ARL₀ from a control limit
With the limit width in σ units in cell B1 (e.g. 3):
=1/(2*(1-NORM.S.DIST(B1,TRUE)))
Returns ≈ 370 for B1 = 3, and ≈ 22 for B1 = 2 — the exact contrast the Part 3 simulation demonstrates.
Signal or Noise — ARL, Nelson Rules & Control Limit Design
ARL, α and β, the eight Nelson tests, why 3-sigma won out, and why CUSUM beats it at catching small shifts — with every probability derived and simulated, not asserted.
370ARL₀ at 3σ (points between false alarms)
8Nelson tests, each tuned to a distinct pattern
4.4×faster shift detection with a matched CUSUM chart
In this lesson
01ARL & error ratesDerive 1/α and 1/(1−β) from first principles, with a live power calculator.
02The 8 Nelson testsEvery rule's probability worked out step by step, with pattern sparklines.
032σ vs 3σRun a real Monte Carlo simulation to see the false-alarm trade-off yourself.
04CUSUM vs ShewhartWatch a matched CUSUM chart out-detect a 3σ chart on identical data.
05Summary & self-checkOne reference table, key takeaways, and a short quiz to test retention.
06Statistics ImplementationExcel functions, Minitab menu paths, and exam tips for everything above.
ARL, 1/α, and 1/(1−β)
Every control chart rule you'll ever use is a bet against a stable, random process. Average Run Length (ARL) is how you keep score: how many points, on average, before the chart signals — under a given process condition.
In-control: ARL₀ = 1/α
α is the probability that a single point falsely signals when nothing is actually wrong. For a normal distribution and a 3σ limit, α is just the tail area beyond ±3 standard deviations.
α = P(Z < −3) + P(Z > 3)
α = 0.00135 + 0.00135
α ≈ 0.0027
ARL₀ = 1 / α
ARL₀ = 1 / 0.0027
ARL₀ ≈ 370 points
Why 1/α works
Each point is an independent Bernoulli trial: "signal" or "no signal," with probability α of signaling. The number of trials until the first success follows a geometric distribution, and the mean of a geometric distribution is exactly 1/p. So ARL₀ isn't a rule of thumb — it falls straight out of the geometric mean.
Out-of-control: ARL₁ = 1/(1−β)
β is the probability of missing a real shift on any given point — the point still falls inside the limits even though the process has moved. (1−β) is power: the chance you catch the shift on that point. Power depends entirely on how big the shift is, in σ units.
Try it
Drag the shift size below. Watch β and ARL₁ move together — small shifts are sneaky; big shifts get caught almost immediately.
1.5σ
same shift, same limit — just a fresh random path
Notice the trap: β (miss probability) and ARL₁ (points-to-detect) both explode as the shift shrinks toward zero, and both collapse toward 1 as the shift grows large. There's no free lunch — sensitivity to small shifts always trades against false-alarm rate, which is exactly why the Nelson tests exist: they buy sensitivity to small/moderate shifts without touching the primary 3σ limit.
Don't confuse these two lines
The chart above marks two different things, and it's easy to blur them together:
The shaded region starts at the control limit (±3σ from the original mean) — fixed, decided in advance, never moves. This is the chart's actual decision rule, and it's what power (1−β) is measured against.
The dashed violet line marks where the two curves cross — where the shifted and original distributions have equal height. This point slides with the shift (it sits at exactly half the shift size) but it is not a decision boundary the chart uses.
Sliding the "catches the shift" boundary from the control limit to the crossing point would badly overstate power — a real 0.5σ drift is only caught ~0.6% of the time at the true 3σ limit, but would look like ~60% if you measured from the crossing point instead. The crossing point is the boundary you'd use if you already knew the exact shift size ahead of time (a Neyman–Pearson-style optimal test); a real control chart can't do that because it has to watch for shifts of every possible size with one fixed rule. That constraint is exactly why the Nelson zone tests exist as a workaround, rather than the chart adapting its limit.
The 8 Nelson tests
Every Shewhart chart splits each side into three bands. All eight tests are just different questions asked about which band points land in, and how many in a row.
Zone reference probabilities (one side, standard normal) — every derivation below is built from these four numbers.
| Region | P (one side) | P (both sides) |
|---|---|---|
| Beyond 3σ (past Zone A) | 0.00135 | 0.0027 |
| Beyond 2σ (Zone A or worse) | 0.02275 | 0.0455 |
| Beyond 1σ (Zone B or worse) | 0.1587 | 0.3174 |
| Within 1σ (Zone C) | 0.3413 | 0.6826 |
2σ vs 3σ — run the simulation yourself
Don't take α ≈ 0.0455 vs 0.0027 on faith. Generate a genuinely stable, in-control process below and watch how often each limit width cries wolf.
