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Summary Central Limit Theorem
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The Central Limit Theorem is an integrated theory in statistics principally charting the behaviour of the sum (average) of a large collection of independent identically distributed random variables. It is asserted that irrespective of the parent probability distribution of these variables, the pare...

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  • September 27, 2024
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  • 2024/2025
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CENTRAL LIMIT THEOREM




Submitted By: Abhishek Gautam
Statistics for ML Cohort 2
VNIT Nagpur

, Introduction

The Central Limit Theorem is an integrated theory in statistics principally charting the
behaviour of the sum (average) of a large collection of independent identically
distributed random variables. It is asserted that irrespective of the parent probability
distribution of these variables, the parent probability distribution of their sum will tend
towards normality with an increase in sample size. This rule, of convergence to
normality, would even apply acceptable given that the characteristics of the
population from which the sample is drawn is not normal if the sample is large
enough.

A Limit of Inference is one of the indispensable prerequisites in mathematics or
statistics. It gives way to making assumptions about specific population
characteristics on the basis of sample statistics within the inferential environment.
The CLT makes it possible for researchers to ignore the shape of original data and
successfully analyze data within a normal framework which includes hypothesis
testing and estimation of confidence intervals. As a result CLT is a basic requirement
to anybody working with any kind of data, hence it is a fundamental aspect for both
pure mathematics and its applied aspect too.


Theory

The Central Limit Theorem (CLT) is a fundamental concept in statistics that
describes how the distribution of sample means behaves as the sample size
increases. Specifically, it states that if you have a population with a finite mean 𝜇 and
a finite variance 𝜎 ! , the distribution of the sample means will approach a normal
distribution as the sample size 𝑛 becomes large, regardless of the shape of the
population distribution.

Mathematically, if 𝑋" , 𝑋! , … , 𝑋# are independent and identically distributed (i.i.d.)
random variables drawn from a population with mean 𝜇 and variance 𝜎 ! , the mean of
these samples can be expressed as:
#
¯ 1
𝑋 = ) 𝑋%
𝑛
%&"

¯
As 𝑛 approaches infinity, the distribution of the sample means 𝑋 will converge in
'
distribution to a normal distribution with mean 𝜇 and standard deviation :
√#

¯
𝑋−𝜇
𝑍= ∼ 𝑁(0,1)
𝜎/√𝑛

This theorem is significant for several reasons:

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