Sampling and collecting data
Key takeaways
- A population is the complete set of items a study is interested in and a sample is the subset actually observed; a statistic calculated from the sample is used to estimate a population parameter.
- Probability methods (simple random, systematic, stratified, cluster) need a full list of the population to sample from; convenience or judgmental sampling is quicker but cannot support statistically valid inference back to the whole population.
- Yamane's formula, n = N / (1 + N·e²), sets the sample size from a known population N and an acceptable margin of error e.
- The standard error of the mean, SE = s / √n, falls as the sample grows, so a larger sample gives a more reliable estimate.
- Accuracy is how close a measurement is to the true value and precision is how close repeated measurements are to each other; quality control aims for both together.
Quality control decisions are rarely based on inspecting every unit produced, they are based on a sample, so this chapter sets out how a sample is chosen, sized and trusted before any of the analysis in the chapter on analysing data can mean anything.
Statistical terms and concepts
Sampling
Sampling is how a population is reduced to a manageable, representative subset. Done well, it lets a quality technician draw conclusions about an entire production run from checking only a fraction of it.
What sampling is
Sampling is the process of selecting a subset of units from a population so that the subset can be used to draw conclusions about the whole population.
Types and limits
Every sampling method carries a limitation: probability methods (simple random, systematic, stratified, cluster) require a full list of the population to sample from and can be costly to administer; non-probability methods are quicker but cannot support statistically valid inference back to the whole population.
Why sampling matters
- Saves time and cost compared with inspecting an entire population (a census)
- Makes inspection of destructive tests possible at all, since testing every unit would destroy the whole batch
- Allows conclusions about a large population from a manageable amount of data
- Can, if done well, be more accurate than a rushed full census
Sample size
A commonly used formula for determining sample size from a known population is Yamane's formula: n = N / (1 + N·e²), where N is the population size and e is the acceptable margin of error.
Sample size calculation
A batch of N = 500 units is produced; the technician accepts a margin of error e = 0.05 (5%).
n = 500 / (1 + 500 × 0.05²) = 500 / (1 + 500 × 0.0025) = 500 / (1 + 1.25) = 500 / 2.25 ≈ 222
A sample of approximately 222 units gives the required confidence at a 5% margin of error.
Confidence levels
Sampling errors
Precision and accuracy
A measurement process can be precise without being accurate (consistently wrong by the same amount, a calibration problem) or accurate without being precise (correct on average, but scattered). Quality control aims for both together.
Sampling versus a census
A census is preferred where the population is small, the stakes of missing a defect are very high, or where legal/regulatory requirements demand 100% inspection; sampling is preferred everywhere routine, high-volume monitoring is needed.
Standard error
The standard error of the mean measures how much a sample mean is expected to vary from the true population mean, SE = s / √n, where s is the sample standard deviation and n is the sample size. A larger sample produces a smaller standard error, a more reliable estimate.
Standard error calculation
A sample of n = 25 units has a standard deviation s = 10.
SE = 10 / √25 = 10 / 5 = 2
Types of data
Collection methods
- Direct observation and physical measurement
- Questionnaires and surveys
- Interviews
- Existing records and secondary data (production logs, past inspection reports)
Self-check
(a) Distinguish a population from a sample, and a parameter from a statistic. (4 marks)
(b) Name three types of probability sampling and one non-probability method. (4 marks)
(c) A batch of N = 1,200 units is produced and a technician accepts a margin of error e = 0.05. Using Yamane's formula, compute the required sample size. (3 marks)
(d) A sample of n = 36 units has a standard deviation s = 15. Compute the standard error of the mean. (2 marks)
(e) Distinguish precision from accuracy, and state whether a set of measurements that cluster tightly but consistently 3mm above the true value is precise, accurate, both or neither. (3 marks)
(f) Explain when a census is preferred over sampling. (2 marks)
Show worked answer
(a) A population is the complete set of items a study is interested in; a sample is a subset of it actually observed. A parameter is a numerical characteristic of the population; a statistic is a numerical characteristic calculated from a sample, used to estimate a parameter.
(b) Probability: simple random sampling, systematic sampling, stratified sampling (also cluster sampling). Non-probability: convenience/judgmental sampling.
(c) n = N / (1 + N e^2) = 1,200 / (1 + 1,200 x 0.05^2) = 1,200 / (1 + 3) = 1,200 / 4 = 300.
(d) SE = s / sqrt(n) = 15 / sqrt(36) = 15 / 6 = 2.5.
(e) Accuracy is how close a measurement is to the true value; precision is how close repeated measurements are to each other, regardless of the true value. Measurements clustering tightly but consistently 3mm off are precise but not accurate.
(f) A census is preferred where the population is small, the stakes of missing a defect are very high, or legal/regulatory requirements demand 100% inspection.
Exam tip
The precision/accuracy distinction is a classic exam question, use the dartboard analogy to explain it: precise-but-inaccurate shots cluster tightly but off-centre, accurate-but-imprecise shots scatter around the centre without clustering.