370
0FALSE ALARMS AT 3σ
0FALSE ALARMS AT 2σ
—2σ ALARM RATE ÷ 3σ ALARM RATE
Why not just tighten to 2σ?
At 2σ, ARL₀ collapses to about 22 points — roughly 1 false alarm every 22 subgroups even when the process hasn't moved at all. Run this on a high-volume line (every heat, every coil) and operators either chase noise constantly or, more dangerously, learn to ignore the chart. 3σ was chosen as the economic balance: rare enough false alarms that a signal can be trusted, without waiting forever to catch real shifts.
The layered design
This is why real SPC systems don't just pick one limit width and stop. They keep the primary limit at 3σ (protecting ARL₀ ≈ 370) and layer the Nelson zone tests on top, each aimed at a specific shift signature the lone 3σ rule would catch too slowly. You get most of the detection speed of tighter limits without paying for it in constant false alarms — as long as you're deliberate about which tests you actually turn on (see the combined-α note below).
The catch
The 8 tests are calibrated individually, but they run simultaneously on every point and aren't independent of each other. Treating them as roughly independent, the combined chance that at least one test fires on a stable point is noticeably higher than any single test's α — summing the both-sides probabilities above gives roughly 2.2% per point with all 8 active (a union-bound upper estimate, since overlapping tests can't simply be added) — dragging combined ARL₀ down toward the 45–70 range. That's the real reason most production charts run a curated subset (often 1, 2, 3, 5) rather than all eight blindly.
CUSUM — catching small shifts faster
Every rule so far — 3σ, the Nelson zone tests — looks at a handful of recent points and forgets everything before them. A CUSUM chart (Cumulative Sum) never forgets: it adds up every deviation from target, forever, so even a shift too small to trip any single-point rule eventually accumulates into an unmistakable signal.
How it works
Two running sums track evidence of an upward shift (C⁺) and a downward shift (C⁻). Each resets to zero the moment it has no evidence left, but as long as the process keeps leaning one direction, it keeps climbing:
C⁺ᵢ = max(0, C⁺ᵢ₋₁ + (xᵢ − target) − k)
C⁻ᵢ = max(0, C⁻ᵢ₋₁ − (xᵢ − target) − k)
Signal the moment C⁺ᵢ > H or C⁻ᵢ > H
k (the "allowance" or reference value) is how much deviation gets forgiven each point before it counts toward the sum — conventionally set to half the shift size you most want to catch quickly (k = Δ/2). H (the decision interval) is the threshold that triggers a signal, chosen to hit a target false-alarm rate — exactly the same design decision as choosing 3σ for a Shewhart chart, just with different knobs.
Matched comparison, not a rigged one
To compare fairly, both charts below are tuned to the same false-alarm rate: the Shewhart chart uses the standard 3σ limit (ARL₀ ≈ 370), and this CUSUM uses k = 0.5σ with H calibrated by simulation to ARL₀ ≈ 370 as well. Any difference you see in detection speed is a real difference in power, not one chart being given an easier bar to clear.
Watch them race on the same data
Both charts below read the same random sequence of points — same pre-shift phase, same shift point, same noise. Only the decision logic differs.
0.75σ
k = 0.5σ, H ≈ 4.77σ (both matched to ARL₀ ≈ 370)
SHEWHART (3σ) — resets every point
CUSUM — accumulates evidence
The systematic result
One random draw is a story, not proof. Here's what a large simulation (thousands of trials per row, both methods calibrated to ARL₀ ≈ 370) says about the average number of points needed to catch a shift of each size:
| Shift size | Shewhart ARL₁ (exact) | CUSUM ARL₁ (simulated) | CUSUM is |
|---|---|---|---|
| 0.25σ | 281.2 | 120.7 | 2.3× faster |
| 0.50σ | 155.2 | 35.0 | 4.4× faster |
| 1.00σ | 43.9 | 9.9 | 4.4× faster |
| 1.50σ | 15.0 | 5.5 | 2.7× faster |
| 2.00σ | 6.3 | 3.8 | 1.6× faster |
| 3.00σ | 2.0 | 2.5 | 0.8× — Shewhart wins |
Where CUSUM stops winning
Notice the last row. For a huge, sudden jump (3σ or more), Shewhart is actually just as fast or slightly faster — a single point that extreme trips the 3σ rule almost immediately, while CUSUM still has to accumulate at least a little evidence. CUSUM's advantage is specifically for small and moderate sustained shifts — exactly the range where a Shewhart chart is weakest and the Nelson zone tests were invented to compensate. CUSUM is a different, often better, way to solve the same problem those zone tests target.
