﻿ Central limit theorem

# Central limit theorem

## Central limit theorem

A key idea in statistics, the central limit theorem, or CLT, is important in many industries, including banking. It offers a crucial perspective on the behaviour of sample means, making it an essential tool for making decisions and testing hypotheses. Understanding the central limit theorem is essential for analysts, researchers, and practitioners who work with data analysis since it forms the basis for many statistical techniques. Here, we look at CLT, its formula, essential elements, and useful examples.

## What is CLT?

The CLT states that, regardless of the shape of the available population distribution, the sampling distribution of the mean of a sufficiently large number of independent random samples from that population will be approximately normally distributed. This remarkable property holds regardless of whether the original population follows a normal distribution.

## Understanding CLT

Although the CLT may initially seem difficult to understand, its implications are crucial for understanding how samples behave and are distributed.

Consider a population with a mean and a standard deviation (σ) to understand the CLT. Consider that we randomly select (n) samples from this population, each with a sample size (n). According to the CLT, as the sample size (n) rises, the distribution of sample means will likely resemble a normal distribution with a standard deviation (σ/√n) that is centred on the population mean (μ).

The CLT is a powerful statistical tool, allowing analysts and researchers to conclude population parameters using sample means, even in the presence of non-normal populations.

By understanding the significance of sample size, independence, and its ability to accommodate diverse population distributions, one can harness the full potential of this theorem in making accurate and meaningful inferences from data. As a cornerstone of statistical theory, the CLT continues to underpin various research studies and practical applications across numerous domains, ensuring robust and reliable data-driven decision-making.

## Significance of sample size (n):

According to the theory, the distribution of sample means tends to follow a normal distribution as the sample size rises. As a result, more accurate estimations of the population mean may be obtained from samples of greater sizes.

Consider sampling a population as being analogous to repeatedly flipping a coin. The distribution of heads and tails tends to equalise when the number of coins reversed (50% heads, 50% seats). Similar to this, regardless of the population’s underlying distribution, the sample mean approaches the genuine population mean when the sample size is sufficiently big.

## Formula

The formula of the CLT is given below:

If X1, X2, …, Xn are independent and identically distributed random variables with a mean μ and a finite variance σ^2, then the sample mean (X̄) of these random variables has an approximately normal distribution with mean μ and standard deviation σ/√n, as n (the sample size) becomes large.

Mathematically, it can be written as:

X̄ ~ N(μ, σ^2/n)

Where:

X̄ = Sample mean

μ = Population mean

σ = Population standard deviation

n = Sample size

## Key components of CLT:

The CLT’s key components include

• Sample size (n):

As the sample size increases, the sampling distribution approaches normality. Larger sample sizes yield a better approximation of the normal distribution.

• Independence

The samples drawn must be independent, ensuring one sample does not influence another.

• Population distribution

The CLT does not require the population to be normally distributed. It holds for various population distributions.

## Examples:

Let’s have a look at an illustration of CLT. Consider a population with an average IQ of 100 and a standard deviation of 15. Let’s compute the means of several random samples from this population of size 30. With a mean of 100 (the same as the overall mean) and an average deviation of 15/√30, or around 2.74, these sample means will have a roughly normal distribution.

CLT is crucial in inferential statistics, where we often work with sample data rather than the entire population. It allows us to infer the population mean using the sample imply and apply parametric tests, like t-tests and Z-tests, to conclude the people.

CLT can be summarised in three essential rules:

• The sampling distribution of the sample means will be approximately normally distributed.
• The mean of the sample means will be equal to the population mean (μ).
• The population standard deviation (σ) divided by the square root of the sample size (n) will give the standard deviation of the sample means.

The central limit theorem is of paramount importance in statistics and data analysis. It provides a foundation for many statistical techniques, allowing us to use normal distribution properties even when dealing with non-normally distributed populations. This simplifies calculations and aids in making more accurate statistical inferences.

The CLT is the cornerstone for several statistical methods, including confidence intervals and hypothesis testing. Predictive modelling, machine learning, and quality control are just some areas where it has applications. The CLT offers a strong and broadly applicable framework for dealing with real-world data and making reliable statistical choices by permitting normality assumptions for sample means.

CLT is a statistical concept that states the distribution of sample means becomes approximately normal, regardless of the population distribution, as the sample size increases. Its conditions include having a sufficiently large sample size, independent samples, and the absence of any skewness or heavy tails in the population distribution.

CLT is a statistical principle stating that when we take many random samples from any population, the average values from these samples will be normally distributed, even if the original population itself does not follow a normal distribution.

## Category

### Read the Latest Market Journal

#### Weekly Updates 26/2/24 – 1/3/24

Published on Feb 28, 2024 47

This weekly update is designed to help you stay informed and relate economic and company...

