5 Control Charts

5.1 Learning Outcomes

Explain the statistical basis for control charts.
Distinguish between common causes and special causes of variation.
Differentiate between variable and attribute control charts.
Select rational subgroups appropriately when designing control charts.

5.2 Types of Control Charts

Control charts are among the most important tools in Statistical Process Control (SPC). They provide a simple graphical method for distinguishing between random variation and variation that signals a real change in the process.

Control charts can be broadly divided into two categories:

Control charts for variables
- Many quality characteristics can be measured numerically, such as weight, length, or volume.
- These are known as variables.
- Commonly used variable control charts include:
  - \(\bar{X}\)-Chart: monitors the process mean (average).
  - R-Chart: monitors process variability using the sample range.
  - S-Chart: monitors process variability using the standard deviation.
Application:
- Typically based on 20–25 preliminary samples, each of size \((n = 3)\) to \((5)\).
- If points fall outside trial limits, investigate assignable causes.
- After revising limits (if necessary), adopt the control limits for monitoring.
Control charts for attributes
- Not all quality characteristics can be measured numerically. Sometimes, items are simply classified as conforming or nonconforming.
- These are known as attributes.
- Commonly used attribute control charts include:
  - p-Chart: fraction nonconforming.
  - np-Chart: number nonconforming (when sample size is constant).
  - c-Chart: number of defects per unit.
  - u-Chart: defects per unit, adjusted for varying sample sizes.

5.3 Causes of Variation

Every process exhibits variation. The role of a control chart is to help distinguish between:

Common causes (chance variation):
- The natural, inherent background noise of the process.
- Small, random fluctuations that are unavoidable.
- Example: small differences in bread loaf weights due to natural mixing variability.
Special causes (assignable variation):
- Large, identifiable sources of variation that disrupt the process.
- Usually caused by specific factors such as equipment failure, operator error, or poor-quality raw materials.
- Example: mixer malfunction causing consistently underweight loaves.

A control chart signals potential special cause variation when a point falls outside the control limits, or when a non-random pattern (trend, cycle, or run) is detected.
Prompt detection and removal of special causes helps reduce process variability and improve quality in the long run.

5.4 Statistical Basis for Control Charts

Control charts are built on the principles of probability and sampling distributions.

Suppose a quality characteristic is normally distributed with mean \(\mu\) and standard deviation \((\sigma)\). If we take a sample of size \((n)\), the sample mean \((\bar{X})\) has:

\[ \mu_{\bar{X}} = \mu, \quad \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} \]

By the empirical rule, nearly all values of \((\bar{X})\) (99.73%) will fall within three standard deviations of the true mean:

\[ \mu \pm 3 \frac{\sigma}{\sqrt{n}} \]

Thus, the Upper Control Limit (UCL) and Lower Control Limit (LCL) are set at:

\[ \text{UCL} = \mu + 3\frac{\sigma}{\sqrt{n}}, \quad \text{LCL} = \mu - 3\frac{\sigma}{\sqrt{n}} \]

If points fall within the limits → variation is due to common causes, process is in control.
If points fall outside the limits → likely special causes, requiring investigation.

This statistical basis ensures that control charts have a low false alarm rate while still being sensitive to meaningful changes.

5.5 Rational Subgroup Selection

The concept of rational subgroups is central to designing effective control charts.

A rational subgroup is a sample of items chosen so that:
- Within-subgroup variation reflects only common causes (random noise).
- Between-subgroup variation reflects potential special causes (shifts in process).

Guidelines for selecting rational subgroups:
- For \((\bar{X})\) charts: choose samples close together in time (e.g., consecutive bottles filled) so that shifts in the mean appear between subgroups.
- For R or S charts: samples should reflect the natural variability within the process at a given point in time.
- Poor subgroup selection can mask special causes or exaggerate common causes.

5.5.1 Example — Rational Subgroup Selection

Suppose we want to monitor the diameter of metal rods produced by a machine.

