Central Limit Theorem

The central limit theorem states that if we consider numerous random samples from a population, the sample mean distribution will form a normal distribution, which is a bell curve. The distribution of the original population can be skewed, Poisson, or binomial, but the mean distribution of the sample will be normal when the sample size is greater than or equal to 30. So, if the sample size is larger, it allows for better estimation of population characteristics.

So far, we have discussed what the central limit theorem states. Let us now learn about the key components of the theorem. The key concepts are: 

• Sampling is successive: It means that some samples may repeat. For instance, in opinion polls, the population selected can be a part of different studies. 

• Sampling is random: The sample here must be selected randomly, as it gives equal chance to all the individuals in the population. For example, in a political poll, if we select the sample based on a specific group, the result can be biased. 

• Samples should be independent: Each sample should be selected individually; the selection of one should not influence the other. The sample should be independent to follow a true, normal pattern. 

• Large sample size: If the sample is large, the sampling distribution will be normal. According to the central limit theorem, an ideal sample size should be 30 or more. 

As we know, the central limit theorem is a fundamental concept in statistics and probability; it helps us understand how the population estimates the behavior under repeated sampling. Now we will discuss how it works and the formulas. 

• When the samples are repeated from the population, the mean will follow a normal distribution, regardless of the type of population distribution (uniform, skewed, or binomial).  
  
   
• The larger the sample size, the narrower the bell curve.
  
   
• As the sample size increases, the mean of the sample distribution approaches the population mean.

Now let’s look at the formula for the central limit theorem. Let X be a random variable with a known or unknown probability distribution. The standard deviation is σ, and the mean of X is μ. According to the central limit theorem, if the large number sample are drawn of size n, then the new random variable is x̄, then,

\(\bar{x} \sim N\!\left(\mu, \frac{\sigma}{\sqrt{n}}\right)\), where σ / √n is the standard error. The z score of the random variable x̄ is \( z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \)

The central limit theorem is used in the real world to analyze the population characteristics. Let’s discuss the real-world applications of the central limit theorem. 

 

• In economics and data science, we use central limit theorem to understand the statistical model and to draw conclusion.
  
  
   
• To understand the public opinion, we use the central limit theorem to analyze and understand the polls and surveys.