Module 11 · Analyze

Hypothesis testing

Two suppliers, two fixtures, two ovens: is the difference you see real, or is it the noise you would see anyway? A hypothesis test is the disciplined way to ask that question. It is also the most misread tool in the Green Belt kit, because a p-value does not say what most people think it says. This module builds the logic once, applies it to the six tests you will actually use, and insists on the two numbers that go next to every p-value: the size of the effect and the interval around it.

Learning objectives

• State a null and an alternative hypothesis for a manufacturing question and say what a Type I and a Type II error would cost.
• Explain a p-value and a confidence interval correctly, and recognise the six common misreadings the ASA statement warns against.
• Run and read a two-sample t-test (Welch), a paired t-test, a one-sample t-test, a one-way ANOVA and a chi-square test of independence from printed data.
• Check the assumptions behind each test (normality, equal variances, independence) and choose a nonparametric alternative when they fail.
• Report an effect size and a confidence interval alongside every p-value, and separate statistical from practical significance.
• Compute the power of a planned comparison and the sample size it needs, and explain why a small study that "found nothing" proves nothing.

Before you start

Three questions on Module 1. They do not count toward completion.

Why this matters

In 2016 the American Statistical Association did something it had never done: it issued a formal statement on a single statistical method, because the method was being misused so widely that the association judged it a threat to the credibility of science. The statement on p-values sets out six principles. A p-value indicates how incompatible the data are with a specified statistical model. It does not measure the probability that the hypothesis is true, or the probability that the data were produced by chance alone. Scientific and business conclusions should not be based only on whether a p-value passes a threshold. Proper inference requires full reporting. A p-value does not measure the size of an effect or the importance of a result. And by itself a p-value is not a good measure of evidence.^[1] Every one of those six has a manufacturing version, and this module is organised so that you cannot commit any of them without noticing.

The following is an illustrative failure with invented details, not a real program. A Green Belt project changes a braze fixture. The leak rate over the two weeks before was 2.3 %, and over the two weeks after it is 1.9 %. The team's report shows two bars and the word "improved". A quality engineer asks how many cores were tested: about 800 in each period, so 18 leakers before and 15 after. A two-proportion test gives p = 0.6; the 95 % interval on the change runs from about −1.8 to +1.0 percentage points, that is, from a substantial improvement to a moderate worsening. The fixture may well be better. The data cannot tell, and the report should have said so, together with the number of cores it would take to find out (Module 8 did that arithmetic for a similar case: several thousand). The test is not there to bless the bar chart. It is there to tell you whether you have enough evidence to act, and if not, how much more you need.

Everything that follows is taught with the ASQ Green Belt body of knowledge in mind, which lists one- and two-sample tests, paired tests, ANOVA, chi-square, tests of proportions and variances, and nonparametric methods.^[14] This is not a certification course, but you will find nothing here that a Green Belt exam would not expect you to know.

The logic of a test

A hypothesis test starts from a statement you would like to disprove, the null hypothesis H₀: the two suppliers have the same mean pull strength; the fixture change did nothing; defect type does not depend on shift. The alternative H₁ is what you would conclude if the null fails. You collect data, compute a test statistic that measures how far the data sit from what H₀ predicts, in units of the noise, and then ask one precise question:

Before looking at the data you choose a significance level α, conventionally 0.05, and you reject H₀ if p < α. That decision can go wrong in two ways, and the two are not symmetric.

In a plant, a Type I error means changing a process, a supplier or a fixture for no reason: cost, disruption, and a story that "the data said so". A Type II error means leaving a real problem in place because the study was too small to see it. Six Sigma culture worries about the first; in practice the second is far more common, because studies are small and nobody computes β. The section on power at the end of this module is the antidote.

p-values and confidence intervals, read correctly

A p-value answers a yes/no question and throws away the magnitude. The confidence interval keeps it. A 95 % confidence interval for a difference is a range computed from the data by a procedure that, in repeated sampling from the same process, would cover the true difference 95 % of the time. It is not a statement that the true difference has a 95 % probability of lying in this particular range; the true difference is a fixed number, and the interval is what varies. In practice the interval is read as the set of true differences that the data are compatible with at the 5 % level, which is exactly the information a decision needs: if the whole interval is inside the range you do not care about, the difference does not matter even if p is small; if the interval spans both "harmless" and "serious", you do not know yet, whatever p says.

