speeds up small-shift detection but drops the false-alarm interval to about every 92 points ^[4]^[7]. A CUSUM designed with k = 0.5σ and h = 5σ catches that same 1-sigma shift in about 10 points while false-alarming only about every 460 — several times faster than Shewhart at a better false-alarm rate, which is why CUSUM and EWMA own the small-sustained-drift problem while the Shewhart chart keeps the large-shift, read-it-at-a-glance job ^[5]^[6]. (All four of those run lengths check out by direct simulation.)

The assumption that breaks SPC most often in test work is independence. The math prices each point as fresh evidence. Chart a slow thermal process sampled every second, or any autocorrelated DAQ stream, and successive points mostly repeat each other — the moving-range sigma estimate collapses, the limits hug the data, and the chart alarms constantly on nothing. Sample slower than the process correlation time, or use a time-series-aware method, before trusting any limit.

History

The starting document is one page long. Walter Shewhart, a physicist who had joined Western Electric's inspection engineering group at the Hawthorne Works in 1918, sent his boss George Edwards a memo dated May 16, 1924, containing a short argument and a hand-drawn diagram that any quality engineer today would recognize as a control chart ^[2]^[3]. Edwards later said that page held essentially all the principles of modern process quality control. Shewhart moved to the new Bell Telephone Laboratories in 1925 and developed the memo into the 1931 book Economic Control of Quality of Manufactured Product, still the field's founding text ^[1]^[2]^[3].

The method spread through two channels. Inside Western Electric, a committee codified the practical zone rules into the 1956 Statistical Quality Control Handbook, which became a standard text and gave the floor its pattern tests ^[7]. Outside, W. Edwards Deming — Shewhart's collaborator and editor — carried the framework to Japan, where the Union of Japanese Scientists and Engineers brought him over in the summer of 1950 to teach SPC to hundreds of engineers and executives ^[1]^[10]. Japanese manufacturers institutionalized the charts at a moment when much of American industry had shelved them, a divergence both industrial bases spent the following decades measuring.

The sequential refinements came from the other side of the Atlantic. E. S. Page at Cambridge published "Continuous Inspection Schemes" in Biometrika in 1954, introducing the cumulative-sum chart as a way to accumulate evidence of a parameter change instead of judging each point alone ^[8]^[9]. S. W. Roberts followed in 1959 with the exponentially weighted moving average chart, trading the CUSUM's hard reset for a geometric fade ^[6]^[11]. Between Shewhart's one-page memo and those two papers, essentially every process-monitoring scheme running on a factory network today was already on paper.

Related tools

/tools/rms-peak — the σ̂ under every control limit is the RMS of the deviations from center; same computation, different context
/tools/snr-enob — deciding whether a reading is signal or noise floor is the instrumentation version of assignable versus common cause
/tools/strain-gauge-bridge — the kind of bench measurement chain a daily check-standard chart is built to watch
/tools/bearing-life-l10 — the other everyday case of making an engineering decision from a statistical population instead of one number

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →