Lesson 5: Hypothesis tests with non-normal data

Every test in Lesson 4 assumed the underlying data were approximately normally distributed. Real housing-operations data often is not: work-order cycle times, cost overruns, and inspection scores are frequently skewed, with a long tail of unusually slow or expensive cases. When that assumption fails, a different family of tests takes over, ones that do not depend on the shape of the distribution and typically compare medians or overall distributions rather than means. This lesson covers when to suspect non-normality and which nonparametric test replaces each normal-data test from Lesson 4.

Deciding whether data are non-normal

Before defaulting to a nonparametric test, check whether normality actually fails, since parametric tests are generally more powerful when their assumptions hold. A normal probability plot is the standard visual check: points falling close to a straight line support normality, while a clear curve or S-shape does not. A formal goodness-of-fit test backs that visual judgment, most commonly the Anderson-Darling test, which compares the sample's cumulative distribution to the normal distribution and returns a p-value; a small p-value (typically below 0.05) is evidence against normality. Small sample sizes make any normality test underpowered, so a Green Belt should also consider whether the process itself has a physical reason to be skewed, such as a hard lower bound at zero on a cycle-time measurement.

Mann-Whitney test

The Mann-Whitney test (also called the Mann-Whitney U test or Wilcoxon-Mann-Whitney test) is the nonparametric counterpart to the 2-sample t test. It compares two independent, non-normal samples by ranking all the observations together and comparing the sum of ranks between groups, testing whether one population tends to produce larger values than the other. For example, GCHS could use it to compare rehab cost overrun amounts between two neighborhoods when the overrun data are heavily skewed by a few very expensive outlier jobs. The result is usually reported in terms of medians rather than means, since the median is not pulled around by extreme values the way a mean is.

Kruskal-Wallis and Mood's median test

When three or more independent, non-normal groups need comparing, two options replace one-way ANOVA:

• Kruskal-Wallis test: extends the Mann-Whitney logic to three or more groups by ranking all observations together and comparing the average rank in each group. It is the more commonly used default and is more powerful than Mood's median test when the data are only moderately skewed.
• Mood's median test: compares the groups' medians directly and is more robust to outliers than Kruskal-Wallis, at some cost in power, because it only asks whether each observation falls above or below the overall median rather than using full rank information.

A practical rule of thumb: if a boxplot or dotplot of the data shows a few extreme outliers driving the story, Mood's median test is the safer choice; otherwise Kruskal-Wallis is typically preferred.

Friedman test

The Friedman test is the nonparametric counterpart to a randomized block design (a two-way ANOVA where one factor is a blocking variable used to remove known nuisance variation, not a factor of interest). It applies when the same set of "blocks", such as the same five inspectors, each rate several different conditions, and the response is non-normal. The test ranks the observations within each block, then tests whether the average rank differs across conditions. GCHS could use it, for example, to have the same five inspectors each score three competing punch-list checklist formats on the same properties, removing individual inspector severity as a source of noise.

1-sample sign test and 1-sample Wilcoxon test

Both tests replace the 1-sample t test when the data are non-normal and the question concerns a single sample's central tendency versus a target value.

• 1-sample sign test: the more general and more robust of the two. It only looks at whether each observation is above or below the hypothesized median, ignoring how far above or below, and makes no assumption about the shape of the distribution beyond continuity. This robustness costs it statistical power.
• 1-sample Wilcoxon test: uses more information than the sign test, ranking the size of each observation's distance from the hypothesized median as well as its direction, which generally gives it more power. Its trade-off is an added assumption: the distribution of the data must be reasonably symmetric around the median, even if not normal.

Worked example, choosing between them. GCHS wants to test whether the median cost overrun on rehab change orders differs from a targeted $500, using a small sample of 10 jobs. A histogram of the overruns shows a long right tail from two very large change orders, so the distribution is clearly not symmetric. Because the Wilcoxon test requires symmetry and the sign test does not, the sign test is the defensible choice here even though it has less power.