Three numbers therefore go on every report of a comparison: the estimate of the difference in engineering units, its confidence interval, and the p-value. A fourth is useful when the units are unfamiliar: a standardised effect size. For a difference of means the usual one is Cohen's d, the difference divided by the pooled standard deviation. Cohen's descriptive labels (0.2 small, 0.5 medium, 0.8 large) come from behavioural science and were offered by Cohen himself as a last resort when nothing better is known; in engineering the better reference is the tolerance or the customer's requirement, not the label.^[13]

Two independent samples: the t-test

The workhorse. Two groups, each a sample of a continuous characteristic; is the mean different? The statistic is the observed difference divided by its standard error. Welch's version, which does not assume the two groups have the same variance, is the default in this course and in most modern software, because it costs almost nothing when the variances are equal and protects you when they are not.^[4]

Worked example 1: pull strength from two suppliers

The data below is a constructed example generated by a seeded script, not a real production run. Setting: the pull strength in N of a soldered terminal, 12 joints made with supplier A's terminals and 12 with supplier B's, pulled on a tester reading to 0.1 N, in randomised order on the same day.

Step 1, look. Supplier A: mean 46.07 N, s = 2.277 N, range 42.4 to 50.4. Supplier B: mean 49.02 N, s = 2.799 N, range 43.5 to 52.8. B is about 3 N stronger on average and slightly more variable. Dot plots of the two groups (Module 9) overlap substantially; the question is whether a 3 N gap between means of 12 is more than noise.

Step 2, assumptions. The t-test assumes each group is a random sample from a roughly normal distribution and that the groups are independent. Anderson-Darling on A: A² = 0.57, p = 0.11; on B: A² = 0.21, p = 0.81. No evidence against normality in either group, which with 12 values is also not strong evidence for it; the t-test is tolerant of moderate non-normality at these sizes. Levene's test for equal variances: W = 1.22, p = 0.28; the F ratio of the variances is 0.66 (p = 0.50). Nothing suggests unequal spread, and Welch's test does not need the assumption anyway.

Step 3, the test. Difference A − B = −2.958 N. Standard error = √(5.186/12 + 7.837/12) = √(0.4322 + 0.6530) = 1.0417 N. t = −2.958 / 1.0417 = −2.840 on ν = 21.1 degrees of freedom, p = 0.0098. The pooled test gives the same t on 22 degrees of freedom and p = 0.0095; the two agree because the variances are similar and the groups are equal in size.

Step 4, size and interval. The 95 % interval on the difference is −2.958 ± 2.079 × 1.0417 = −2.958 ± 2.17, that is −5.12 to −0.79 N. Pooled s_p = √6.511 = 2.552 N, so d = −2.958 / 2.552 = −1.16: a difference of more than one standard deviation, large by any label.

Step 5, the sentence. "Supplier B's joints pulled −3.0 N stronger on average (95 % interval −5.1 to −0.8 N in favour of B; Welch t = −2.84, ν = 21.1, p = 0.010; d = −1.16). Data this far apart would arise in about 1 % of studies of this size if the suppliers were truly equal." Whether −3.0 N matters is an engineering question: if the drawing minimum is 40 N and both suppliers sit well above it, the difference is real and irrelevant; if the minimum is 45 N, supplier A's lower tail is a problem and B is the answer. The test does not know which. You do.

Calculator: t-tests and one-way ANOVA

Pre-loaded with Table 1. The output prints the p-value sentence in the correct form, both the Welch and pooled results, the interval and d. Switch the mode to run the paired and one-sample tests and ANOVA on the same page.

