Central limit Theorem | Management Optional Notes for UPSC PDF Download

Introduction

• The Central Limit Theorem holds significant importance within inferential statistics. This theorem, developed by Abraham de Moivre, a French mathematician, utilizes the normal distribution to approximate the distribution of outcomes, such as the number of heads resulting from multiple tosses of a fair coin. It elucidates the relationship between the sampling distribution of sample means and the population from which the samples are derived.
• The Central Limit Theorem elucidates the characteristics of the "population of means," derived from an infinite number of random population samples of size 'N' drawn from a specific "parent population." Regardless of the distribution of the parent population:
  • The mean of the population of means always equals the mean of the parent population.
  • The standard deviation of the population of means equals the standard deviation of the parent population divided by the square root of the sample size ('N').
  • The distribution of means progressively approaches a normal distribution as the sample size ('N') increases.
• When random samples of varying sizes are taken from a population, the distribution of sample means tends towards a normal distribution. If the sample size of the sampling distribution of the mean is 30 or more, it is considered to be normally distributed.
• The central limit theorem suggests that the sampling distribution of any statistic will be normal or nearly normal if the sample size is sufficiently large. Typically, a sample size is deemed "large enough" under the following conditions:
  • The population distribution is normal.
  • The population distribution is approximately symmetric, unimodal, without outliers, and the sample size is 15 or less.
  • The population distribution is moderately skewed, unimodal, without outliers, and the sample size ranges between 16 and 40.
  • The sample size exceeds 40 and is devoid of outliers.

Central Limit Theorem Description

• The Central Limit Theorem (CLT) states that when random samples of 'n' observations are drawn from any population with a finite mean 'µ' and standard deviation 'σ', then, as 'n' becomes large, the sampling distribution of the mean (X¯) approximates a normal distribution with a mean of 'µ' and a standard deviation of 'σ/√n'. According to this theorem, the mean of a sampling distribution of means serves as an unbiased estimator of the population mean (Naval, 2010).
  µx = µ
• Similarly, the standard deviation of a sampling distribution of means is given by the formula:
  σx = σ/√n
• The central limit theorem has various versions. In its typical form, the random variables must be identically distributed. However, in alternate versions, convergence to the normal distribution occurs for non-identical distributions or non-independent observations, provided they meet specific conditions (Hoffman, 2001).
• The central limit theorem pertains to the sampling distribution of sample means. Researchers often inquire about the overall shape of this sampling distribution. As per the central limit theorem, this distribution is approximately normal, commonly known as a bell curve. This approximation improves as researchers increase the size of the simple random samples used to construct the sampling distribution.

Main Aspect of the Central Limit Theorem

• A fundamental aspect of the Central Limit Theorem (CLT) is its ability to produce a normal distribution regardless of the original distribution. Even when the population exhibits a skewed distribution, such as in cases involving variables like incomes or individuals' weights, a sampling distribution derived from a sufficiently large sample size will tend towards normality (Fischer, 2010). 
• It is evident that the CLT elucidates the rationale behind many non-normal distributions gradually converging towards a normal distribution as the sample size 'N' increases. This encompasses uniform, triangular, inverse, and even parabolic distributions.

Significance of the Central Limit Theorem

• The significance of the Central Limit Theorem within statistical philosophy cannot be overstated. The majority of hypothesis testing and sampling theories are grounded in this theorem. Moreover, it offers an explanation for utilizing the normal curve as a model for numerous naturally occurring phenomena. 
• For instance, if a characteristic, such as intelligence, can be viewed as a blend of relatively independent events—both genetic and environmental—then one would anticipate the trait to exhibit a normal distribution within a population.

Conclusion

The Central Limit Theorem asserts that even when a population distribution is notably non-normal, the sampling distribution of means will approximate normality for large sample sizes (typically over 30). This theorem facilitates the utilization of probabilities associated with the normal curve to address queries pertaining to the means of sufficiently large samples. Widely regarded as one of the most pivotal results in probability theory, the theorem essentially posits that the sum of a considerable number of independent observations from the same distribution will, under certain general conditions, possess an approximate normal distribution. Furthermore, this approximation improves progressively with an increasing number of observations. Thus, the Central Limit Theorem stands as a cornerstone of probability theory.