Summary reference
One table, all eight tests, the math behind each probability.
| # | Rule | P (one side) | P (both) | Detects |
|---|---|---|---|---|
| 1 | 1 point beyond 3σ | 0.00135 | 0.0027 | Sudden large shift, outlier |
| 2 | 9 in a row, same side of mean | 0.00195 | 0.0039 | Sustained small shift |
| 3 | 6 in a row, steadily trending | 0.00139 | 0.0028 | Trend / drift |
| 4 | 14 in a row, alternating | — | ≈0.00024 | Overcontrol / interleaved streams |
| 5 | 2 of 3 in Zone A+, same side | 0.00153 | 0.0031 | Moderate shift |
| 6 | 4 of 5 in Zone B+, same side | 0.00277 | 0.0055 | Small sustained shift |
| 7 | 15 in a row within Zone C | — | 0.0033 | Stratification / hidden mixture, averaged out |
| 8 | 8 in a row, none in Zone C | — | 0.0001 | Mixture of two distributions |
Practical tip — orientation pivoting
When you're checking these rules across a dataset with multiple test orientations per sample (LPA vs TWA, for instance), pivot by sample and heat first so each row shows both orientations side by side before you scan for Nelson patterns — a shift that only shows up in one orientation is a very different root cause than one showing up in both.
df.groupby(['sample_no','heat_no','test_orientation_code']).first().unstack('test_orientation_code')
Key takeaways
1
ARL is just a geometric-distribution mean. ARL₀ = 1/α (false alarms under a stable process) and ARL₁ = 1/(1−β) (points to catch a real shift) trade off against each other — tightening one always loosens the other.
2
3σ limits exist because they balance that trade-off economically: rare enough false alarms (ARL₀ ≈ 370) to trust a signal, without an unreasonable wait to catch real shifts.
3
The 8 Nelson tests each target a different failure signature (jump, drift, oscillation, moderate/small shift, stratification, mixture) by asking how surprising a run of points would be under a stable process — but running all 8 at once inflates the combined false-alarm rate well past any single test's α.
4
A control limit is a fixed decision boundary, decided before you know if or how much the process will shift. It is not the point where two hypothetical distributions cross — that's a different, shift-dependent concept.
5
CUSUM accumulates evidence across points instead of judging each one alone, making it 2–4× faster than a matched Shewhart chart at catching small/moderate sustained shifts — though Shewhart holds its own (or edges ahead) for a single huge jump.
Check your understanding
Five quick questions. Click a question to reveal the answer and reasoning — no grading, just a gut check before you move on.
Statistics Implementation
Every number in this lesson — α, β, ARL, the eight Nelson-test probabilities, CUSUM's k and H — can be reproduced in Excel or navigated to directly in Minitab. This section is the "how do I actually build this" reference.
Excel Functions
| Function | Syntax | Purpose | When to use it |
|---|---|---|---|
| NORM.S.DIST | =NORM.S.DIST(z,TRUE) | Standard normal CDF, Φ(z). | Compute one-sided tail area beyond a control limit, e.g. =1-NORM.S.DIST(3,TRUE) for the 3σ tail (≈ 0.00135), the building block of α and ARL₀. |
| NORM.S.INV | =NORM.S.INV(probability) | Inverse standard normal CDF (z from a probability). | Back-solve for a control limit width given a target α, or find the crossing point between two shifted normal curves. |
| NORM.DIST | =NORM.DIST(x,mean,sd,TRUE) | CDF of a normal distribution with any mean/SD. | Compute β directly for a shifted process: =NORM.DIST(limit,shift,1,TRUE)-NORM.DIST(-limit,shift,1,TRUE) reproduces the β calculation used throughout Part 1. |
| BINOM.DIST | =BINOM.DIST(k,n,p,FALSE) | Exact binomial point probability. | Derive the Nelson zone-test probabilities (Rules 5 and 6), e.g. =BINOM.DIST(2,3,0.02275,FALSE) for "2 of 3 beyond Zone A." |
| COMBIN | =COMBIN(n,k) | Number of combinations, C(n,k). | Build the same zone-test binomial terms by hand instead of relying on BINOM.DIST, useful when showing the derivation step by step (as in the "Show derivation" panels above). |
| FACT | =FACT(n) | Factorial, n!. | Compute the trend-test denominator (6! = 720) used to derive Rule 3's P ≈ 0.0028. |
| GEOMDIST / NEGBINOM.DIST | =NEGBINOM.DIST(0,1,p,TRUE) | Geometric distribution mean/CDF (Excel exposes it as a special case of the negative binomial). | Confirm that ARL is exactly the geometric-distribution mean, 1/p — the identity behind ARL₀ = 1/α and ARL₁ = 1/(1−β). |
Excel Use Cases
ARL₁ for a given shift
With limit in B1 and shift size (σ units) in B2:
=1/(1-(NORM.DIST(B1,B2,1,TRUE)-NORM.DIST(-B1,B2,1,TRUE)))
This is exactly the calculation driving the live power calculator in Part 1 — change B2 and watch the answer move the same way the slider does.