Paired samples

When the same units are measured twice, before and after a change, the two measurements are not independent: a station that was slow before is probably slow after. Comparing the two columns with a two-sample t-test throws that information away and buries the change under the station-to-station spread. The paired t-test works on the differences instead, one per unit, and tests whether their mean is zero. It is simply a one-sample t-test on the differences.

Worked example 2: cycle time before and after a fixture change

Constructed data, not a real production run. Setting: the manual assembly cycle time in seconds at 12 stations, each timed with a stopwatch to 0.1 s before and after a new locating fixture was fitted, same operators, same shift.

Mean before 49.16 s, mean after 47.87 s. The differences have mean d̄ = −1.292 s and s_d = 1.056 s; standard error 1.056/√12 = 0.3049 s; t = −1.292 / 0.3049 = −4.24 on 11 degrees of freedom, p = 0.0014. The 95 % interval on the mean saving is −1.292 ± 2.201 × 0.3049 = −1.96 to −0.62 s; d (on the differences) = −1.22. The fixture saves about 2.6 % of the cycle, and the data are quite clear about it: eleven of the twelve stations got faster, and the one that did not changed by less than half a second.

Run the same numbers as a two-sample test (the "before" column against the "after" column) and the station-to-station spread of more than 3 s swamps a 1.3 s change: the p-value rises to about 0.4 and you would report no evidence of improvement. The pairing is the analysis. Note also what the test does not tell you: whether the saving persists after the novelty wears off, and whether the operators worked faster because they were being timed. A control chart of cycle time over the following weeks (Module 16) answers the first.

One sample against a target

Is the process on target? The one-sample t-test compares a sample mean with a stated value μ₀: t = (x̄ − μ₀)/(s/√n) on n − 1 degrees of freedom. The NIST handbook's worked example of this test is the reference for its form.^[3] The interesting thing about it in a plant is how often it is significant and irrelevant at the same time.

Constructed data: the Module 1 torque exercise. Thirty tightening torques with a target of 12.0 N·m and a tolerance of ±1.0 N·m: mean 12.1287 N·m, s = 0.2854, standard error 0.0521; t = 0.1287 / 0.0521 = 2.47, p = 0.020; 95 % interval for the mean 12.022 to 12.235 N·m; d = 0.45. Statistically, the mean is not 12.0: the offset of 12.13 − 12.0 = 0.13 N·m is about two and a half standard errors from zero. Practically, the offset is 13 % of the half-tolerance, and whether that is worth a tool adjustment depends on the joint, not on the p-value. This is the ASA's fifth principle in one example: p measures the evidence for an offset, not its size.^[1]

Several groups: one-way ANOVA

Three machines, four cavities, five operators: are the means all the same? Running t-tests on every pair inflates the Type I error (six comparisons at α = 0.05 gives roughly a one in four chance of at least one false alarm). Analysis of variance tests all the means at once by splitting the total variation into a between-group part and a within-group part and comparing them.

Worked example 3: cycle time on three machines

Constructed data, not a real production run. Setting: the cycle time in s of the same operation on three nominally identical machines, ten cycles each, timed to 0.1 s.

Means 42.24, 41.85 and 42.95 s; grand mean 42.347 s. F = 3.110 / 0.5553 = 5.60 against a 5 % critical value of 3.35 on (2, 27) degrees of freedom; p = 0.0092. η² = 6.221 / 21.215 = 0.29: the machine identity explains about 29 % of the variation in cycle time, and the within-machine standard deviation (√MS_within) is 0.745 s. Levene's test on the three spreads: p = 0.42, so the equal-variance assumption behind the F test is not contradicted.

ANOVA says the means are not all equal. It does not say which ones differ. Machine 3 is 1.10 s slower than machine 2 and the obvious next step is to compare pairs, but each pairwise comparison after a significant F must use a multiple-comparison method (Tukey's is the standard) so that the family of comparisons keeps its 5 % error rate; any statistics package provides it (this course's calculator stops at the F test and η², deliberately, and says so). The engineering question, as always, is different: why is machine 3 slower, and does 1.1 s matter against the takt time (Module 2)?

