mathematics

Hartley's test

Hartley's 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 Hartley's test rather than just read about it. In short: In statistics, Hartley's test, also known as the Fmax test or Hartley's Fmax, is used in the analysis of variance to verify that different groups have a similar variance, an assumption needed for other statistical tests. It was developed by H.

Key takeaways

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

Reference excerpt

In statistics, Hartley's test, also known as the Fmax test or Hartley's Fmax, is used in the analysis of variance to verify that different groups have a similar variance, an assumption needed for other statistical tests. It was developed by H. O. Hartley, who published it in 1950. The test involves computing the ratio of the largest group variance, max(sj2) to the smallest group variance, min(sj2). The resulting ratio, Fmax, is then compared to a critical value from a table of the sampling distribution of Fmax. If the computed ratio is less than the critical value, the groups are assumed to have similar or equal variances. Hartley's test assumes that data for each group are normally distributed, and that each group has an equal number of members. This test, although convenient, is quite sensitive to violations of the normality assumption. Alternatives to Hartley's test that are robust to violations of normality are O'Brien's procedure, and the Brown–Forsythe test.

Related tests Hartley's test is related to Cochran's C test in which the test statistic is the ratio of max(sj2) to the sum of all the group variances. Other tests related to these, have test statistics in which the within-group variances are replaced by the within-group range. Hartley's test and these similar tests, which are easy to perform but are sensitive to departures from normality, have been grouped together as quick tests for equal variances and, as such, are given a commentary by Hand & Nagaraja (2003).

See also Bartlett's test Brown–Forsythe test

Notes

References Bliss, C.I., Cochran, W.G., Tukey, T.W. (1956) A Rejection Criterion Based upon the Range. Biometrika, 43, 418–422. Cochran, W.G. (1941). The distribution of the largest of a set of estimated variances as a fraction of their total. Annals of Eugenics, 11, 47–52 Hand, H.A. & Nagaraja, H.N. (2003) Order Statistics, 3rd Edition. Wiley. ISBN 0-471-38926-9 Hartley, H.O. (1950). The maximum F-ratio as a short cut test for homogeneity of variance, Biometrika, 37, 308-312. David, H.A. (1952). "Upper 5 and 1% points of maximum F-ratio." Biometrika, 39, 422–424. O'Brien, R.G. (1981). A simple test for variance effects in experimental designs. Psychological Bulletin, 89, 570–574. Keppel, G. and Wickens, T.D. (2004). Design and analysis (4th ed.). Englewood Cliffs, NJ: Prentice-Hall. Pearson, E.S., Hartley, H.O. (1970). Biometrika Tables for Statisticians, Vol 1, CUP ISBN 0-521-05920-8

External links Table of critical values for the Fmax test [1] Archived 2011-06-01 at the Wayback Machine

Worked examples

Example 1 — a first encounter with Hartley's test

Start with the simplest possible case. Write down what Hartley's 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 Hartley's 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 Hartley's 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 Hartley's test

In research: Hartley's 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 Hartley's 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: Hartley's test is common in secondary-school and first-year university syllabi. It links to neighbouring topics Analysis of variance, Statistical tests, so understanding it makes those chapters shorter.
In everyday life: Look for Hartley's 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 Hartley's test in 20 minutes

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

Frequently asked questions

What is Hartley's test in simple terms?

Why does Hartley's 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 Hartley's 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 Hartley's test.