Illustration of the central limit theorem

Key takeaways

Illustration of the central limit theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
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Reproduce the core statement of Illustration of the central limit theorem from memory before moving on to harder problems.

Reference excerpt

In probability theory, the central limit theorem (CLT) states that, in many situations, when independent and identically distributed random variables are added, their properly normalized sum tends toward a normal distribution. This article gives two illustrations of this theorem. Both involve the sum of independent and identically-distributed random variables and show how the probability distribution of the sum approaches the normal distribution as the number of terms in the sum increases. The first illustration involves a continuous probability distribution, for which the random variables have a probability density function. The second illustration, for which most of the computation can be done by hand, involves a discrete probability distribution, which is characterized by a probability mass function.

Illustration of the continuous case The density of the sum of two independent real-valued random variables equals the convolution of the density functions of the original variables. Thus, the density of the sum of m+n terms of a sequence of independent identically distributed variables equals the convolution of the densities of the sums of m terms and of n term. In particular, the density of the sum of n+1 terms equals the convolution of the density of the sum of n terms with the original density (the "sum" of 1 term). A probability density function is shown in the first figure below. Then the densities of the sums of two, three, and four independent identically distributed variables, each having the original density, are shown in the following figures. If the original density is a piecewise polynomial, as it is in the example, then so are the sum densities, of increasingly higher degree. Although the original density is far from normal, the density of the sum of just a few variables with that density is much smoother and has some of the qualitative features of the normal density. The convolutions were computed via the discrete Fourier transform. A list of values y = f(x0 + k Δx) was constructed, where f is the original density function, and Δx is approximately equal to 0.002, and k is equal to 0 through 1000. The discrete Fourier transform Y of y was computed. Then the convolution of f with itself is proportional to the inverse discrete Fourier transform of the pointwise product of Y with itself.

Original probability density function We start with a probability density function. This function, although discontinuous, is far from the most pathological example that could be created. It is a piecewise polynomial, with pieces of degrees 0 and 1. The mean of this distribution is 0 and its standard deviation is 1.

Probability density function of the sum of two terms Next we compute the density of the sum of two independent variables, each having the above density. The density of the sum is the convolution of the above density with itself. The sum of two variables has mean 0. The density shown in the figure at right has been rescaled by 2 {\displaystyle {\sqrt {2}}} , so that its standard deviation is 1. This density is already smoother than the original. There are obvious lumps, which correspond to the intervals on which the original density was defined.

Probability density function of the sum of three terms We then compute the density of the sum of three independent variables, each having the above density. The density of the sum is the convolution of the first density with the second. The sum of three variables has mean 0. The density shown in the figure at right has been rescaled by √3, so that its standard deviation is 1. This density is even smoother than the preceding one. The lumps can hardly be detected in this figure.

Probability density function of the sum of four terms Finally, we compute the density of the sum of four independent variables, each having the above density. The density of the sum is the convolution of the first density with the third (or the second density with itself). The sum of four variables has mean 0. The density shown in the figure at right has been rescaled by √4, so that its standard deviation is 1. This density appears qualitatively very similar to a normal density. No lumps can be distinguished by the eye.

Illustration of the discrete case This section illustrates the central limit theorem via an example for which the computation can be done quickly by hand on paper, unlike the more computing-intensive example of the previous section.

Original probability mass function Suppose the probability distribution of a discrete random variable X puts equal weights on 1, 2, and 3:

X = { 1 with probability 1 / 3 , 2 with probability 1 / 3 , 3 with probability 1 / 3. {\displaystyle X=\left\{{\begin{matrix}1&{\mbox{with}}\ {\mbox{probability}}\ 1/3,\\2&{\mbox{with}}\ {\mbox{probability}}\ 1/3,\\3&{\mbox{with}}\ {\mbox{probability}}\ 1/3.\end{matrix}}\right.}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Illustration of the central limit theorem

Start with the simplest possible case. Write down what Illustration of the central limit theorem 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 Illustration of the central limit theorem 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 Illustration of the central limit theorem 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 Illustration of the central limit theorem

In research: Illustration of the central limit theorem 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 Illustration of the central limit theorem 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: Illustration of the central limit theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Central limit theorem, so understanding it makes those chapters shorter.
In everyday life: Look for Illustration of the central limit theorem 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 Illustration of the central limit theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Illustration of the central limit theorem 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.
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Frequently asked questions

What is Illustration of the central limit theorem in simple terms?

In probability theory, the central limit theorem (CLT) states that, in many situations, when independent and identically distributed random variables are added, their properly normalized sum tends toward a normal distribution. This article gives two illustrations of this theorem.

Why does Illustration of the central limit theorem 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 Illustration of the central limit theorem?

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 Illustration of the central limit theorem.