Counts in categories: the chi-square test

Defect type by shift, leak location by fixture, pass/fail by supplier: when both variables are categories the data are a table of counts, and the question is whether the row variable and the column variable are associated. Pearson's chi-square test compares each observed count with the count expected if the rows and columns were independent.^[10]

Worked example 4: defect type by shift

Constructed data, not a real production run. Setting: every reject on a brazed assembly line over a month, classified by defect type and by the shift that made it.

Row totals 120, 110 and 95; column totals 115, 90, 78 and 42; N = 325. If defect type did not depend on shift, the day shift would show 120 × 115 / 325 = 42.46 porosity rejects (it shows 42) and the night shift 95 × 78 / 325 = 22.80 gap rejects (it shows 32). The largest contributions to χ² come from exactly those cells: night-shift gap (3.71) and evening-shift porosity (2.61). In all, χ² = 15.15 on 6 degrees of freedom against a 5 % critical value of 12.59; p = 0.019; the smallest expected count is 12.3, so the approximation is safe. Cramér's V = √(15.15 / 650) = 0.15, a weak association overall.

What it means: the mix of defects differs by shift, and the table says where. Gap defects are 18.3 % of day-shift rejects, 21.8 % of evening and 33.7 % of night; porosity runs the other way (35.0, 44.5, 25.3 %). That is a lead, not a cause: the night shift may set fixtures differently, or run a different furnace, or simply classify defects differently (Module 4's attribute agreement analysis is how you check the last one). Note also what the test compares: the mix of defect types among rejects, not the reject rate. A shift with twice the rejects and the same mix would show nothing here; that question is a p chart or a two-proportion test.

Tests for variances

Sometimes the spread is the question: does the new fixture reduce variation, are two gauges equally repeatable? The classical F test compares two variances by their ratio, F = s₁²/s₂², against the F distribution with (n₁ − 1, n₂ − 1) degrees of freedom. It is exact for normal data and badly wrong for anything else, because a variance ratio is very sensitive to tails. Levene's test, which runs an ANOVA on the absolute deviations of each value from its group centre (the median, in the robust version used here), is far less sensitive to non-normality and is the one to report.^[5] In worked example 1 the F ratio was 0.66 (p = 0.50) and Levene's W was 1.22 (p = 0.28); they agree here because the data are close to normal. Because Welch's t-test does not assume equal variances, the variance test is not a gate before the t-test; it is a question in its own right.

Normality tests, and what they are for

Two tests dominate the software menus. The Shapiro-Wilk test compares the ordered sample with what a normal sample would look like and is the most powerful general test at small n.^[6] The Anderson-Darling test measures the distance between the empirical and the fitted normal cumulative distribution, weighted toward the tails, and is the default in most SPC software; the p-values in this course come from the adjusted statistic A* = A²(1 + 0.75/n + 2.25/n²) with the approximation tabulated by Stephens.^[7][8]

Three cautions. A normality test can only reject; a p of 0.8 on 12 values says the sample is not obviously non-normal, which is faint praise. With hundreds of values the test rejects for departures too small to matter to a t-test, whose robustness comes from the central limit theorem (Module 1). And a rejection is a question, not a verdict: Module 8's flatness data failed the test because the characteristic is bounded, and Wheeler's point that non-normal-looking data are usually inhomogeneous data applies to test inputs as much as to capability studies. Look at the histogram and the time order before you believe either the test or its p-value.

When the assumptions fail: rank-based tests

For skewed data, data with outliers, or ordinal ratings, the mean is a poor summary and the t-test's p-value is unreliable at small n. Rank-based tests replace the values by their ranks and ask whether one group tends to have larger values than the other. The two-sample version is the Wilcoxon rank-sum test, equivalently the Mann-Whitney U test; the paired version is the Wilcoxon signed-rank test; the several-group version is the Kruskal-Wallis test.^[9] They lose a little power on truly normal data and gain a great deal on anything else.

Worked example 5: flatness from two fixtures

Constructed data, not a real production run. Setting: the flatness in µm of a milled face on 15 parts from each of two fixtures, CMM to 0.1 µm. Flatness is bounded at zero and right-skewed (Modules 1 and 8), and the data were constructed that way.

