mathematics

Median test

Median 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 Median test rather than just read about it. In short: The median test (also Mood’s median-test, Westenberg-Mood median test or Brown-Mood median test) is a special case of Pearson's chi-squared test. It is a nonparametric test that tests the null hypothesis that the medians of the populations from which two or more samples drawn are identical.

Key takeaways

Median 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 Median test to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Median test from memory before moving on to harder problems.

Reference excerpt

The median test (also Mood’s median-test, Westenberg-Mood median test or Brown-Mood median test) is a special case of Pearson's chi-squared test. It is a nonparametric test that tests the null hypothesis that the medians of the populations from which two or more samples drawn are identical. The data in each sample is assigned to two groups, one consisting of data whose values are higher than the median value in the two groups combined, and the other consisting of data whose values are at the median or below. A Pearson's chi-squared test is then used to determine whether the observed frequencies in each sample differ from expected frequencies derived from a distribution combining the two groups.

Relation to other tests The test has low power (efficiency) for moderate to large sample sizes. The Wilcoxon–Mann–Whitney U two-sample test or its generalisation for more samples, the Kruskal–Wallis test, can often be considered instead. The relevant aspect of the median test is that it only considers the position of each observation relative to the overall median, whereas the Wilcoxon–Mann–Whitney test takes the ranks of each observation into account. Thus the other mentioned tests are usually more powerful than the median test. Moreover, the median test can only be used for quantitative data. However, the null hypothesis verified by the Wilcoxon–Mann–Whitney U (and so the Kruskal–Wallis test) is not only about medians. The test is sensitive also to differences in scale parameters and symmetry. As a consequence, if the Wilcoxon–Mann–Whitney U test rejects the null hypothesis, one cannot say that the rejection was caused only by the shift in medians. It is easy to prove by simulations, where samples with equal medians, yet different scales and shapes, lead the Wilcoxon–Mann–Whitney U test to fail as a test of medians. However, although the alternative Kruskal-Wallis test does not assume normal distributions, it does assume that the variance is approximately equal across samples. Hence, in situations where that assumption does not hold, the median test is an appropriate test. Moreover, Siegel & Castellan (1988, p. 124) suggest that there is no alternative to the median test when one or more observations are "off the scale."

See also Sign test – a paired alternative to the median test.

References

Corder, G.W. & Foreman, D.I. (2014). Nonparametric Statistics: A Step-by-Step Approach, Wiley. ISBN 978-1118840313. Siegel, S., & Castellan, N. J. Jr. (1988, 2nd ed.). Nonparametric statistics for the behavioral sciences. New York: McGraw–Hill. Friedlin, B. & Gastwirth, J. L. (2000). Should the median test be retired from general use? The American Statistician, 54, 161–164.

Worked examples

Example 1 — a first encounter with Median test

Start with the simplest possible case. Write down what Median 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 Median 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 Median 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 Median test

In research: Median 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 Median 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: Median 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 Median 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 Median test in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Median 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 Median test out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Median test in simple terms?

Why does Median 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 Median 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 Median test.