# 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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### Bayesian and frequentist connections regarding the central limit theorem

I have been wondering how the central limit theorem may be useful in Bayesian statistics with potentially misspecified model distribution. Suppose $x$ is a random variable that follows an unknown (and ...
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### How do we know the distribution of regression coefficients

I'm reading up on asymptotics and hypothesis testing and was thinking about how they link together with regression coefficients. I have read that the CLT shows that the standardised sample mean ...
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### Sample size in Sample Proportions

I am in high school learning about sample proportions and they say that $n$ is the sample size. The example they gave is you spin a spinner board where the chance of landing on a 1 in 0.6, and 0 is 0....
32 views

### In a skewed sample with a large n, does Central Limit Theorem dictate that a t-test can be used, even if the mean cannot be interpreted? [duplicate]

I understand that, in the case of a highly skewed population and sample, the sampling distribution of the mean can still be normally distributed if the sample size is large, according to Central Limit ...
41 views

### Testing the Anderson–Darling and Central Limit Theorem

I was trying to simulate the Central Limit Theorem in R. Unfortunately, even in large samples (e.g., 80), the Anderson–Darling test could not recognize normality. Therefore, I wrote the following code ...
1 vote
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### Papers or documents about the central limit theorem and its possible extensions: what happen when the sample size is big? [closed]

The central limit theorem in its most popular form states that (without being too formal) for a set of random variables $X_1,X_2,...,X_n$ independent and identically distributed with mean $\mu$ and ...
26 views

### Distribution of the mean of samples taken from two different distributions

Consider you have some distributions $Z_1$ and $Z_2$ of which you select $n_1$ samples from $Z_1$ and $n_2$ samples from $Z_2$. We now add up these samples and take the mean. The question becomes, ...
27 views

### Are all examples of normal distribution in nature only a consequence of CLT? [duplicate]

I have read that many bell curves we see in nature is just a consequence of the CLT, because those things are just the result of many small additive causes (e.g. human height). Then my question is: ...
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### CTL vs normal distribition of residuals

Hello, I submitted a study with a linear regression model. My sample size is >1000, so I invoked CTL. The statistics reviewer is requesting that we check assumptions of normal distribution (...
316 views

### What is the probability space that the CLT is really being applied to?

Can someone please walk me through (or cite a reference to) exactly which fixed probability space is used in an application of the CLT, especially in the finite probability space case? My question ...
1 vote
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### What is the rigorous justification for applying LLN or CLT to finite probability spaces?

Both CLT and LLN are stated in terms of a fixed probability space that admits an infinite sequence of IID RVs. It is a common-place in many probability and statistics texts/notes that such a sequence ...
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### How to prove that if the bootstrapped sample means are approximately normal, then the sample mean is approximately normal?

I'm considering the scenario described in this question, namely: we have IID $X_i, i = 1, ..., n$ sampled from a population with mean $\mu$, variance $\sigma^2$ unknown the goal is to put a ...
1 vote
37 views

### Why does the normal distribution show up everywhere? [closed]

I know that the common explanation for this is the Central Limit Theorem being applicable to most probability distributions. However, I can't seem to apply the Central Limit Theorem to simple examples ...
1 vote
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### Could the numerical value of mean square error (or root mean square error) tell us something about the rate of convergence?

Suppose I have an estimator $\widehat{\theta}$ for $\theta_0$ that is root-n consistent and asymptotically normal. In the monte carlo simulations of many papers that I've read, consistency is usually ...
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### Does the Central Limit Theorem Apply to All Finite Samples Even If They Come From Distributions That Don't Have a Finite Variance?

Some distributions, like the Cauchy distribution, don't have a finite variance, and therefore the central limit theorem does not apply to them. If I have a thousand randomly selected observations from ...
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### Correctness of derivation for binary F1 variance for F1 confidence intervals

I'm developing a python library for confidence intervals for common accuracy metrics, with both analytic and bootstrap computations. Following this paper, I implemented the Macro and Micro F1 scores ...
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### Techniques/diagnostics for gaining confidence in normality assumptions and resulting confidence intervals

I have data that is reasonably assumed to be iid samples from some distribution. Our goal is to put a confidence interval on the population mean and have something similar for the population variance. ...
465 views

### Why don't we use normal distribution in every problem? [closed]

I was reading about normal distributions and the Central Limit Theorem (CLT) and I came up with a question. Why do we bother ourselves to use machine learning techniques when the CLT gives us the ...
40 views

### Interpreting the Concept of 'Single Sample Normality' in the Context of the Central Limit Theorem

In the context of the Central Limit Theorem (CLT), which postulates that the distribution of sample means will approximate a normal distribution given a sufficiently large number of samples and sample ...
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### Monte Carlo simulations and Central Limit theorem

I am simulating the revenues of a portfolio of items using one input variable. This variable is randomly extracted from a normal distribution n times, where n is the number of Monte Carlo simulations. ...
1 vote
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### The basic requirements and CLT for one sample t-test