Medians 8.7 and 11.2 µm; means 9.12 and 12.48 µm, pulled up by the tails. Rank sums R₁ = 177 and R₂ = 288 (they add to 465, as 30 ranks must). U₁ = 177 − 15 × 16/2 = 177 − 120 = 57; U₂ = 225 − 57 = 168. Mean of U under H₀ = 112.5; σ_U = 24.10 (it would be 24.11 without the 3 tied pairs); z = (168 − 112.5 − 0.5) / 24.10 = 2.28; p = 0.022. Fixture 2 tends to give larger flatness: the probability that a random F1 part is flatter than a random F2 part is 1 − 0.25 = about three in four, and the Hodges-Lehmann shift estimate is −3.1 µm (F1 minus F2).

Now the same data in Welch's t-test: t = −1.57, p = 0.129. Not significant. The t-test is comparing means that a few large values dominate, with standard errors those same values inflate; the rank test sees that most F2 parts sit above most F1 parts. Neither test is "right" in the abstract. The rank test answers the better question for a skewed, bounded characteristic ("does one fixture tend to make flatter parts?"), and it should be the one on the report, with the medians and the shift estimate. A log transformation followed by a t-test, as in Module 8, is the other legitimate route and usually agrees with the rank test.

Power and sample size

Every "no significant difference" result raises the question the ASA's sixth principle points at: was there enough evidence to find one? Power is the probability that the test rejects H₀ when a difference of a stated size δ really exists. It rises with the sample size, falls with the noise σ, and depends on the α you chose. The NIST handbook gives the sample-size formula and its worked example, which the course harness reproduces exactly.^[2]

The NIST check. One-sample, one-sided, α = 0.05, β = 0.10, δ = σ: (1.645 + 1.282)² = 2.926² = 8.56, so n = 9; the t-iteration raises it to 11. Both match the handbook's published example.^[2]

Worked example 6: how much could the pull-strength study have missed?

Parameters from the constructed pull-strength study. Suppose the engineering question had been whether the suppliers differ by 2.0 N or more, with σ taken as 2.65 N (the pooled standard deviation of the study, rounded) and 12 joints per supplier. The standardised difference is d = 2.0 / 2.65 = 0.755; the noncentrality is d √(12/2) = 0.755 × 2.449 = 1.849; the power is Φ(1.849 − 1.960) = Φ(−0.111) = 0.46. A coin toss. Had the true difference been 2.0 N, this study would have missed it more often than not; it found a 3.0 N difference, which is why p came out small, but a "not significant" result from the same design would have meant almost nothing.

To have an 80 % chance of detecting 2.0 N: n = 2 × (1.960 + 0.842)² × (2.65/2.0)² = 2 × 7.849 × 1.7556 = 27.6, so 28 per group by the normal formula and 29 after NIST's t-iteration. Roughly two and a half times the study that was run. This is the number to compute before the study, from the difference that would change a decision and the σ from a pilot or a control chart, and to write into the project plan. A study designed for 80 % power that finds nothing has earned the right to say "no difference of 2 N or more"; a study of 12 that finds nothing has not.

Calculator: sample size for a two-sample t-test

Pre-loaded with the pull-strength parameters. Change δ to the difference that would change your decision and σ to your process's noise.

