mathematics

Siegel–Tukey test

Siegel–Tukey test is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Siegel–Tukey test rather than just read about it. In short: Siegel–Tukey test, named after Sidney Siegel and John Tukey, is a non-parametric test which may be applied to data measured at least on an ordinal scale. It tests for differences in scale between two groups.

Key takeaways

Siegel–Tukey test belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Siegel–Tukey test to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Siegel–Tukey test from memory before moving on to harder problems.

Reference excerpt

Siegel–Tukey test, named after Sidney Siegel and John Tukey, is a non-parametric test which may be applied to data measured at least on an ordinal scale. It tests for differences in scale between two groups. The test is used to determine if one of two groups of data tends to have more widely dispersed values than the other. In other words, the test determines whether one of the two groups tends to move, sometimes to the right, sometimes to the left, but away from the center (of the ordinal scale). The test was published in 1960 by Sidney Siegel and John Wilder Tukey in the Journal of the American Statistical Association, in the article "A Nonparametric Sum of Ranks Procedure for Relative Spread in Unpaired Samples."

Principle The principle is based on the following idea: Suppose there are two groups A and B with n observations for the first group and m observations for the second (so there are N = n + m total observations). If all N observations are arranged in ascending order, it can be expected that the values of the two groups will be mixed or sorted randomly, if there are no differences between the two groups (following the null hypothesis H0). This would mean that among the ranks of extreme (high and low) scores, there would be similar values from Group A and Group B. If, say, Group A were more inclined to extreme values (the alternative hypothesis H1), then there will be a higher proportion of observations from group A with low or high values, and a reduced proportion of values at the center.

Hypothesis H0: σ2A = σ2B & MeA = MeB (where σ2 and Me are the variance and the median, respectively) Hypothesis H1: σ2A > σ2B

Method Two groups, A and B, produce the following values (already sorted in ascending order):

A: 33 62 84 85 88 93 97 B: 4 16 48 51 66 98 By combining the groups, a group of 13 entries is obtained. The ranking is done by alternate extremes (rank 1 is lowest, 2 and 3 are the two highest, 4 and 5 are the two next lowest, etc.).

The sum of the ranks within each W group:

WA = 5 + 12 + 11 + 10 + 7 + 6 + 3 = 54 WB = 1 + 4 + 8 + 9 + 13 + 2 = 37 If the null hypothesis is true, it is expected that the average ranks of the two groups will be similar. If one of the two groups is more dispersed its ranks will be lower, as extreme values receive lower ranks, while the other group will receive more of the high scores assigned to the center. To test the difference between groups for significance a Wilcoxon rank sum test is used, which also justifies the notation WA and WB in calculating the rank sums. From the rank sums the U statistics are calculated by subtracting off the minimum possible score, n(n + 1)/2 for each group:

UA = 54 − 7(8)/2 = 26 UB = 37 − 6(7)/2 = 16 According to H 0 {\displaystyle H_{0}} the minimum of these two values is distributed according to a Wilcoxon rank-sum distribution with parameters given by the two group sizes:

min ( U A , U B ) ∼ Wilcoxon ( m , n ) {\displaystyle \min(U_{A},U_{B})\sim {\text{Wilcoxon}}(m,n)\!}

Which allows the calculation of a p-value for this test according to the following formula:

p = Pr [ X ≤ min ( U A , U B ) ] {\displaystyle p=\Pr \left[X\leq \min(U_{A},U_{B})\right]\,\!}

X ∼ Wilcoxon ( m , n ) {\displaystyle X\sim {\text{Wilcoxon}}(m,n)\,\!}

a table of the Wilcoxon rank-sum distribution can be used to find the statistical significance of the results (see Mann–Whitney_U_test for more explanations on these tables). For the example data, with groups of sizes m=6 and n=7 the p-value is:

p = Pr [ x ≤ 16 ] = 0.2669. {\displaystyle p=\Pr \left[x\leq 16\right]=0.2669.\,\!}

indicating little or no reason to reject the null hypothesis that the dispersion of the two groups is the same.

See also Non-parametric statistics Statistical hypothesis testing

References

External links an R implementation of Siegel-Tukey test

Worked examples

Example 1 — a first encounter with Siegel–Tukey test

Start with the simplest possible case. Write down what Siegel–Tukey test claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Siegel–Tukey test before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Siegel–Tukey test ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Siegel–Tukey test

In research: Siegel–Tukey test appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Siegel–Tukey test in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Siegel–Tukey test is common in secondary-school and first-year university syllabi. It links to neighbouring topics Nonparametric statistics, Statistical tests, so understanding it makes those chapters shorter.
In everyday life: Look for Siegel–Tukey test outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Siegel–Tukey test in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Siegel–Tukey test means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Siegel–Tukey test out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Siegel–Tukey test in simple terms?

Why does Siegel–Tukey test matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Siegel–Tukey test?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Siegel–Tukey test.