Control charts: action and warning lines (Higher tier)

Introduction to Control Charts

Imagine you are managing a factory that produces bags of crisps. You want every bag to weigh exactly \(30g\), but in the real world, machines aren't perfect. Some bags will be \(29.9g\), others \(30.1g\). How do you know if the machine is just "being a bit wobbly" or if it is actually broken and needs fixing?

This is where Control Charts come in. They are a special type of time series graph used in Quality Assurance to monitor a process over time. By plotting sample results (like the mean weight of 5 bags every hour), we can decide when to leave the machine alone and when to take action.

Note: This is a Higher Tier topic. In this section, we assume that sample means are more closely distributed (packed together) than individual values from the same population.

The Anatomy of a Control Chart

A control chart consists of a horizontal "target" line in the middle and four boundary lines (two above and two below). In the exam, you might see charts for the sample mean, median, or range.

1. The Target Value (Expected Value)

The center line represents the value we expect to get if everything is working perfectly. For our crisps, this would be \(30g\).

2. Warning Lines (\(\pm 2\) Standard Deviations)

These lines are drawn at \(2\) standard deviations above and below the target value.
Mathematically, we place them at:
\( \text{Target} + 2\sigma \) and \( \text{Target} - 2\sigma \)

The Rule: In a healthy process, about 95% of your samples should fall within these lines. This means there is only a 1 in 20 chance (5%) of a sample falling outside these lines just by "bad luck."

3. Action Lines (\(\pm 3\) Standard Deviations)

These are the "emergency" lines. They are drawn at \(3\) standard deviations above and below the target value.
Mathematically: \( \text{Target} + 3\sigma \) and \( \text{Target} - 3\sigma \)

The Rule: Almost all (99.7%) of your samples should fall within these lines. If a point falls outside these, it is extremely unlikely to be a fluke.

Quick Review:

• Warning Lines: \(\pm 2\) standard deviations. 1 in 20 chance of being outside.
• Action Lines: \(\pm 3\) standard deviations. Almost all values should be inside.

Interpreting the Chart: What do we do?

As a quality controller, you plot your sample mean on the chart and look at where it lands. Here is the protocol you need to know for the exam:

Situation A: The point is inside the Warning Lines

Interpretation: The process is "in control." The small variations are just natural, random fluctuations.
Action: Do nothing. Continue to monitor the process.

Situation B: The point is between the Warning and Action Lines

Interpretation: This is a "warning" signal. While it could be a 1 in 20 fluke, it might also mean the machine is starting to drift away from the target.
Action: Take another sample immediately.
If the next sample is back inside the warning lines, you can carry on. If it's outside again, you have a problem!

Situation C: The point is outside the Action Lines

Interpretation: The process is "out of control." It is mathematically "almost impossible" for this to happen by chance.
Action: Stop the process immediately. Investigate the cause and reset or repair the machinery.

Common Mistake to Avoid:

Don't confuse the lines! Warning = 2 SDs, Action = 3 SDs. A common exam error is swapping these numbers. Remember: "Action" is further away because it requires a more "extreme" result to justify stopping the whole factory!

Why do we use Standard Deviations?

You might wonder why we use 2 and 3 standard deviations specifically. This links to the Normal Distribution (which you study in the Probability section).
- 95% of data in a normal distribution lies within \( \pm 2 \sigma \).
- 99.7% of data lies within \( \pm 3 \sigma \).

Did you know? Even if the machine is working perfectly, 1 out of every 20 samples will statistically fall outside the warning lines. This is why we don't stop the machine for just one warning—we take another sample to double-check!

Summary Table for Revision

Key Takeaways

1. Control charts monitor sample statistics (mean, median, or range) over time.
2. Warning lines are at \( \pm 2 \) standard deviations; they suggest a 1 in 20 chance of error.
3. Action lines are at \( \pm 3 \) standard deviations; they suggest the process has failed.
4. If a point hits the Warning zone, you sample again. If it hits the Action zone, you stop the process.