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Statistics · Chapter 13

Checking the Conditions

Both tests in this chapter hand you a p‑value, and both of them earn that p‑value only when a short list of conditions holds. The conditions are not decoration — when they fail, the number the test prints is simply wrong, confident and wrong. The good news is that each test has one condition you can actually check from the data in front of you: for chi‑square it is whether the expected cell counts are large enough for the approximation to hold; for ANOVA it is whether the groups share a common spread. Pick a test below and move the sliders. Every count, every standard deviation, every verdict is computed live from the numbers you build.

A two‑way table of counts

The three conditions, checked live

One of these is a fact about how the data were gathered and can never be recovered from the numbers. The other two are properties of the numbers themselves — and those are the ones recomputing below as you move the sliders.
Pick a test to begineverything below recomputes from your settings

The table you are running

Two groups compared across several outcome categories. The expected count in a cell is its row total times its column total, divided by the grand total — the count you would predict if the two variables were unrelated. That is what the χ² approximation needs to be large; the observed counts can be anything.
Sample size
120
Shape of the table
3
20%
even
Push N down, add categories, or make the last category rare — each one thins the expected counts until some cell drops below 5 and the approximation can no longer be trusted.

The groups you are comparing

Several groups measured on one outcome. ANOVA pools every group's scatter into a single estimate of the common variance, so it only makes sense if there really is one common spread to estimate. The working rule of thumb: the largest group's standard deviation should be less than about twice the smallest.
The design
3
10
The spread inside each group
7.0
off
At zero inflation every group shares the same true spread and the ratio hovers near 1 (it wobbles from sampling alone). Push the red slider and one group's scatter grows — watch the largest‑over‑smallest ratio cross 2 and the condition fail.

Say it right

Statements people make about test conditions. Some are sound; some are habits worth breaking. Tap each one.
Pick a statementEach is either a fair reading of the conditions or a common slip. Work through all of them.

What to carry out of this section

The chi‑square condition is about expected counts, never observed ones. A cell can hold an observed count of zero and cause no trouble at all, as long as the count you would expect there is large enough. The reason is arithmetic: χ² divides by the expected count, and dividing by a tiny expected value makes a single ordinary cell explode the statistic. The common rule is that every expected count should be at least 5; a looser version tolerates a few between 1 and 5 as long as none fall below 1. When cells are too thin, collapse categories together or gather more data — or switch to Fisher's exact test, which needs no such approximation.
ANOVA's equal‑variance condition cannot be fixed by collecting more data. More people sharpen every estimate, but they do not pull unequal spreads together. If one group is genuinely more variable, that is a fact about the world, and the honest responses are a variance‑stabilising transformation or a version of the test built for unequal spreads (Welch's ANOVA) — not a bigger sample.
Normality matters most when the groups are small. The F test leans on the sampling behaviour of group means, and as each group grows the central limit theorem makes those means well‑behaved even when the raw data are not. With a handful of observations per group, though, a skewed or outlier‑ridden distribution can distort the result, and there is little data with which to check. Small groups are exactly where normality deserves your attention.
Independence is a property of the design, not of the numbers. No table of counts and no set of standard deviations can tell you whether one observation influenced another — that is decided by how the data were collected. Measuring the same person twice, clustering by household or clinic, or letting one response sway the next all break independence invisibly, and no on‑screen check will catch it. It is the condition you must reason about before the data ever arrive.