Lesson 2: Inferential statistics

GCHS cannot re-inspect every home it has ever rehabbed before deciding whether the program's quality has improved. What it can do is inspect a manageable sample and use inferential statistics, the branch of statistics that draws conclusions about a whole population from a properly drawn sample, to say something trustworthy about the whole program. Inferential statistics is the foundation under nearly every tool in the rest of this Green Belt tier: hypothesis tests, confidence intervals, regression, and control charts are all, at bottom, ways of using limited data to make a defensible statement about something larger than the data itself.

Population, sample, parameter, statistic

A population is the complete set of items, homes, or events a question is really about, such as every rehab GCHS has completed. A sample is the subset actually measured. A number that describes the population, such as its true mean (μ) or true standard deviation (σ), is a parameter; a number calculated from a sample, such as a sample mean (x̄) or sample standard deviation (s), is a statistic. Inference is the disciplined process of using a statistic, plus knowledge of how much sample statistics vary from sample to sample, to make a defensible claim about the unknown parameter. A Green Belt who reports "x̄ = 3.2 days" has reported a statistic; a Green Belt who says "we are 95 percent confident the true average cycle time is between 2.9 and 3.5 days" has performed inference.

Sampling techniques

How a sample is drawn determines whether inference from it can be trusted. The common techniques:

• Simple random sampling: every item in the population has an equal chance of selection, for example numbering all 500 completed rehabs and drawing 30 at random.
• Stratified sampling: the population is split into meaningful subgroups (strata), such as county, and a sample is drawn from each stratum in proportion to its size, guaranteeing every county is represented.
• Systematic sampling: items are selected at a fixed interval from an ordered list, such as inspecting every 10th work order in the sequential log.
• Cluster sampling: whole naturally occurring groups (clusters) are randomly selected and every item within a selected cluster is measured, such as randomly picking five neighborhoods and inspecting every home in each.
• Subgroup (rational subgroup) sampling: samples are drawn so that variation within a subgroup reflects only common-cause variation, isolating a specific source, such as sampling five consecutive units from one crew's output rather than mixing crews together.

Two different problems can affect any of these techniques. Sampling error is the ordinary, unavoidable difference between a sample statistic and the true population parameter, caused purely by chance; it shrinks as sample size grows. Bias is a systematic distortion caused by a flawed sampling method, such as only inspecting homes that are easy to reach; it does not shrink with a bigger sample and can only be fixed by changing how the sample is drawn.

The Central Limit Theorem

The Central Limit Theorem (CLT) is the reason inference about a mean works at all. It states that as sample size n grows, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the underlying population distribution, and it is centered on the true population mean μ. The spread of that sampling distribution is measured by the standard error of the mean:

SE = σ / √n

As n increases, √n grows, so SE shrinks: larger samples produce sample means that cluster more tightly around the true population mean, which is exactly what the narrowing curves in the figure above show. A common rule of thumb is that n ≥ 30 is usually enough for the sampling distribution of the mean to be treated as approximately normal, but this is a guideline, not a law: a population that is heavily skewed or has extreme outliers may need a considerably larger n before the approximation is trustworthy, while a population that is already close to normal can behave well with a much smaller n.

Worked example. Suppose the wall-gap measurement from Lesson 1 has a true population standard deviation σ = 0.15 inch. For a sample of n = 5 walls, SE = 0.15 / √5 = 0.15 / 2.236 ≈ 0.067 inch. For a sample of n = 30 walls, SE = 0.15 / √30 = 0.15 / 5.477 ≈ 0.027 inch. Increasing the sample sixfold, from 5 to 30, cuts the standard error by more than half, which is why the n = 30 curve in the figure is so much narrower than the n = 5 curve.

Confidence intervals: a first look

A confidence interval uses a sample statistic plus its standard error to build a range of plausible values for the unknown population parameter, instead of reporting a single number as if it were exact. The general form is:

point estimate ± (critical value × standard error)

For example, x̄ ± z × (σ / √n) when σ is known, or x̄ ± t × (s / √n) when it is estimated from the sample, using the t-distribution instead of the normal distribution to account for that extra uncertainty. A correctly stated 95 percent confidence interval means that if the same sampling procedure were repeated many times, about 95 percent of the intervals constructed that way would contain the true population parameter. It is a statement about the reliability of the method over many repetitions, not a 95 percent probability that this one specific interval happens to contain the true value, which is a common misreading worth avoiding both on the exam and in practice. The hypothesis-testing lessons that follow rest on this same standard-error logic.