Common mistakes

1. Reading p as the probability that H₀ is true. Consequence: "p = 0.03, so there is a 97 % chance the fixture works." Fix: p is the probability of data this extreme if H₀ were true; say it that way, every time.
2. Reporting p without the difference and its interval. Consequence: a 0.13 N·m offset with p = 0.02 gets a tool adjustment nobody needed, and a 2.9 N difference with p = 0.06 gets ignored. Fix: estimate, interval, p, effect size, in that order.
3. "Not significant" read as "no difference". Consequence: a real 2 N supplier difference is declared absent by a study with 46 % power. Fix: compute the power, or better, the sample size, before the study; report what the study could have detected.
4. Unpaired analysis of paired data. Consequence: a 1.3 s saving at every station disappears under 3 s of station-to-station spread. Fix: if the same units were measured twice, analyse the differences.
5. Many t-tests instead of ANOVA, or ANOVA without a multiple-comparison method after it. Consequence: false alarms at a rate far above 5 %. Fix: ANOVA first, Tukey (or equivalent) for the pairs.
6. A t-test on skewed data with outliers. Consequence: the mean is a poor summary and the p-value swings with one part. Fix: look first; use a rank test or a transformation, and report medians.
7. Testing the hypothesis the data suggested. Consequence: you noticed the night shift looked worse, then tested night against the rest and got p = 0.02; the test's α no longer means 5 %, because you chose the comparison after seeing it. Fix: state the hypothesis before looking, or treat a data-suggested finding as a lead to confirm with new data.
8. Chi-square on rates that are really counts of different sizes. Consequence: comparing "defects per shift" when the shifts made different quantities. Fix: the chi-square table needs the counts, and the denominators need a p chart or a two-proportion test.

Exercises

Exercise 1: cure time in two ovens

Constructed data, not a real production run. Setting: the time in minutes for a structural adhesive to reach handling strength, ten samples cured in each of two ovens, timed to 0.1 min.

Tasks. (a) Summarise both groups and check the assumptions. (b) Run Welch's t-test; give t, ν, p, the 95 % interval on the difference and d. (c) The process owner wants to know whether oven 2 is slower. Write the sentence. (d) A difference of 1.0 min would matter to the line balance. Could this study have detected it? How many samples per oven would be needed for 80 % power?

Exercise 2: leak location by braze fixture

Constructed data, not a real production run. Setting: 112 heat-exchanger cores that failed the leak test over a quarter, each traced to the braze fixture it was built on and to the joint that leaked (header, tube or baffle).

Tasks. (a) State H₀. (b) Compute the expected counts, χ², df and p. (c) Fixture A shows a higher share of header leaks; does the test support acting on that? (d) What question does this table not answer about the fixtures?

Quiz

Ten questions. Score 70 % or more to mark the module complete on this device.

Key takeaways

• A p-value is the probability of data at least this extreme if the null hypothesis were true. It is not the probability the hypothesis is true, and it does not measure the size or importance of an effect.
• Report four things: the estimated difference in engineering units, its confidence interval, the p-value, and an effect size. Decide on the interval against the tolerance or the customer's need, not on p alone.
• Welch's t-test for two independent groups, the paired t-test for the same units measured twice, one-way ANOVA (then Tukey) for several groups, chi-square for counts in categories.
• Check assumptions by looking first: histograms, time order, then Anderson-Darling or Shapiro-Wilk and Levene. A normality test can only reject, and at large n it rejects for departures that do not matter.
• For skewed or outlier-prone data use rank-based tests (Mann-Whitney, Wilcoxon signed-rank, Kruskal-Wallis) with medians and the Hodges-Lehmann shift, or transform and test.
• "Not significant" is not "no difference". Compute power or the required sample size before the study, from the difference that would change a decision and a σ from a pilot or a control chart.
• State the hypothesis before looking at the data. A comparison chosen because it looked interesting has an α nobody can compute; confirm it with new data.

References

All web sources accessed 2026-09-09 or 2026-09-10. Sources marked "secondary" or "catalogue" were not read in full by the course author; the claim is taken from the part shown. Standards and papers are paraphrased.

