Half-normal distribution

Key takeaways

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

Reference excerpt

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution. Let X {\displaystyle X} follow an ordinary normal distribution, N ( 0 , σ 2 ) {\displaystyle N(0,\sigma ^{2})} . Then, Y = | X | {\displaystyle Y=|X|} follows a half-normal distribution. Thus, the half-normal distribution is a fold at the mean of an ordinary normal distribution with mean zero.

Properties Using the σ {\displaystyle \sigma } parametrization of the normal distribution, the probability density function (PDF) of the half-normal is given by

f Y ( y ; σ ) = 2 σ π exp ⁡ ( − y 2 2 σ 2 ) y ≥ 0 , {\displaystyle f_{Y}(y;\sigma )={\frac {\sqrt {2}}{\sigma {\sqrt {\pi }}}}\exp \left(-{\frac {y^{2}}{2\sigma ^{2}}}\right)\quad y\geq 0,}

where E [ Y ] = μ = σ 2 π {\displaystyle E[Y]=\mu ={\frac {\sigma {\sqrt {2}}}{\sqrt {\pi }}}} . Alternatively using a scaled precision (inverse of the variance) parametrization (to avoid issues if σ {\displaystyle \sigma } is near zero), obtained by setting θ = π σ 2 {\displaystyle \theta ={\frac {\sqrt {\pi }}{\sigma {\sqrt {2}}}}} , the probability density function is given by

f Y ( y ; θ ) = 2 θ π exp ⁡ ( − y 2 θ 2 π ) y ≥ 0 , {\displaystyle f_{Y}(y;\theta )={\frac {2\theta }{\pi }}\exp \left(-{\frac {y^{2}\theta ^{2}}{\pi }}\right)\quad y\geq 0,}

where E [ Y ] = μ = 1 θ {\displaystyle E[Y]=\mu ={\frac {1}{\theta }}} . The cumulative distribution function (CDF) is given by

F Y ( y ; σ ) = ∫ 0 y 1 σ 2 π exp ⁡ ( − x 2 2 σ 2 ) d x {\displaystyle F_{Y}(y;\sigma )=\int _{0}^{y}{\frac {1}{\sigma }}{\sqrt {\frac {2}{\pi }}}\,\exp \left(-{\frac {x^{2}}{2\sigma ^{2}}}\right)\,dx}

Using the change-of-variables z = x / ( 2 σ ) {\displaystyle z=x/({\sqrt {2}}\sigma )} , the CDF can be written as

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Half-normal distribution

Start with the simplest possible case. Write down what Half-normal distribution 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 Half-normal distribution 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 Half-normal distribution 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 Half-normal distribution

In research: Half-normal distribution 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 Half-normal distribution 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: Half-normal distribution is common in secondary-school and first-year university syllabi. It links to neighbouring topics Continuous distributions, Normal distribution, so understanding it makes those chapters shorter.
In everyday life: Look for Half-normal distribution 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 Half-normal distribution in 20 minutes

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

Frequently asked questions

What is Half-normal distribution in simple terms?

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution. Let X {\displaystyle X} follow an ordinary normal distribution, N ( 0 , σ 2 ) {\displaystyle N(0,\sigma ^{2})} .

Why does Half-normal distribution 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 Half-normal distribution?

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 Half-normal distribution.