| name | inferential-statistics |
| description | Drawing conclusions about populations from sample data. Covers sampling distributions, confidence intervals, hypothesis testing (z-tests, t-tests, chi-squared tests, ANOVA), p-values, significance levels, power, Type I and Type II errors, effect sizes, and the logic connecting sample statistics to population parameters. Emphasizes the distinction between statistical significance and practical significance. Use when testing hypotheses, constructing confidence intervals, designing studies, or interpreting inferential results. |
| type | skill |
| category | statistics |
| status | stable |
| origin | tibsfox |
| modified | false |
| first_seen | "2026-04-11T00:00:00.000Z" |
| first_path | examples/skills/statistics/inferential-statistics/SKILL.md |
| superseded_by | null |
Inferential Statistics
Inferential statistics is the bridge from sample to population. A researcher observes 200 patients and wants to draw conclusions about all patients. A factory tests 50 parts and wants to guarantee the quality of 10,000. The logical machinery that makes this possible -- sampling distributions, confidence intervals, hypothesis tests, and their attendant concepts of error and power -- forms the core of this skill.
Agent affinity: pearson (chi-squared, test design), gosset (t-tests, small-sample inference), wasserstein (p-value interpretation, communication), george (pedagogy)
Concept IDs: stat-hypothesis-testing, stat-sampling-bias, stat-descriptive-statistics
The Logic of Inference
From sample to population
A parameter is a fixed but unknown number describing a population (mu, sigma, p). A statistic is a number computed from sample data (x-bar, s, p-hat) that estimates the parameter.
The key question: how much can a statistic vary from sample to sample? The sampling distribution of a statistic describes this variability. The standard deviation of a sampling distribution is called the standard error (SE).
The sampling distribution of the mean
If samples of size n are drawn from a population with mean mu and SD sigma:
- E(X-bar) = mu (unbiased).
- SE(X-bar) = sigma / sqrt(n).
- By the CLT, X-bar is approximately normal for large n.
This is the foundation of virtually every test and interval for means.
Confidence Intervals
Construction
A confidence interval for a parameter has the form: point estimate +/- margin of error.
For a population mean with known sigma: X-bar +/- z* (sigma / sqrt(n)), where z* is the critical value from the standard normal distribution.
For a population mean with unknown sigma: X-bar +/- t* (s / sqrt(n)), where t* comes from the t-distribution with n-1 degrees of freedom.
Interpretation
A 95% confidence interval means: if we repeated this sampling procedure many times, about 95% of the resulting intervals would contain the true parameter. It does NOT mean "there is a 95% probability that the parameter is in this interval." The parameter is fixed; the interval is random.
Common confidence intervals
| Parameter | Conditions | Formula |
|---|
| Mean (sigma known) | n >= 30 or normal population | X-bar +/- z*(sigma/sqrt(n)) |
| Mean (sigma unknown) |