1. Wasserstein, R. L., and Lazar, N. A. "The ASA's Statement on p-Values: Context, Process, and Purpose." The American Statistician 70(2):129 to 133, 2016. https://www.stat.berkeley.edu/~aldous/Real_World/ASA_statement.pdf
2. NIST/SEMATECH. "7.2.2.2. Sample sizes required." e-Handbook of Statistical Methods. The z-based formula, the t-iteration, and the worked example (α 0.05, β 0.10, δ = σ: 9, then 11) reproduced by this course's harness. https://www.itl.nist.gov/div898/handbook/prc/section2/prc222.htm
3. NIST/SEMATECH. "7.2.2. Are the data consistent with the assumed process mean?" e-Handbook of Statistical Methods. One-sample t and z tests. https://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm
4. Welch, B. L. "The Generalization of 'Student's' Problem when Several Different Population Variances are Involved." Biometrika 34(1/2):28 to 35, 1947. https://academic.oup.com/biomet/article-abstract/34/1-2/28/210174 (abstract read)
5. Levene, H. "Robust Tests for Equality of Variances." In Olkin, I. (ed.), Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling, pp. 278 to 292. Stanford University Press, 1960. (catalogue; the median-centred version used here is the common modification)
6. Shapiro, S. S., and Wilk, M. B. "An Analysis of Variance Test for Normality (Complete Samples)." Biometrika 52(3/4):591 to 611, 1965. https://academic.oup.com/biomet/article-abstract/52/3-4/591/336553 (abstract read)
7. Anderson, T. W., and Darling, D. A. "Asymptotic Theory of Certain 'Goodness of Fit' Criteria Based on Stochastic Processes." Annals of Mathematical Statistics 23(2):193 to 212, 1952. https://projecteuclid.org/.../10.1214/aoms/1177729437.full (abstract read)
8. Stephens, M. A. "Tests Based on EDF Statistics." In D'Agostino, R. B., and Stephens, M. A. (eds.), Goodness-of-Fit Techniques. Marcel Dekker, 1986. https://openlibrary.org/books/OL26834757M/Goodness-of-Fit_Techniques (catalogue; source of the adjusted Anderson-Darling statistic and p-value approximation implemented in the course harness, as reproduced in standard statistical software)
9. Wilcoxon, F. "Individual Comparisons by Ranking Methods." Biometrics Bulletin 1(6):80 to 83, 1945; Mann, H. B., and Whitney, D. R. "On a Test of Whether One of Two Random Variables is Stochastically Larger than the Other." Annals of Mathematical Statistics 18(1):50 to 60, 1947; Kruskal, W. H., and Wallis, W. A. "Use of Ranks in One-Criterion Variance Analysis." Journal of the American Statistical Association 47(260):583 to 621, 1952. https://garfield.library.upenn.edu/classics1987/A1987K083100001.pdf (citation-classic commentary read; originals not read)
10. Pearson, K. "On the Criterion that a Given System of Deviations from the Probable in the Case of a Correlated System of Variables is such that it can be Reasonably Supposed to have Arisen from Random Sampling." Philosophical Magazine, Series 5, 50(302):157 to 175, 1900. https://www.tandfonline.com/doi/abs/10.1080/14786440009463897 (catalogue; origin of the chi-square test)
11. Cramér, H. Mathematical Methods of Statistics. Princeton University Press, 1946. https://archive.org/details/in.ernet.dli.2015.223699 (catalogue; origin of the association measure known as Cramér's V)
12. Hodges, J. L., and Lehmann, E. L. "Estimates of Location Based on Rank Tests." Annals of Mathematical Statistics 34(2):598 to 611, 1963. https://projecteuclid.org/.../10.1214/aoms/1177704172.full (abstract read)
13. Cohen, J. Statistical Power Analysis for the Behavioral Sciences, 2nd ed. Lawrence Erlbaum, 1988. https://www.routledge.com/Statistical-Power-Analysis-for-the-Behavioral-Sciences/Cohen/p/book/9780805802832 (catalogue; the 0.2 / 0.5 / 0.8 labels are behavioural-science conventions)
14. ASQ. Certified Six Sigma Green Belt (CSSGB) Body of Knowledge Map 2014 to 2022. ASQ, 2022. https://www.asq.org/cert/resource/pdf/certification/2022-CSSGB-BoK-Map.pdf

Further reading

• NIST/SEMATECH e-Handbook, Chapter 7 (Product and Process Comparisons), for the full set of comparison procedures with worked examples.
• Module 8 for the two-proportion arithmetic on attribute data; Module 13 for designed experiments, where the same F tests reappear with several factors at once.