6.9 Population Parameters, Sample Statistics, and Valid Statistical Conclusions

Parameters and statistics

A parameter is a fixed numerical property of a whole population. A statistic is a computed property of a sample, used to estimate the corresponding parameter. Parameters are usually unknown; statistics are always known but vary from sample to sample.

The convention is that Greek letters denote parameters and Latin letters denote statistics. It is worth internalizing, because exam questions frequently hinge on whether a value given is a known population value or an estimate carrying sampling error.

Why the sample standard deviation divides by n minus 1

$s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n - 1}} \qquad \sigma = \sqrt{\frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N}}$

Deviations are taken from $\bar{x}$, which was itself computed from the same data and is by construction the value that minimizes the sum of squared deviations. Sample scatter is therefore systematically smaller than scatter about the true mean $\mu$, and dividing by $n$ would give a downward-biased estimate. Dividing by $n - 1$, the degrees of freedom, corrects the bias.

Degrees of freedom count the independent pieces of information remaining: once $\bar{x}$ is fixed, only $n - 1$ of the deviations are free to vary, because the deviations must sum to zero.

Sampling distribution and standard error

A statistic computed from a sample is itself a random variable with its own distribution. The standard deviation of that sampling distribution is the standard error:

$SE_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \qquad \text{estimated by} \qquad \frac{s}{\sqrt{n}}$

Two consequences that matter constantly:

• Precision improves with the square root of sample size. Halving the standard error requires quadrupling $n$. This is why moving from 25 to 30 observations buys almost nothing and moving from 25 to 100 buys a great deal.
• The standard error is not the standard deviation of the process. Reporting $s/\sqrt{n}$ as the process spread understates variation dramatically, and it is a favourite exam trap.

Descriptive versus inferential studies

A descriptive statement about a sample that is then used as though it described the process has made an inferential claim without the machinery that justifies it.

Enumerative versus analytic studies

Deming's distinction is the deeper one, and it is where Six Sigma differs from classical survey statistics.

• An enumerative study aims to describe an existing, bounded population. The action is taken on the population studied. Counting inventory in a warehouse, or auditing a fixed batch of 5,000 invoices, is enumerative. Random sampling from the frame is sufficient to justify the conclusion.
• An analytic study aims to predict and act on a process that will generate future output. The action is taken on the process, and the units of interest have not been produced yet. Almost all Six Sigma work is analytic.

The consequence is decisive: no amount of random sampling from past output justifies a prediction about future output unless the process is stable. A perfect random sample from a process that shifted last Tuesday describes a mixture of two processes and predicts neither.

This is why the Measure phase establishes statistical control before computing capability, and it is the reason the Body of Knowledge places capability studies after control charts conceptually. Stability is the licence to extrapolate.

Drawing valid conclusions

The Body of Knowledge tags this at the Evaluate level. The checklist below is the practical form of that requirement.

1. Is the measurement system adequate? No conclusion survives a failed MSA.
2. Is the sample representative of the population or process of interest? Convenience samples, day shift only, or a single lot are all common invalidators.
3. Is the process stable over the study period? For analytic studies this is prerequisite, not optional.
4. Are the test's assumptions met? Normality, independence, equal variances, as required by the procedure used.
5. Is the sample size adequate for the effect being claimed? A non-significant result from an underpowered study is not evidence of no effect.
6. Is the effect practically as well as statistically significant? With a large enough sample, trivial differences become significant.
7. Is the conclusion within the range of the data? Extrapolating a regression beyond the observed factor range is unsupported.
8. Has causation been claimed from observational data? Correlation alone does not establish it.

Correlation and causation

Three alternatives must be excluded before an observed association supports a causal claim:

• Confounding: a third variable drives both. Ice cream sales correlate with drowning; temperature causes both.
• Reverse causation: the presumed effect is the cause. Machines that receive more maintenance may show more failures because failures trigger maintenance.
• Selection effects: the way the data was obtained creates the association. Studying only units that reached final inspection excludes those scrapped earlier.

The instruments that do establish causation are designed experiments, where factors are deliberately manipulated and randomization neutralizes unknown confounders, and, more weakly, careful observational designs with control of known confounders and a plausible mechanism.

Common invalid conclusions