Quality assurance and control charts (Edexcel GCSE Statistics): Revision Notes

Quality control

Introduction to quality control

When manufacturing companies produce items on a production line, they must regularly check the quality of their products. Rather than testing every single item (which would be impractical and expensive), companies take samples at regular intervals and test these sample items. This might involve weighing products, measuring their dimensions, or checking other important characteristics.

The key principle is that by monitoring these samples carefully, companies can detect when something goes wrong with their production process before too many faulty items are produced.

Understanding control charts and sample means

What is a control chart?

A control chart is a graph that helps monitor the quality of a production process. Here's how it works:

• The horizontal axis shows the test number (sample 1, sample 2, etc.)
• The vertical axis shows the sample mean for each test
• The chart has several important horizontal lines marked on it

Key lines on a control chart

Target mean (μ): This is the ideal average value you want your product to have. It's shown as a solid line in the middle of the chart.

Warning limits: These are drawn at ±2σ (plus or minus 2 standard deviations) from the target mean. They're shown as dashed lines above and below the target mean.

Action limits: These are drawn at ±3σ (plus or minus 3 standard deviations) from the target mean. They're typically shown as solid lines further out from the target mean.

Understanding the distribution

The sample means follow a normal distribution with mean μ. This means that most sample means will cluster around the target value, with fewer samples appearing further away from the target.

The three-step action process

When you plot a sample mean on your control chart, you need to take different actions depending on where it falls:

Action 1: Sample mean within warning limits

What to do: Do nothing - continue normal production.

Why: This is expected behaviour. The sample mean is close enough to the target that no intervention is needed.

Action 2: Sample mean between warning and action limits

What to do: Take a second test immediately.

Why: This suggests the process might be starting to drift away from the target. A second test helps confirm whether this is just normal variation or the beginning of a problem.

Action 3: Sample mean outside action limits

What to do: Stop the machine, check it, and reset it.

Why: This strongly suggests something has gone wrong with the production process. Continuing production would likely result in many faulty items.

What to expect when everything works correctly

Understanding the statistical expectations helps you interpret control charts properly:

The 95% rule (±2σ)

When the production process is working correctly, approximately 95% of sample means should fall within the warning limits (±2σ of the target mean). This means about 19 out of every 20 samples should be within these limits.

The 99.7% rule (±3σ)

When everything is working properly, approximately 99.7% of sample means should fall within the action limits (±3σ of the target mean). This means about 999 out of every 1000 samples should be within these limits.

Probability of action

If the machine is working correctly, there's only a 0.2% probability that a sample mean will fall outside the action limits. When this happens, it's much more likely that something has actually gone wrong rather than this being normal variation.

Worked example: Precision needles

Let's work through a complete example to see how quality control works in practice.

Problem: A factory tests samples of 10 precision needles every 30 minutes to check their lengths. For the first three samples, the total lengths were:

• Sample 1: ΣL = 180 mm
• Sample 2: ΣL = 183.5 mm
• Sample 3: ΣL = 185 mm

Step 1: Calculate the sample means

• Sample 1 mean = 180 ÷ 10 = 18.0 mm
• Sample 2 mean = 183.5 ÷ 10 = 18.35 mm
• Sample 3 mean = 185 ÷ 10 = 18.5 mm

Step 2: Plot these means on the control chart Looking at the chart, we can see where each point falls relative to the warning and action limits.

Step 3: Determine what action to take for each sample

• Sample 1 (18.0 mm): No action - within warning limits
• Sample 2 (18.35 mm): Test again - between warning and action limits
• Sample 3 (18.5 mm): Stop the process and reset - outside action limits

Worked example: Bottle filling machine

Here's another practical example to reinforce these concepts.

Problem: A machine fills bottles with liquid. Every 10 minutes, 12 bottles are tested. The sample means follow a normal distribution with μ = 330 ml and σ = 2.4 ml.

Part (a): Work out the warning and action limits

• Warning limits: μ ± 2σ = 330 ± 2(2.4) = 330 ± 4.8
  • Lower warning limit = 325.2 ml
  • Upper warning limit = 334.8 ml
• Action limits: μ ± 3σ = 330 ± 3(2.4) = 330 ± 7.2
  • Lower action limit = 322.8 ml
  • Upper action limit = 337.2 ml

Part (b): One sample has ΣV = 3998 ml. What action should be taken?

• Sample mean = 3998 ÷ 12 = 333.17 ml
• This falls between the warning limit (334.8) and action limit (337.2)
• Action: Take a second test immediately

Key formulas to remember

• Sample mean = Total of sample values ÷ Number of items in sample
• Warning limits = Target mean ± 2σ
• Action limits = Target mean ± 3σ
• 95% of sample means fall within ±2σ when process is working correctly
• 99.7% of sample means fall within ±3σ when process is working correctly

Common exam tips and traps

Exam tips

• Always clearly identify which limit a sample mean has crossed
• Show your calculation of the sample mean step by step
• State the specific action required, don't just say "investigate"
• Remember that crossing warning limits triggers retesting, while crossing action limits stops production

Common traps

• Don't confuse warning limits (±2σ) with action limits (±3σ)
• Remember to divide the sum by the sample size to get the mean
• Don't assume that one unusual result means the process is broken - this is why we have the three-step process

Remember!

• Quality control uses sample means plotted on control charts to monitor production processes efficiently • Warning limits are set at ±2σ and action limits at ±3σ from the target mean • The three-step process: do nothing (within warning limits), test again (between limits), stop and reset (outside action limits) • When working correctly, 95% of samples fall within ±2σ and 99.7% fall within ±3σ • Always calculate sample means by dividing the total by the number of items in the sample