Midhinge

Key takeaways

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

Reference excerpt

In statistics, the midhinge (MH) is the average of the first and third quartiles and is thus a measure of location. Equivalently, it is the 25% trimmed mid-range or 25% midsummary; it is an L-estimator. The midhinge MH is defined as MH ⁡ ( X ) = Q 1 , 3 ( X ) ¯ = Q 1 ( X ) + Q 3 ( X ) 2 = P 25 ( X ) + P 75 ( X ) 2 = M 25 ( X ) . {\displaystyle {\begin{aligned}\operatorname {MH} (X)&={\overline {Q_{1,3}(X)}}\\&={\frac {Q_{1}(X)+Q_{3}(X)}{2}}\\&={\frac {P_{25}(X)+P_{75}(X)}{2}}\\&=M_{25}(X).\end{aligned}}}

The midhinge is related to the interquartile range (IQR), the difference of the third and first quartiles (i.e. IQR = Q3 − Q1), which is a measure of statistical dispersion. The two are complementary in sense that if one knows the midhinge and the IQR, one can find the first and third quartiles. The use of the term hinge for the lower or upper quartiles derives from John Tukey's work on exploratory data analysis in the late 1970s, and midhinge is a fairly modern term dating from around that time. The midhinge is slightly simpler to calculate than the trimean (TM), which originated in the same context and equals the average of the median (~X = Q2 = P50) and the midhinge:

MH ⁡ ( X ) = 2 TM ⁡ ( X ) − med ⁡ ( X ) = 2 Q 1 + 2 Q 2 + Q 3 4 − Q 2 . {\displaystyle {\begin{aligned}\operatorname {MH} (X)&=2\operatorname {TM} (X)-\operatorname {med} (X)\\&=2\;{\frac {Q_{1}+2Q_{2}+Q_{3}}{4}}-Q_{2}.\end{aligned}}}

Worked examples

Example 1 — a first encounter with Midhinge

Start with the simplest possible case. Write down what Midhinge claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Midhinge 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 Midhinge 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 Midhinge

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

How to study Midhinge in 20 minutes

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

Frequently asked questions

What is Midhinge in simple terms?

In statistics, the midhinge (MH) is the average of the first and third quartiles and is thus a measure of location. Equivalently, it is the 25% trimmed mid-range or 25% midsummary; it is an L-estimator.

Why does Midhinge matter?

Because it connects several science 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 Midhinge?

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 Midhinge.