Statistical Process Control (SPC)

Statistical Process Control (SPC) is a method of quality control that uses statistical techniques to monitor and control a process. The goal of SPC is to ensure that the process operates efficiently, producing more specification-conforming products with less waste (rework or scrap).

Really the question comes down to, imagine that the proportion of claimants receiving late payments rose from 1% to 3%, SPC can help us determine if this is a cause for concern or just a random occurrence.

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Conceptual Foundation

Let

• $X_t$ = process output at time $t$
• $\mu =\mathbb{E}[X_t]$
• ${\sigma }^2=\mathrm{Var}(X_t)$

A process is in control if:

• $\mu$ is constant over time
• ${\sigma }^2$ is constant over time
• Observations are independent (or weakly dependent in a stable way)

That is,

$$X_t\sim F(\mu ,{\sigma }^2)\text{for all }t$$

If the distribution changes (mean shift, variance increase, structural change), the process is out of control.

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Common Vs Special Cause Variation

Common Cause Variation

Inherent, random variation in the system.

Mathematically:

$$X_t=\mu +{\epsilon }_t$$

where

$${\epsilon }_t\sim \text{i.i.d. }(0,{\sigma }^2)$$

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Special Cause Variation

A structural shift in parameters:

• Mean shift: $\mu \to {\mu }^{\prime }$
• Variance shift: ${\sigma }^2\to {{\sigma }^{\prime }}^2$

Example:

$$X_t={\mu }^{\prime }+{\epsilon }_t\text{for }t\ge T$$

SPC attempts to detect such changes.

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Control Charts

The primary SPC tool is the control chart.

A control chart consists of:

• Center Line (CL)
• Upper Control Limit (UCL)
• Lower Control Limit (LCL)

Typically:

$$\text{UCL}=\mu +3\sigma$$ $$\text{LCL}=\mu -3\sigma$$

If $X_t$ exceeds these limits, the probability under normality is:

$$P(|Z|>3)\approx 0.0027$$

Thus, false alarm rate ≈ 0.27%.

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$\bar{X}$-Chart (Monitoring The Mean)

Suppose we take samples of size $n$.

Let:

$$\bar{X}=\frac{1}{n}\sum _{i=1}^n{X}_i$$

Then:

$$\mathbb{E}[\bar{X}]=\mu$$ $$\mathrm{Var}(\bar{X})=\frac{{\sigma }^2}{n}$$

Standard deviation of $\bar{X}$:

$${\sigma }_{\bar{X}}=\frac{\sigma }{\sqrt{n}}$$

Control limits:

$$\text{UCL}=\mu +3\frac{\sigma }{\sqrt{n}}$$ $$\text{LCL}=\mu -3\frac{\sigma }{\sqrt{n}}$$

If $\sigma$ is unknown, estimate using:

$$s=\sqrt{\frac{1}{n-1}\sum (X_i-\bar{X}{)}^2}$$

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$p$-Chart (Monitoring Proportions)

Used when data are binary (e.g., late vs on-time payments).

Let:

• $n$ = sample size
• $X\sim \mathrm{Binomial}(n,p)$
• $\hat{p}=X/n$

Then:

$$\mathbb{E}[\hat{p}]=p$$ $$\mathrm{Var}(\hat{p})=\frac{p(1-p)}{n}$$

Standard deviation:

$${\sigma }_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}$$

Control limits:

$$\text{UCL}=p+3\sqrt{\frac{p(1-p)}{n}}$$ $$\text{LCL}=p-3\sqrt{\frac{p(1-p)}{n}}$$

(If LCL < 0, set to 0.)

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Example: Late Payments (1% → 3%)

Suppose historical rate:

$$p=0.01$$

New observed sample:

$$\hat{p}=0.03$$

Assume $n=500$.

Standard deviation:

$${\sigma }_{\hat{p}}=\sqrt{\frac{0.01(0.99)}{500}}\approx 0.00445$$

3-sigma limit:

$$\text{UCL}=0.01+3(0.00445)\approx 0.0234$$

Observed value:

$$0.03>0.0234$$

Thus the increase is statistically unlikely under random variation → strong evidence of special cause variation.

Equivalent z-score:

$$z=\frac{0.03-0.01}{0.00445}\approx 4.49$$

Since $|z|>3$, the shift is statistically significant.

SPC concludes: process likely out of control.

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Hypothesis Testing Interpretation

SPC can be viewed as repeated hypothesis testing.

Null hypothesis:

$$H_0:\mu ={\mu }_0$$

or

$$H_0:p=p_0$$

Control limits define a rejection region with type I error approximately:

$$\alpha \approx 0.0027$$

SPC is essentially a sequential testing procedure.

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Average Run Length (ARL)

Average Run Length (ARL) = expected number of samples taken before a signal.

If false alarm probability per sample is $\alpha$, then:

$${\text{ARL}}_0=\frac{1}{\alpha }$$

For 3-sigma charts:

$$\alpha =0.0027$$ $${\text{ARL}}_0\approx 370$$

Meaning: on average, one false alarm every 370 samples.

source: Session 15+16 - Quality Analytics Simulation