Every control chart eventually gives you a signal. The question is: how long do you have to wait?
That waiting time has a name. It is called the Average Run Length, or ARL. It is one of the most useful ideas in statistical process control, and one of the most misunderstood.
What Is Average Run Length?
The run length is the number of points plotted on a control chart before the chart signals an out-of-control condition.
The Average Run Length (ARL) is the expected value of that number.
Two versions matter:
- \( ARL_0 \) is the in-control ARL. The process is stable, so any signal is a false alarm. You want this number to be large.
- \( ARL_1 \) is the out-of-control ARL. A real shift has occurred. You want this number to be small.
Good chart design is the balance between the two.
The Basic Formula
Assume the plotted points are independent. Let \( p \) be the probability that any single point falls outside the control limits.
The run length then follows a geometric distribution:
$$ P(RL = n) = (1 - p)^{n-1} p, \qquad n = 1, 2, 3, \ldots $$The mean of a geometric distribution gives us the ARL:
$$ ARL = \frac{1}{p} $$That is the whole idea in one line. A smaller signalling probability means a longer wait.
In-Control ARL for a Shewhart Chart
For a standard Shewhart chart with 3-sigma limits and a normally distributed statistic:
$$ p = 2\left[1 - \Phi(3)\right] = 0.0027 $$Therefore:
$$ ARL_0 = \frac{1}{0.0027} \approx 370 $$On average, a perfectly stable process produces a false alarm once every 370 points. This is the number behind the familiar 3-sigma convention.
The general form for \( k \)-sigma limits is:
$$ ARL_0 = \frac{1}{2\left[1 - \Phi(k)\right]} $$| Limits | \( p \) | \( ARL_0 \) |
|---|---|---|
| 2.0 sigma | 0.0455 | 22 |
| 2.5 sigma | 0.0124 | 81 |
| 3.0 sigma | 0.0027 | 370 |
| 3.5 sigma | 0.00047 | 2149 |
This table explains why 2-sigma limits are not used as action limits. A false alarm every 22 points destroys operator confidence in the chart.
Out-of-Control ARL
Now suppose the process mean shifts. Let \( \beta \) be the probability of missing the shift on any one point. That is the Type II error.
The probability of detection on a single point is \( 1 - \beta \), which is the power of the chart. So:
$$ ARL_1 = \frac{1}{1 - \beta} $$For an \( \bar{X} \) chart with 3-sigma limits and a mean shift of \( \delta \) standard deviations of \( \bar{X} \):
$$ 1 - \beta = \left[1 - \Phi(3 - \delta)\right] + \Phi(-3 - \delta) $$Detection Speed for Common Shifts
| Shift \( \delta \) | Power \( 1 - \beta \) | \( ARL_1 \) |
|---|---|---|
| 0.0 | 0.0027 | 370.4 |
| 0.5 | 0.0064 | 155.2 |
| 1.0 | 0.0228 | 43.9 |
| 1.5 | 0.0668 | 15.0 |
| 2.0 | 0.1587 | 6.3 |
| 2.5 | 0.3085 | 3.2 |
| 3.0 | 0.5000 | 2.0 |
| 4.0 | 0.8413 | 1.2 |
Read the table carefully. It carries the single most important lesson about Shewhart charts.
Large shifts are caught almost immediately. A 3-sigma shift is detected in 2 points on average.
Small shifts are not. A 0.5-sigma shift takes about 155 points. If you sample once per hour, that is nearly a week of production before the chart reacts.
The Effect of Sample Size
Subgrouping is your main lever for small shifts. The standard error of the subgroup mean is:
$$ \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} $$So a process shift of \( \delta \) process standard deviations becomes an effective shift of \( \delta\sqrt{n} \) on the chart.
Take a 1-sigma shift:
- \( n = 1 \): effective shift 1.0, \( ARL_1 \approx 44 \)
- \( n = 4 \): effective shift 2.0, \( ARL_1 \approx 6.3 \)
- \( n = 5 \): effective shift 2.24, \( ARL_1 \approx 4.5 \)
- \( n = 9 \): effective shift 3.0, \( ARL_1 \approx 2.0 \)
Larger subgroups detect faster. They also cost more. That trade-off is the heart of control chart design.