Nelson Rule 5 probability (2 of 3 in Zone A+)
With p = 0.02275 (one-side Zone A tail) in B1:
=(COMBIN(3,2)*B1^2*(1-B1)+B1^3)*2
The *2 doubles the one-side result for "either side," matching the both-sides column in the Part 2 zone-probability table.
CUSUM decision interval, by simulation
Excel has no closed-form CUSUM ARL function; build a small Monte Carlo column: =MAX(0,A2+(RAND_NORM()-target)-k) chained down a column for C⁺, flag the first row where it exceeds a trial H, and repeat/average across many trials (or use Data Table sensitivity) to calibrate H to a target ARL₀ — exactly what produced the H ≈ 4.77 value used in Part 4.
Minitab Navigation
| Task | Menu path | Notes |
|---|---|---|
| Build a Shewhart individuals chart (3σ limits) | Stat → Control Charts → Variables Charts for Individuals → I-MR | Minitab plots the individuals (I) chart with 3σ limits by default and the moving-range (MR) chart alongside it — the direct equivalent of the Part 1/3 simulations. |
| Turn on the Nelson (Western Electric) zone tests | Stat → Control Charts → I-MR → I-MR Options → Tests tab | Minitab lists all 8 tests by number, matching this lesson's numbering exactly, and lets you select a subset. Selecting all 8 reproduces the inflated combined-α effect discussed in Part 3 — most practitioners enable only 1, 2, 3, and 5. |
| Tighten or loosen the control limit width | Stat → Control Charts → I-MR → I-MR Options → Estimate tab (or Parameters, to fix limits manually) | Minitab defaults to 3σ; to reproduce the Part 3 comparison, run the chart twice — once at the default and once with sigma limits manually set to 2 — and compare the number of flagged points. |
| Build a CUSUM chart | Stat → Control Charts → Time-Weighted Charts → CUSUM | The dialog asks directly for the reference value (k, as "shift to detect" in σ or raw units) and the decision interval (H) — the same two knobs derived in Part 4. Minitab can also estimate them from a target ARL, which is how the k = 0.5σ, H ≈ 4.77 pairing in this lesson was chosen. |
Which option should you choose?
Start from what you're trying to catch: a single Shewhart I-MR chart with a curated subset of Nelson tests (1, 2, 3, 5) covers sudden shifts, trends, and moderate sustained shifts without inflating the false-alarm rate too far past ARL₀ ≈ 370. Add a CUSUM chart specifically when the process risk is a small, slow sustained drift — exactly the case Part 4 shows Shewhart is weakest against.
Exam Tips
1
ARL questions are geometric-mean questions in disguise. ASQ CSSBB/CQE items that ask "how many subgroups on average before a false alarm" are testing whether you know ARL₀ = 1/α, not a memorized constant — recompute it from the tail area if the limit width in the question isn't the standard 3σ.
2
Know which Nelson rule number matches which pattern. Exam scenarios describe a pattern in words ("nine consecutive points on one side of the centerline") and expect you to name the rule and what it detects — drift, oscillation, stratification, or a mixture — not just recite probabilities.
3
Don't confuse the control limit with the crossing point. A common exam trap describes "where the shifted and original distributions overlap" and asks you to compute power from it — the correct boundary is always the fixed control limit, never the shift-dependent crossing point.
4
CUSUM vs. Shewhart is a recurring comparison question. Remember the shape of the trade-off: CUSUM wins for small/moderate sustained shifts, Shewhart holds even or wins for one huge jump — and both are typically compared at a matched ARL₀ so the comparison is fair.
5
Combined false-alarm rate is a favorite "gotcha." If a question implies all 8 Nelson tests run simultaneously, the effective ARL₀ is well below 370 — don't answer with the single-rule 0.27% α unless the question specifies only the primary 3σ rule is active.