Option A:
- Take 5 rods consecutively at 10:00 AM.
- These rods likely reflect only short-term variation (machine, operator, material at that moment).
- Subgroup represents the within-sample variability (chance/common causes only).
Option B:
- Take 1 rod at 10:00, 11:00, 12:00, 1:00, 2:00 and combine them into a single subgroup.
- These rods reflect long-term variation (different shifts, temperatures, operators).
- Subgroup now mixes special-cause and common-cause variation.

Which is better?
- For an X̄ chart (monitoring process average), Option A is preferred, because subgroup means will highlight shifts between groups.
- For an R chart (monitoring variability within subgroups), Option A again makes sense, because within-sample variation should reflect only common causes.

flowchart TB
    A["All Products Produced"] --> B["Divide into Subgroups"]
    B --> C1["Option A: Consecutive Items (Short-term)"]
    B --> C2["Option B: Widely Spaced Items (Long-term)"]

    C1 --> D1["Captures Common Cause Variation"]
    C2 --> D2["Mixes Common + Special Causes"]

    C1,D1:::good
    C2,D2:::bad

5.6 Exercises

Exercise 4.1 — Common vs. Special Causes
A bakery monitors the weight of its bread loaves. The average target weight is 500 g.
- One day, random samples show weights of 499 g, 501 g, 498 g, and 502 g.
- Another day, the oven malfunctioned, and loaves came out at 470 g, 468 g, 472 g, 469 g.

Task:
1. Which case illustrates common cause variation?
2. Which case illustrates special cause variation?
3. How should the manager respond in each case?

Tip

Tip: Small random shifts around 500 g are common causes. A big drop caused by equipment failure is a special cause. Managers investigate and fix special causes, but tolerate common variation within limits.

Exercise 4.2 — Attribute vs. Variable Charts
A factory produces 200 pens daily. Defective pens are recorded each day for two weeks.
- Day 1: 8 defectives, Day 2: 6 defectives, Day 3: 9 defectives … (data continues).

Task:
1. Which type of control chart is suitable for this data (attribute or variable)? Why?
2. Sketch how this chart would look.

Tip

Tip: The data are counts of defectives, not measurements. Use an np-chart (if sample size is fixed) or a p-chart (if sample size varies).

Exercise 4.3 — Statistical Basis
A filling machine dispenses detergent into 1-liter bottles. The process standard deviation is known to be 0.02 L. Sample size \(n=5\).
- Target mean = 1.00 L.

Task:
1. Calculate the control limits for the \(\bar{X}\)-chart using 3σ limits.
2. Interpret what would happen if a sample mean of 1.06 L is recorded.

Tip

Tip: Compute \(\sigma_{\bar{X}} = \sigma / \sqrt{n} = 0.02 / \sqrt{5}\). Then apply \(\mu \pm 3\sigma_{\bar{X}}\). If 1.06 L lies outside, it signals a special cause.

Exercise 4.4 — Rational Subgroups
A factory inspects 4 bolts every 30 minutes. Sometimes, the operator mistakenly collects 1 bolt every 7 minutes instead.

Task:
1. Which method is the correct rational subgroup selection? Why?
2. What problem occurs if bolts are sampled every 7 minutes?

Tip

Tip: Collecting bolts consecutively (every 30 minutes) keeps within-subgroup variation random. Sampling across 7 minutes mixes in process shifts and hides special causes.

Exercise 4.5 — Control vs. Specification Limits
A production process is designed to produce rods of length 50 cm ± 0.5 cm. Control limits are set at 49.6 cm and 50.4 cm.

Task:
1. Explain the difference between specification limits and control limits in this case.
2. If a rod measures 49.7 cm, is it within specification? Is it in control? Explain.

Tip

Tip: Specification = what the customer requires (49.5–50.5).
Control = what the process can achieve (49.6–50.4). At 49.7 cm, the rod meets both.