Central Limit Theorem

Central Limit Theorem

The central limit theorem is a key statistical concept that helps describe how a sample’s mean behaves when samples are repeatedly drawn from a population. This is usually regardless of how the original population distribution is shaped. The theorem is important in inferential statistics, especially for large sample sizes. The central limit theorem asserts that as the size of the sample is increased, the mean of the samples will have a distribution that closely resembles a normal distribution, despite how the population distribution is shaped (Dmitry Dolgopyat et al., 2022). This remains valid regardless of whether the population distribution deviates from normality or if its shape is uncertain.

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Key Properties

The first essential characteristic or property of a sampling distribution is normality. With this property, an increase in the sample results means that the sample will also become normally distributed. However, if multiple samples are extracted from a specific population along with their means, sample means’ distribution will result in a bell-shaped normal distribution. The sampling distribution’s mean and standard deviation are important properties (Gupta et al., 2019). In this case, the population mean and the sampling distribution’s mean will be the same. Consequently, it is essential to note that the standard deviation of the sampling distribution of the mean is usually determined by dividing the population standard deviation by the sample size’s square root (Gupta et al., 2019). The formula for standard error, which is key in sampling distribution, is usually calculated through the division of the standard deviation by the sample size’s square root.

Minimum Sample Size Requirement

The central limit theorem applies to all sample sizes. However, when there is a smaller sample size, the approximation of the sampling distribution to a normal distribution could be less accurate if the original population distribution is highly skewed. Using a sample size of at least 30 is often recommended if one is looking to achieve a normal sampling distribution’s mean. However, there are some cases where the sample size is small and where the original population distribution’s shape can impact the sampling distribution’s mean. A normal population distribution usually suggests a normal sampling distribution of the mean, even when with a small sample size, in cases where the population distribution is highly skewed or contains several outliers, a larger sample size is needed to ensure that the sampling distribution of the mean becomes more regular.

References

Dmitry Dolgopyat, Dong, C., Kanigowski, A., & Péter Nándori. (2022). Flexibility of statistical properties for smooth systems satisfying the central limit theorem. 230(1), 31–120. https://doi.org/10.1007/s00222-022-01121-0

Gupta, A., Mishra, P., Pandey, C., Singh, U., Sahu, C., & Keshri, A. (2019). Descriptive Statistics and Normality Tests for Statistical Data. Annals of Cardiac Anaesthesia, 22(1), 67. https://doi.org/10.4103%

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Question: Central Limit Theorem

Click the ‘scenario’ button below to review the topic and then answer the following question:

Explain the central limit theorem. Describe the two properties of the sampling distribution of the mean. Include the minimum sample size requirement and how it relates to the shape of the original population distribution.

Central Limit Theorem

Scenario: Inferential Stats
The recent pandemic greatly affected the working environment. According to a survey, post-pandemic employees are more than ever demanding changes to policies and benefits in the workplace.

Your organization, XYZ Company, notes several hurdles to attracting and retaining staff in the post-pandemic climate. One example is the employer’s inability to adjust to the remote work arrangement, and another is a lack of pay equity commitment.

In keeping with XYZ Company’s goal to treat employees well, management is looking for feedback to measure the company’s performance as the employer of choice.

You work as the Manager of Workforce Analytics within the Human Resources department. The company is growing rapidly, and your boss, Jane, who is the Chief People Officer (CPO), wants to make sure that the company is treating its employees equitably. She is analysis-driven, so she taps you to work on several projects to uncover any potential issues. Her objectives are to:

Describe the current state of salary data using the measures of central tendency and variability.
Give a point estimate and construct a confidence interval of the number of employees who want to work remotely.
Conduct a hypothesis test from two populations, male employees and female employees, of a claim that they have the same mean salary.
Test a claim that employee pay is on par with industry standards with a hypothesis test from one population.
Apply a normal distribution to review the current dental insurance plan expenses.
Identify any correlation between seniority and pay
Conduct a regression analysis between employee age and salary
Let’s get started and gather insights into the workforce data.