what is the central limit theorem clt 
in probability theory the central limit theorem clt states that the distribution of a sample variable approximates a normal distribution i e a bell curve as the sample size becomes larger regardless of the population s actual distribution shape put another way clt is a statistical premise that given a sufficiently large sample size from a population with a finite level of variance the mean of all sampled variables from the same population will be approximately equal to the mean of the whole population furthermore these samples approximate a normal distribution with their variances being approximately equal to the variance of the population as the sample size gets larger according to the law of large numbers although this concept was first developed by abraham de moivre in 1733 it was not formalized until 1920 when noted hungarian mathematician george p lya dubbed it the central limit theorem 12investopedia jiaqi zhouunderstanding the central limit theorem clt according to the central limit theorem the mean of a sample of data will be closer to the mean of the overall population in question as the sample size increases notwithstanding the actual distribution of the data in other words the data is accurate whether the distribution is normal or aberrant as a general rule sample sizes of 30 are typically deemed sufficient for the clt to hold meaning that the distribution of the sample means is fairly normally distributed therefore the more samples one takes the more the graphed results take the shape of a normal distribution 3the central limit theorem is often used in conjunction with the law of large numbers which states that the average of the sample means will come closer to equaling the population mean as the sample size grows which is extremely useful in accurately predicting the characteristics of populations investopedia sabrina jiangkey components of the central limit theoremthe central limit theorem is comprised of several key characteristics these characteristics largely revolve around samples sample sizes and the population of data the central limit theorem in financethe clt is useful when examining the returns of an individual stock or broader indices because the analysis is simple due to the relative ease of generating the necessary financial data consequently investors of all types rely on the clt to analyze stock returns construct portfolios and manage risk say for example an investor wishes to analyze the overall return for a stock index that comprises 1 000 equities in this scenario that investor may simply study a random sample of stocks to cultivate estimated returns of the total index to be safe at least 30 50 randomly selected stocks across various sectors should be sampled for the central limit theorem to hold furthermore previously selected stocks must be swapped out with different names to help eliminate bias 