I am having trouble learning the relevant concepts of t-test. Even if you search the web or books, the explanation is slightly different, it causes confusion. (Please understand that I am not a ...
1 vote
39 views

### What should be the sample size in a stratified random sample to get the same precision of a simple random sample?

I was reading example 3.2 on stratified sampling in Sampling: Design and Analysis by Lohr where a stratified random sampling design is proposed and compared with a simple random sample one. The point ...
88 views

### Asymptotic normality in central limit theorem

I am a bit confused by Classical CLT section of the central limit theorem on Wikipedia. It basically says at the sample size gets larger, the difference between the sample mean and true mean ...
117 views

### Does asymptotic normality of the likelihood function follow from the central limit theorem?

In the book Mathematical Methods for Physics and Engineering it is said that the likelihood function tends to a Gaussian (centred on the maximum-likelihood estimate) in the large sample limit. The way ...
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### Approximating the Distribution of Xbar using the Central Limit Theorem

Is the following statement correct? The Central Limit Theorem (CLT) states that as the sample size tends to infinity, the standardized sample mean distribution approaches the standard normal ...
54 views

### Estimate how many wrong measures there is in a sample

This study shows that the average penis mean is 13.24cm and the standard deviation is 1.89cm. Let's suppose we have a population with this mean and standard deviation for penis length. Suppose we ask ...
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### Central limit theorem : relaxing assumption of all finite moments

Consider $S_n = \sum_{i = 1}^n b_{i,n} X_{i,n}$ where $X_{i,n}$ are random variable neither independent neither identically distribution and $b_{i,n}$ are weights satisfying the Lindeberg condition. I ...
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### T-test for paired differences between sample means

Suppose I have pre-treatment samples and post-treatment samples for a group of patients. Let the treatment effect on each patient be the difference between the two sample means. Now I have a new ...
57 views

### t-test on non normal data: type I/II error vs validity

First, I don't believe this is a duplicate post even though this topic has been brought up a million times. If it is, please point me to the relevant post and I will remove this one. I am basically ...
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### $MA(q)$ : Show $\sqrt{n}\hat{\rho}(q + l)\overset{d}{\to} N(0, 1 + 2\sum _{j=1}^q\rho ^2\left(j\right)), l\ge 1$

I am trying to show that for an $MA(q)$ process, $$\sqrt{n}\hat{\rho}(q + l)\overset{d}{\to} N(0, 1 + 2\sum _{j=1}^q\rho ^2\left(j\right)), \quad l\ge 1.$$ I'm having a hard time doing this. I'm not ...
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### Equivalence of Logistic regression to Gaussian naive bayes

I was revisiting the differences between logistic regression and Naive Bayes, and had a conceptual question. A logistic regression classifier makes intuitive sense to me as a classifier that directly ...
1 vote
110 views

### Distribution of a sample drawn from distinct Poisson distributions

I have a sample $X = \{\mathbf{x}_i\}$, where each $\mathbf{x}_i$ has a set of discrete features $x_{j}$ and a value $y_i$. I'm interested in the distribution of the sample mean, $\bar{\mathbf{y}}$ - ...
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### Why is the Central Limit applicable in A/B testing?

I am having trouble understanding why the Central Limit Theorem (CLT) is applicable in A/B testing. As a beginner in statistics, I am trying to grasp the intuition behind it. The CLT states that as we ...
85 views

### Central Limit Theorem and Normal Distribution Approximation [duplicate]

The central limit theorem states that: for identically distributed independent samples, the standardized sample mean tends towards the standard normal distribution even if the original variables ...
1 vote
81 views

### How to apply the central limit theorem on higher order moments?

If I have a set of $N$ independent samples from a probability distribution $P(X)$, $X_i\sim P(X)$, then I know from the central limit theorem (assuming the distribution is well behaved) that the ...
373 views

### Why do we need normality test if we already have CLT?

Why do we need normality test if the sample size is large enough and hence, the distribution of the sample mean is approximately normal based on central limit theorem?
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### Should you bootstrap when doing F-test on small sample

I want to conduct an F-test on a linear model with 5 groups each with around 20-25 samples. I know I could perform an anova() in R, but the residuals aren't quite normally distributed. If I perform a ...
1 vote
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### In practice, is there a $n$-value that we use to apply CLT?

For i.i.d. $X_i$ for $i \in \mathbb N$, denote $\bar X_n := \frac{1}{n}\sum_{i=1}^n X_i$. We know that $\frac{\sqrt{n}\bar X_n}{\sigma}$ converges (in distributional sense) to $\mathcal N(0,1)$, but ...
Let $\{X_n\}_n$ be a sequence of i.i.d. RV's, where each $X_n$ is a copy of RV $X: \Omega \rightarrow \mathbb R$. For $n > 0$, I can define $\bar X_n := \frac{1}{n}\sum_{i=1}^n X_i$. Then, central ...
Can you tell me if my understanding of the CLT is correct? Maybe it's just a matter of notation. The classical CLT states: Let $X_1,...,X_i,...,X_n$ be a sequence of iid random variables drawn from a ...