Chapter 7 — Probability and statistics
Lesson map
Lesson 07 published. Map, example, EX019.
flowchart TD
N0["Learning outcomes"]
N1["7.1 Sampling for quality"]
N2["7.2 Confidence intervals"]
N3["7.3 Control charts"]
N4["7.4 Statistics and governance"]
N0 --> N1 --> N2 --> N3 --> N4
N4 --> EX["Worked example"]
EX --> Q["Exercise EX019"]
Logical drawing
flowchart LR
N0["Learning outcomes"]
N1["7.1 Sampling for quality"]
N2["7.2 Confidence intervals"]
N3["7.3 Control charts"]
N4["7.4 Statistics and governance"]
N0 --> N1 --> N2 --> N3 --> N4 --> Q["EX019"]
[Learning outcomes]
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v
[7.1 Sampling for quality]
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v
[7.2 Confidence intervals]
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v
[7.3 Control charts]
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v
[7.4 Statistics and governance]
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v
[Worked example] --> [Exercise EX019]
Learning outcomes
You will (1) estimate error rates from samples; (2) build confidence intervals; (3) use control limits for quality monitoring.
7.1 Sampling for quality
Tidewater cannot inspect every reading by hand, so Kofi samples. He draws 400 synthetic readings at random and checks each against the meter photograph taken by field staff. Eight disagree. The estimated error rate is 2 percent. He explains that a random sample allows a valid estimate of the whole population's error rate, while samples chosen for convenience, such as only easy-to-reach meters, may be biased and mislead decisions about data quality. The sample is drawn by a script with a fixed seed, so anyone can draw the same meters and repeat the checks.
7.2 Confidence intervals
Kofi computes a 95 percent confidence interval for the error rate using the Wilson method, which behaves well for small proportions. The interval runs from about 1.0 to 3.9 percent. He reports the interval rather than only the point estimate, because Tidewater's quality target is 3 percent and the interval includes values above it. The data owner therefore cannot claim with confidence that the target is met, and Kofi recommends a larger sample next quarter. With 400 readings and eight errors, the interval is wide; a sample of 1,600 would narrow it considerably.
7.3 Control charts
For ongoing monitoring, Kofi plots the daily proportion of rejected readings with control limits three standard deviations from the long-run mean. Points outside the limits signal a change worth investigating. In the synthetic data, one day exceeds the upper limit, which turns out to be a firmware update on a batch of meters. He learns that control charts separate normal variation from real problems and stop teams chasing random noise or ignoring real changes. The firmware batch is rolled back, and the next day returns within the limits.
7.4 Statistics and governance
Prof. Varga connects statistics to governance. Quality targets mean little unless measured properly. A target of 3 percent errors needs a sampling plan, an interval method and an owner who acts when limits are crossed. Kofi writes these elements into a measurement standard for Tidewater's meter data. He notes that Batini and colleagues describe assessment as a stage in quality methodologies, and that statistics turns assessment from opinion into evidence that managers and auditors can check. The standard names the owner of each step and the action required.
Worked example
Kofi estimates a 2 percent error rate from 400 sampled readings, reports a Wilson interval of about 1.0 to 3.9 percent, detects a firmware change with a control chart and writes a measurement standard.
Synthetic case. Readings are synthetic; Batini et al. (2009) is real.
Errors and limits
Control limits assume stable processes; recalculate after known changes.
Exercises
- EX019 — Error estimate. What error rate did the sample give? Rubric: 3 = 2 percent;; 2 = partly correct;; 1 = wrong;; 0 = none.
- EX020 — Target judgement. Why can the owner not claim the 3 percent target is met? Rubric: 3 = interval includes values above 3 percent;; 2 = mentions interval;; 1 = vague;; 0 = none.
- EX021 — Control signal. What caused the out-of-limit day? Rubric: 3 = a firmware update on a batch of meters;; 2 = partly correct;; 1 = vague;; 0 = none.
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