#### All-in-One Guide to Investing in China via ETFs

Published on Feb 27, 2024 271

#### Navigating the Post-Inflation Landscape in 2024: Top 10 US Markets Key Events to Look out for

Published on Feb 23, 2024 300

#### From Boom to Bust: Lessons from the Barings Bank Collapse

Published on Feb 23, 2024 60

Barings Bank was one of the oldest merchant banks in England with a long history...

#### Decoding FX CFD 2.0

Published on Feb 20, 2024 66

#### Weekly Updates 19/2/24 – 23/2/24

Published on Feb 19, 2024 89

This weekly update is designed to help you stay informed and relate economic and company...

#### Unlock Prosperity with 5 Sure-Fire Financial Instruments!

Published on Feb 14, 2024 197

In Singapore, the concept of guaranteed returns may evoke the spirit of prosperity, reminiscent perhaps...

Published on Feb 13, 2024 70

This weekly update is designed to help you stay informed and relate economic and company...

POEMS 3 App

• Call Back

• Chat with us

Need Assistance? Share your Details and we’ll get back to you

IMPORTANT INFORMATION

This material is provided by Phillip Capital Management (S) Ltd (“PCM”) for general information only and does not constitute a recommendation, an offer to sell, or a solicitation of any offer to invest in any of the exchange-traded fund (“ETF”) or the unit trust (“Products”) mentioned herein. It does not have any regard to your specific investment objectives, financial situation and any of your particular needs. You should read the Prospectus and the accompanying Product Highlights Sheet (“PHS”) for key features, key risks and other important information of the Products and obtain advice from a financial adviser (“FA“) pursuant to a separate engagement before making a commitment to invest in the Products. In the event that you choose not to obtain advice from a FA, you should assess whether the Products are suitable for you before proceeding to invest. A copy of the Prospectus and PHS are available from PCM, any of its Participating Dealers (“PDs“) for the ETF, or any of its authorised distributors for the unit trust managed by PCM.

An ETF is not like a typical unit trust as the units of the ETF (the “Units“) are to be listed and traded like any share on the Singapore Exchange Securities Trading Limited (“SGX-ST”). Listing on the SGX-ST does not guarantee a liquid market for the Units which may be traded at prices above or below its NAV or may be suspended or delisted. Investors may buy or sell the Units on SGX-ST when it is listed. Investors cannot create or redeem Units directly with PCM and have no rights to request PCM to redeem or purchase their Units. Creation and redemption of Units are through PDs if investors are clients of the PDs, who have no obligation to agree to create or redeem Units on behalf of any investor and may impose terms and conditions in connection with such creation or redemption orders. Please refer to the Prospectus of the ETF for more details.

Investments are subject to investment risks including the possible loss of the principal amount invested. The purchase of a unit in a fund is not the same as placing your money on deposit with a bank or deposit-taking company. There is no guarantee as to the amount of capital invested or return received. The value of the units and the income accruing to the units may fall or rise. Past performance is not necessarily indicative of the future or likely performance of the Products. There can be no assurance that investment objectives will be achieved.

Where applicable, fund(s) may invest in financial derivatives and/or participate in securities lending and repurchase transactions for the purpose of hedging and/or efficient portfolio management, subject to the relevant regulatory requirements. PCM reserves the discretion to determine if currency exposure should be hedged actively, passively or not at all, in the best interest of the Products.

The regular dividend distributions, out of either income and/or capital, are not guaranteed and subject to PCM’s discretion. Past payout yields and payments do not represent future payout yields and payments. Such dividend distributions will reduce the available capital for reinvestment and may result in an immediate decrease in the net asset value (“NAV”) of the Products. Please refer to <www.phillipfunds.com> for more information in relation to the dividend distributions.

The information provided herein may be obtained or compiled from public and/or third party sources that PCM has no reason to believe are unreliable. Any opinion or view herein is an expression of belief of the individual author or the indicated source (as applicable) only. PCM makes no representation or warranty that such information is accurate, complete, verified or should be relied upon as such. The information does not constitute, and should not be used as a substitute for tax, legal or investment advice.

The information herein are not for any person in any jurisdiction or country where such distribution or availability for use would contravene any applicable law or regulation or would subject PCM to any registration or licensing requirement in such jurisdiction or country. The Products is not offered to U.S. Persons. PhillipCapital Group of Companies, including PCM, their affiliates and/or their officers, directors and/or employees may own or have positions in the Products. Any member of the PhillipCapital Group of Companies may have acted upon or used the information, analyses and opinions herein before they have been published.

This advertisement has not been reviewed by the Monetary Authority of Singapore.

Phillip Capital Management (S) Ltd (Co. Reg. No. 199905233W)
250 North Bridge Road #06-00, Raffles City Tower ,Singapore 179101
Tel: (65) 6230 8133 Fax: (65) 65383066 www.phillipfunds.com