Statistics — Introduction and Learning Goals¶

Short summary Hands-on statistics for measurement data: distributions, descriptive stats, hypothesis testing, outliers, and the central limit theorem.

Learning objectives - Compute and interpret mean, variance, confidence intervals. - Apply t-tests, chi-square tests, and identify outliers. - Relate histograms to probability distributions and sampling variability (CLT).

Key concepts (brief) - Sampling distributions and the Central Limit Theorem. - When to use t-distribution vs normal approximation. - Robust statistics and practical outlier handling.

Prerequisites Introductory probability and basic Python (NumPy, matplotlib).

Ordered reading (suggested)¶

Follow this sequence to build statistical tools progressively. The order moves from basic descriptive statistics, through probability distributions and hypothesis testing, and concludes with practical outlier detection methods.

Very basic review of some statistics terms — mean, variance, standard deviation, and fundamental statistical concepts.
Getting started with measurement uncertainty — first uncertainty calculation from repeated data.
From Histogram to Distribution — from binned data to probability models.
Probability Distributions and the Central Limit Theorem — overview of common probability distributions and their properties.
Exploring Different Probability Distributions — hands-on comparison of distribution shapes.
Central limit theorem (or why errors often look Gaussian) — why errors are often Gaussian; the power of averaging many measurements.
Statistical parameters using probability density function — mean, variance, and median from a PDF.
Chi-square Test — probability fundamentals and introduction to distributions.
Chi-square Test: Worked Example — full chi-square workflow on real data.
"Student" t-distribution — Student's t-distribution and its importance for small samples in measurement data.
t-test — hypothesis testing and comparing two datasets.
Outlier Detection: Modified Thompson Test — detecting and handling outliers using the Modified Thompson test.
Outlier Detection: Modified Thompson Test, Second Version — second worked version of the Thompson test.
Outlier Detection: Comparing Tests — comparison of outlier rules.
Outlier Detection in Regression Residuals — outlier detection in regression residuals (bivariate data).

Rationale: Students first learn to compute and interpret basic statistics, then understand how distributions arise from data, then learn hypothesis testing and practical statistical tests. Outlier detection concludes the chapter because it often requires multiple statistical tools.

What comes next?¶

Now that you can quantify uncertainty from data, the Calibration chapter shows you how to use these statistical tools to characterize real instruments. You'll build calibration curves, estimate regression uncertainty, and learn to quantify systematic errors (Type B sources) that complement the statistical (Type A) errors you just learned to compute.