Quantile normalization

Key takeaways

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

Reference excerpt

In statistics, quantile normalization is a technique for making two distributions identical in statistical properties. To quantile-normalize a test distribution to a reference distribution of the same length, sort the test distribution and sort the reference distribution. The highest entry in the test distribution then takes the value of the highest entry in the reference distribution, the next highest entry in the reference distribution, and so on, until the test distribution is a perturbation of the reference distribution. To quantile normalize two or more distributions to each other, without a reference distribution, sort as before, then set to the average (usually, arithmetic mean) of the distributions. So the highest value in all cases becomes the mean of the highest values, the second highest value becomes the mean of the second highest values, and so on. Generally a reference distribution will be one of the standard statistical distributions such as the Gaussian distribution or the Poisson distribution. The reference distribution can be generated randomly or from taking regular samples from the cumulative distribution function of the distribution. However, any reference distribution can be used. Quantile normalization is frequently used in microarray data analysis. It was introduced as quantile standardization and then renamed as quantile normalization.

Example A quick illustration of such normalizing on a very small dataset, organized into columns (1-3) and rows (A-D):

A : B : C : D : 1 _ 2 _ 3 _ 5 4 3 2 1 4 3 4 6 4 2 8 {\displaystyle {\begin{matrix}&{}\\[6pt]&A:\\&B:\\&C:\\&D:\end{matrix}}\quad {\begin{matrix}{\underline {1}}&{\underline {2}}&{\underline {3}}\\[6pt]5&4&3\\2&1&4\\3&4&6\\4&2&8\end{matrix}}}

For each column, rank the entries from lowest to highest (i to iv):

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Quantile normalization

Start with the simplest possible case. Write down what Quantile normalization 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 Quantile normalization 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 Quantile normalization 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 Quantile normalization

In research: Quantile normalization 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 Quantile normalization 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: Quantile normalization is common in secondary-school and first-year university syllabi. It links to neighbouring topics Equivalence (mathematics), Statistical data transformation, so understanding it makes those chapters shorter.
In everyday life: Look for Quantile normalization 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 Quantile normalization in 20 minutes

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

Frequently asked questions

What is Quantile normalization in simple terms?

In statistics, quantile normalization is a technique for making two distributions identical in statistical properties. To quantile-normalize a test distribution to a reference distribution of the same length, sort the test distribution and sort the reference distribution.

Why does Quantile normalization 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 Quantile normalization?

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 Quantile normalization.