Questions tagged [central-limit-theorem]

For questions about the central limit theorem, which states: "Given certain conditions, the mean of a sufficiently large number of iterates of independent random variables, each with a well-defined mean and well-defined variance, will be approximately normally distributed." (Wikipedia)

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166
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8answers
36k views

What intuitive explanation is there for the central limit theorem?

In several different contexts we invoke the central limit theorem to justify whatever statistical method we want to adopt (e.g., approximate the binomial distribution by a normal distribution). I ...
96
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T-test for non normal when N>50?

Long ago I learnt that normal distribution was necessary to use a two sample T-test. Today a colleague told me that she learnt that for N>50 normal distribution was not necessary. Is that true? If ...
60
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5answers
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Central limit theorem for sample medians

If I calculate the median of a sufficiently large number of observations drawn from the same distribution, does the central limit theorem state that the distribution of medians will approximate a ...
51
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3answers
18k views

When combining p-values, why not just averaging?

I recently learned about Fisher's method to combine p-values. This is based on the fact that p-value under the null follows a uniform distribution, and that $$-2\sum_{i=1}^n{\log X_i} \sim \chi^2(2n), ...
49
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3answers
9k views

Consider the sum of $n$ uniform distributions on $[0,1]$, or $Z_n$. Why does the cusp in the PDF of $Z_n$ disappear for $n \geq 3$?

I've been wondering about this one for a while; I find it a little weird how abruptly it happens. Basically, why do we need just three uniforms for $Z_n$ to smooth out like it does? And why does the ...
45
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4answers
93k views

What references should be cited to support using 30 as a large enough sample size?

I have read/heard many times that the sample size of at least 30 units is considered as "large sample" (normality assumptions of means usually approximately holds due to the CLT, ...). Therefore, in ...
44
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4answers
10k views

Where does $\sqrt{n}$ come from in central limit theorem (CLT)?

A very simple version of central limited theorem as below $$ \sqrt{n}\bigg(\bigg(\frac{1}{n}\sum_{i=1}^n X_i\bigg) - \mu\bigg)\ \xrightarrow{d}\ \mathcal{N}(0,\;\sigma^2) $$ which is Lindeberg–Lévy ...
39
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6answers
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Debunking wrong CLT statement

The central limit theorem (CLT) gives some nice properties about converging to a normal distribution. Prior to studying statistics formally, I was under the extremely wrong impression that the CLT ...
39
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3answers
5k views

Why law of large numbers does not apply in the case of Apple share price?

Here is the article in NY times called "Apple confronts the law of large numbers". It tries to explain Apple share price rise using law of large numbers. What statistical (or mathematical) errors does ...
35
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7answers
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Are there any examples of where the central limit theorem does not hold?

Wikipedia says - In probability theory, the central limit theorem (CLT) establishes that, in most situations, when independent random variables are added, their properly normalized sum tends ...
33
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7answers
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How do you convey the beauty of the Central Limit Theorem to a non-statistician?

My father is a math enthusiast, but not interested in statistics much. It would be neat to try to illustrate some of the wonderful bits of statistics, and the CLT is a prime candidate. How would you ...
31
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9answers
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Expectation of 500 coin flips after 500 realizations

I was hoping someone could provide clarity surrounding the following scenario. You are asked "What is the expected number of observed heads and tails if you flip a fair coin 1000 times". Knowing that ...
30
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6answers
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Test for finite variance?

Is it possible to test for finiteness (or existence) of the variance of a random variable given a sample? As a null, either {the variance exists and is finite} or {the variance does not exist/is ...
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3answers
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Asymptotic distribution of sample variance of non-normal sample

This is a more general treatment of the issue posed by this question. After deriving the asymptotic distribution of the sample variance, we can apply the Delta method to arrive at the corresponding ...
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3answers
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What is the sum of squared t variates?

Let $t_i$ be drawn i.i.d from a Student t distribution with $n$ degrees of freedom, for moderately sized $n$ (say less than 100). Define $$T = \sum_{1\le i \le k} t_i^2$$ Is $T$ distributed nearly as ...
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4answers
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Why does Central Limit Theorem break down in my simulation?

Let say I have following numbers: 4,3,5,6,5,3,4,2,5,4,3,6,5 I sample some of them, say, 5 of them, and calculate the sum of 5 samples. Then I repeat that over ...
21
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1answer
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Central limit theorem and the law of large numbers

I have a very beginner's question regarding the Central Limit Theorem (CLT): I am aware that the CLT states that a mean of i.i.d. random variables is approximately normal distributed (for $n \to \...
20
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4answers
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Reasons for data to be normally distributed

What are some theorems which might explain (i.e., generatively) why real-world data might be expected to be normally distributed? There are two that I know of: The Central Limit Theorem (of course), ...
20
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2answers
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How to test for differences between two group means when the data is not normally distributed?

