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# Central Limit Theorem

Central Limit Theorem

The Central Limit Theorem states that “as the sample size gets larger, the sampling distribution of the sample mean approaches normal distributing, regardless of the shape of the population distribution” In other words, as you take larger and larger sample size, the shape of the graph of the sample mean will look more like that of the normal distribution. This theorem could be easily explained by the following graphs.

The graphs above shows a simple test of ‘rolling a fair die’. As the number of times of rolling a die increases, the shape of the distribution of the means tends to appear more like that of the normal distribution graph.

Importance of Central Limit Theorem

There two major applications of central limit theorem-statistical application and practical application. In statistical application, it is applied in data analysis through the statistical method of hypothesis testing and construction of confidence intervals. The central limit theorem also helps in estimation of the population mean from the concept that as the sample size drawn from population is increased, the standard deviation of the sample mean will decrease. Further, the sample mean could be used to create a range of values referred to as the confidence intervals.

The practical application includes in political or election polls. It helps in the calculation of the confidence intervals associated with the number of people supporting a particular candidate. Additionally, the confidence interval, which is a concept of the central limit theorem, is applied in determining the mean family income of a certain region. In finance, it is applied in the examination of the stock or index returns. Since it is easy to generate large financial data, a sample size of 50 observations is considered significant.

Assumptions of Central Limit Theorem

This technique is applied under several assumptions as discussed below:

1. The data used should be obtained randomly – follow a randomized condition
2. The samples used should be independent of each other.
3. When the sampling is carried out without replacement, the sample size obtained should not exceed 10% of the population
4. The sample size should be significantly large, in relation to the population

Generally, if the population considered is symmetric, then the sample size of 30 is considered appropriate. The key aspects associated with the central limit theorem is that the average of the sample mean and standard deviation are equal to the means and standards deviations of the population. Additionally, if the sample size is sufficiently large, it can accurately predict the population characteristics.