Average Time to Signal
ARL counts points, not hours. Managers care about hours. Convert using the sampling interval \( h \):
$$ ATS = ARL \times h $$A chart with \( ARL_1 = 10 \) and hourly sampling signals after about 10 hours. The same \( ARL_1 \) with 15-minute sampling signals in 2.5 hours.
You can improve detection speed in two ways: sample more often, or sample more units. Both cost money.
The Warning Everyone Ignores
ARL is an average, and the run length distribution is heavily skewed. The standard deviation of the run length is:
$$ SDRL = \frac{\sqrt{1 - p}}{p} $$For \( p = 0.0027 \), the SDRL is about 370. The standard deviation is essentially equal to the mean.
Two consequences follow:
- The median run length is only about 257 points, not 370. More than half of all run lengths are shorter than the average.
- The probability of at least one false alarm in the first 100 points is \( 1 - (0.9973)^{100} \approx 24\% \).
So do not treat \( ARL_0 = 370 \) as a promise of 370 quiet points. Early false alarms are common and completely normal.
What Sensitizing Rules Do to ARL
Runs rules such as the Western Electric rules improve sensitivity to small shifts. They also add more chances to signal falsely.
Applying the full Western Electric rule set drops \( ARL_0 \) from roughly 370 to roughly 92. That is a false alarm about four times more often.
The trade is deliberate, not free. Add rules only when small-shift detection genuinely matters to you, and expect more investigations of stable processes.
When Shewhart Charts Are the Wrong Tool
A Shewhart chart uses only the most recent point. It has no memory. That is exactly why it is slow on small shifts.
CUSUM and EWMA charts accumulate information across points. For the same in-control ARL, they detect small shifts far faster.
| Chart | Approx. \( ARL_0 \) | \( ARL_1 \) at 1-sigma shift |
|---|---|---|
| Shewhart \( \bar{X} \), 3-sigma, \( n=1 \) | 370 | 44 |
| CUSUM, \( k = 0.5 \), \( h = 5 \) | 465 | 10 |
| EWMA, \( \lambda = 0.1 \), \( L = 2.8 \) | 500 | 10 |
The rule of thumb is simple:
- Shifts of 1.5 sigma or more: Shewhart charts work well.
- Shifts below 1.5 sigma: use CUSUM or EWMA.
Assumptions Behind the Numbers
Published ARL values rest on assumptions. Check them before quoting the numbers.
- Plotted statistics are independent. Autocorrelated data inflates false alarms badly.
- The underlying distribution is normal, or subgroups are large enough for the central limit theorem to apply.
- Process parameters are known. Limits estimated from a small baseline give an actual \( ARL_0 \) well below 370.
- The shift is sustained, not a single spike.
Practical Takeaways
- Design charts around \( ARL_0 \) and \( ARL_1 \) together, never one alone.
- Decide the smallest shift worth detecting before choosing subgroup size.
- Use \( ATS \) when explaining detection speed to management.
- Expect early false alarms. The run length distribution is skewed.
- Do not stack sensitizing rules without accepting the false alarm cost.
- Switch to CUSUM or EWMA when small shifts matter.
Quick Reference
| Quantity | Formula |
|---|---|
| Run length distribution | \( P(RL = n) = (1-p)^{n-1} p \) |
| General ARL | \( ARL = 1/p \) |
| In-control ARL | \( ARL_0 = 1/\alpha \) |
| Out-of-control ARL | \( ARL_1 = 1/(1-\beta) \) |
| Standard deviation of RL | \( SDRL = \sqrt{1-p}\,/\,p \) |
| Average time to signal | \( ATS = ARL \times h \) |
| Effective shift with subgroups | \( \delta_{\text{eff}} = \delta \sqrt{n} \) |
Closing Thought
A control chart is a detection system. ARL tells you how fast it detects and how often it cries wolf.
Once you can compute both numbers, chart selection stops being habit and becomes engineering.