I'll eliminate all the biological details and experiments and quote just the problem at hand and what I have done statistically. I would like to know if its right, and if not, how to proceed. If the ...
20
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1answer
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Error in normal approximation to a uniform sum distribution

One naive method for approximating a normal distribution is to add together perhaps $100$ IID random variables uniformly distributed on $[0,1]$, then recenter and rescale, relying on the Central Limit ...
19
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3answers
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Example of distribution where large sample size is necessary for central limit theorem

Some books state a sample size of size 30 or higher is necessary for the central limit theorem to give a good approximation for $\bar{X}$. I know this isn't enough for all distributions. I wish ...
19
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4answers
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Why does increasing the sample size of coin flips not improve the normal curve approximation?

I'm reading the Statistics (Freeman, Pisani, Purves) book and I'm trying to reproduce an example where a coin is tossed say 50 times, the number of heads counted and this is repeated say 1,000 times. ...
17
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1answer
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Cauchy Distribution and Central Limit Theorem

In order for the CLT to hold we need the distribution we wish to approximate to have mean $\mu$ and finite variance $\sigma^2$. Would it be true to say that for the case of the Cauchy distribution, ...
16
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3answers
739 views

Why doesn't the CLT work for $x \sim poisson(\lambda = 1) $?

So we know that a sum of $n$ poissons with parameter $\lambda$ is itself a poisson with $n\lambda$. So hypothetically, one could take $x \sim poisson(\lambda = 1) $ and say it is actually $\sum_1^n ...
16
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3answers
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Central limit theorem versus law of large numbers

The central limit theorem states that the mean of i.i.d. variables, as $N$ goes to infinity, becomes normally distributed. This raises two questions: Can we deduce from this the law of large ...
16
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3answers
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A dynamical systems view of the Central Limit Theorem?

(Originally posted on MSE.) I have seen many heuristic discussions of the classical central limit theorem speak of the normal distribution (or any of the stable distributions) as an "attractor" in ...
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4answers
443 views

The operation of chance in a deterministic world

In Steven Pinker's book Better Angels of Our Nature, he notes that Probability is a matter of perspective. Viewed at sufficiently close range, individual events have determinate causes. Even a ...
14
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2answers
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Independent samples t-test: Do data really need to be normally distributed for large sample sizes?

Let's say I want to test if two independent samples have different means. I know the underlying distribution is not normal. If I understand correctly, my test statistic is the mean, and for large ...
14
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2answers
499 views

How can the number of connections be Gaussian if it cannot be negative?

I am analyzing social networks (not virtual) and I am observing the connections between people. If a person would choose another person to connect with randomly, the number of connections within a ...
13
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2answers
203 views

Why is pseudo-random sampling applicable for Monte Carlo integration, even though it does not satisfy the CLT requirements?

Assume we have a function $f\left(x\right)$ defined on $\left[0, 1\right]$ that we want to integrate and estimate the error using Monte Carlo method. We generate realizations of uniformly distributed ...
12
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2answers
238 views

How can we get a normal distribution as $n \to \infty$ if the range of values of our random variable is bounded?

Let's say we have a random variable with a range of values bounded by $a$ and $b$, where $a$ is the minimum value and $b$ the maximum value. I was told that as $n \to \infty$, where $n$ is our sample ...
12
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1answer
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Is there a theorem that says that $\sqrt{n}\frac{\bar{X} - \mu}{S}$ converges in distribution to a normal as $n$ goes to infinity?

Let $X$ be any distribution with defined mean, $\mu$, and standard deviation, $\sigma$. The central limit theorem says that $$ \sqrt{n}\frac{\bar{X} - \mu}{\sigma} $$ converges in distribution to a ...
12
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1answer
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Why does the central limit theorem work with a single sample?

I have always been taught that the CLT works when you have repeated sampling, with each sample being large enough. For example, imagine I have a country of 1,000,000 citizens. My understanding of CLT ...
12
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2answers
599 views

Central Limit Theorem - Rule of thumb for repeated sampling

My question was inspired by this post which concerns some of the myths and misunderstandings surrounding the Central Limit Theorem. I was asked a question by a colleague once and I couldn't offer an ...
12
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2answers
706 views

Are there any distributions other than Cauchy for which the arithmetic mean of a sample follows the same distribution?

If $X$ follows a Cauchy distribution then $Y = \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$ also follows exactly the same distribution as $X$; see this thread. Does this property have a name? Are there ...
12
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1answer
740 views

Assessments of “Approximately Normal” for t-tests

I am testing equality of means using Welch's t-test. The underlying distribution is far from normal (more skewed than the example in a related discussion here). I can obtain more data but would like ...
11
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2answers
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How can the central limit theorem hold for distributions which have limits on the random variable?

