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 that 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.

As we have discussed, the central limit theorem.  Let us now learn about the key components of the central limit 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 a chance to all the individuals in the population. For example, in political poll if we select the sample based on 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:
  The larger the sample, the sampling distribution should be normal. According to the central limit theorem, the perfect sample size should be 30 or more. 

As we know, the central limit theorem is a fundamental concept in statistics and probability, as it helps to 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. Even if the population distribution is uniform, skewed or binomial.  
  
   
• The larger the sample size, the narrower the bell curve
  
   
• The larger the sample size, the distribution mean is equal to 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. 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 application 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.