Chi-square Goodness of Fit Lab

STA-215 · Semester III · Statistics II

intermediate30 minAvailable

Chi-square Goodness of Fit

Do observed counts match the expected distribution?

Observed (Oᵢ)

Expected (Eᵢ)

df = categories − 1 = 1. Add/remove categories by toggling — but chi-square needs Eᵢ ≥ 5 to be valid.

χ² statistic

1.440

Σ (Oᵢ − Eᵢ)² / Eᵢ, df = 1

Fail to reject H₀ — 1.44 ≤ 3.84: fit is plausible.

Critical χ²(0.05, df=1) = 3.84.

Lab brief

Objective

Test whether observed category counts match a proposed model by computing the chi-square statistic and comparing it with the critical value.

Theory & concepts

The goodness of fit test compares observed counts O with expected counts E under a model. The statistic chi-square = sum((O - E)^2 / E) has k - 1 degrees of freedom when all parameters are fixed. Large values indicate the model does not fit the data.

Rendering diagram…

The chi-square goodness-of-fit test

Expected counts can come from a stated model (like a fair die) or from a fitted distribution. The test requires every expected count to be at least 5.

Critical region

The set of chi-square values larger than the critical value chi-square(alpha, k-1). If the observed statistic falls in this region, the model is rejected.

Prerequisites

Lesson: Chi-square goodness of fit test
Lesson: Introduction to non-parametric tests

Instructions

1Load a category dataset (e.g. 120 die rolls) and read the observed counts.
2Set the model probabilities or the expected counts and confirm they sum to the sample size.
3Read the chi-square statistic, degrees of freedom and p-value.
4Compare with the critical value at alpha = 0.05 and note the conclusion.
5Edit a few counts to make the fit worse and watch the statistic grow.

Tasks

Fit the die
Test whether the roll counts are consistent with a fair die.
Manual calculation
Hand-compute one (O - E)^2 / E term and verify it on the panel.
Model comparison
Propose a different model and decide which one the data fit better.

Self-check

The computed chi-square statistic matches hand calculation.
Degrees of freedom are reported correctly as k - 1.
The conclusion distinguishes 'data consistent with the model' from 'model proven true'.

Environment

editable category countsmodel probability editorlive chi-square statisticcritical value reference