I've always taken issue with, and never been given a good answer, for how it is possible that the central limit theorem - the classical version where the distribution of sample means approaches ...
11
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3answers
419 views

Yet another central limit theorem question

Let $\{X_n:n\ge1\}$ be a sequence of independent Bernoulli random variables with $$P\{X_k=1\}=1-P\{X_k=0\}=\frac{1}{k}.$$ Set $$S_n=\sum^{n}_{k=1}\left(X_k-\frac{1}{k}\right), \ B_n^2=\sum^{n}_{...
11
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1answer
3k views

Central Limit Theorem for square roots of sums of i.i.d. random variables

Intrigued by a question at math.stackexchange, and investigating it empirically, I am wondering about the following statement on the square-root of sums of i.i.d. random variables. Suppose $X_1, X_2, ...
11
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2answers
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Central limit theorem proof not using characteristic functions

Is there any proof for the CLT not using characteristic functions, a simpler method? Maybe Tikhomirov or Stein's methods? Something self-contained you can explain to a university student (first year ...
11
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2answers
939 views

Are there any examples of a variable being normally distributed that is *not* due to the Central Limit Theorem?

The normal distribution seems unintuitive until you learn the CLT, which explains why it is so prevalent in real life. But does it ever arise as the "natural" distribution for some quantity?
11
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3answers
865 views

Information theoretic central limit theorem

The simplest form of the information theoretic CLT is the following: Let $X_1, X_2,\dots$ be iid with mean $0$ and variance $1$. Let $f_n$ be the density of the normalized sum $\frac{\sum_{i=1}^n X_i}...
11
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1answer
391 views

Is MLE of $\theta$ asymptotically normal when $(X,Y)\sim e^{-(x/\theta+\theta y)}\mathbf1_{x,y>0}$?

Suppose $(X,Y)$ has the pdf $$f_{\theta}(x,y)=e^{-(x/\theta+\theta y)}\mathbf1_{x>0,y>0}\quad,\,\theta>0$$ Density of the sample $(\mathbf X,\mathbf Y)=(X_i,Y_i)_{1\le i\le n}$ drawn from ...
11
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3answers
277 views

How many of the biggest terms in $\sum_{i=1}^N |X_i|$ add up to half the total?

Consider $\sum_{i=1}^N |X_i|$ where $X_1, \ldots, X_N$ are i.i.d. and the CLT holds. How many of the biggest terms add up to half the total sum ? For example, 10 + 9 + 8 $\approx$ (10 + 9 + 8 $\dots$ +...
11
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1answer
422 views

Asymptotic normality of a quadratic form

Let $\mathbf{x}$ be a random vector drawn from $P$. Consider a sample $\{ \mathbf{x}_i \}_{i=1}^n \stackrel{i.i.d.}{\sim} P$. Define $\bar{\mathbf{x}}_n := \frac{1}{n} \sum_{i=1}^n \mathbf{x}_i$, and $...
10
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3answers
2k views

Does this code demonstrate the central limit theorem?

Does this code demonstrate the central limit theorem? This is not a homework assignment! Au contraire, I'm a faculty teaching some methods to non-stats students. ...
10
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3answers
2k views

In CLT, why $\sqrt{n}\frac{\bar{X}_n-\mu}{\sigma} \rightarrow N(0,1)$ $\implies$ $\bar{X}_n \sim N(\mu, \frac{\sigma^2}{n})$?

Let $X_1,...,X_n$ be independent observations from a distribution that has the mean $\mu$ and variance $\sigma^2 < \infty$, when $n \rightarrow \infty$, then $$\sqrt{n}\frac{\bar{X}_n-\mu}{\sigma} ...
10
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3answers
5k views

Central limit theorem and the Pareto distribution

Can somebody please provide a simple (lay person) explanation of the relationship between Pareto distributions and the Central Limit Theorem (e.g. does it apply? Why/ why not?)? I am trying to ...
10
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2answers
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Central Limit Theorem for Markov Chains

$\newcommand{\E}{\mathbb{E}}$$\newcommand{\P}{\mathbb{P}}$The Central Limit Theorem (CLT) states that for $X_1,X_2,\dots$ independent and identically distributed (iid) with $\E[X_i]=0$ and $\...
10
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2answers
6k views

Expectation of square root of sum of independent squared uniform random variables

Let $X_1,\dots,X_n \sim U(0,1)$ be independent and identicallly distributed standard uniform random variables. $$\text{Let }\quad Y_n=\sum_i^nX_i^2 \quad \quad \text{I seek: } \quad \mathbb{E}\big[\...

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