mathematics

Trimean

Trimean 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 Trimean rather than just read about it. In short: In statistics the trimean (TM), or Tukey's trimean, is a measure of a probability distribution's central tendency defined as a weighted average of the distribution's quartiles: T M = Q 1 + 2 Q 2 + Q 3 4 {\displaystyle TM={\frac {Q_{1}+2Q_{2}+Q_{3}}{4}}} It gives twice as much weight to the median or second quartile Q 2 {\displaystyle Q_{2}} than the first and third quartiles. This is equivalent to the arithmetic mea…

Key takeaways

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

Reference excerpt

In statistics the trimean (TM), or Tukey's trimean, is a measure of a probability distribution's central tendency defined as a weighted average of the distribution's quartiles:

T M = Q 1 + 2 Q 2 + Q 3 4 {\displaystyle TM={\frac {Q_{1}+2Q_{2}+Q_{3}}{4}}}

It gives twice as much weight to the median or second quartile Q 2 {\displaystyle Q_{2}} than the first and third quartiles. This is equivalent to the arithmetic mean of the median and the midhinge:

T M = 1 2 ( Q 2 + Q 1 + Q 3 2 ) {\displaystyle TM={\frac {1}{2}}\left(Q_{2}+{\frac {Q_{1}+Q_{3}}{2}}\right)}

The foundations of the trimean were part of Arthur Bowley's teachings, and later popularized by statistician John Tukey in his 1977 book which has given its name to a set of techniques called exploratory data analysis. Like the median and the midhinge, but unlike the sample mean, it is a statistically resistant L-estimator with a breakdown point of 25%. This beneficial property has been described as follows:

An advantage of the trimean as a measure of the center (of a distribution) is that it combines the median's emphasis on center values with the midhinge's attention to the extremes.

Efficiency Despite its simplicity, the trimean is a remarkably efficient estimator of population mean. More precisely, for a large data set (over 100 points) from a symmetric population, the average of the 18th, 50th, and 82nd percentile is the most efficient 3-point L-estimator, with 88% efficiency. For context, the best single point estimate by L-estimators is the median, with an efficiency of 64% or better (for all n), while using two points (for a large data set of over 100 points from a symmetric population), the most efficient estimate is the 27% midsummary (mean of 27th and 73rd percentiles), which has an efficiency of about 81%. Using quartiles, these optimal estimators can be approximated by the midhinge and the trimean. Using further points yield higher efficiency, though it is notable that only three points are needed for very high efficiency.

See also Truncated mean Interquartile mean Decile mean Tukey median

References

External links Trimean at MathWorld

Worked examples

Example 1 — a first encounter with Trimean

Start with the simplest possible case. Write down what Trimean 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 Trimean 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 Trimean 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 Trimean

In research: Trimean 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 Trimean 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: Trimean is common in secondary-school and first-year university syllabi. It links to neighbouring topics Exploratory data analysis, Means, Robust statistics, so understanding it makes those chapters shorter.
In everyday life: Look for Trimean 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 Trimean in 20 minutes

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

Frequently asked questions

What is Trimean in simple terms?

In statistics the trimean (TM), or Tukey's trimean, is a measure of a probability distribution's central tendency defined as a weighted average of the distribution's quartiles: T M = Q 1 + 2 Q 2 + Q 3 4 {\displaystyle TM={\frac {Q_{1}+2Q_{2}+Q_{3}}{4}}} It gives twice as much weight to the median o…

Why does Trimean 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 Trimean?

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 Trimean.