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. 

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