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Central Limit Theorem states that distribution of sample averages will tend towards a normal distribution as the sample size increases or in other words we can say that irrespective of shape of distribution of population , the distribution of average values of sample drawn from that population will tend toward a normal distribution as the sample size grows. Because of the CLT we can use average of small samples to evaluate any population using the normal distribution. We can see practical application during election . Any time when we see polling results on the news along with confidence interval, it gives an appeal to the central limit theorem.it tells us that larger the sample, better the approximation and this we can see from news channel to channel that sample sizes are different and results changes accordingly. from this we can guess how an election will turn out. We take a poll and find out that in our sample how much % of people would like to vote a candidate over another candidate. We have taken a small sample over a large population but as per Central Limit Theorem if we ran poll over and over again, the resulting guesses would be normally distributed or in other words we can say that we will have a clear picture around the large population and can guess about the winning candidate. So if we take large sample size and repeat again and again we will have a clear idea about the large population.

Benchmark Six Sigma Expert View by Venugopal R Many of the tools used in Six Sigma project, where samples are used for analysis and decision making, apply the principle of Central Limit Theorem (CLT). As per the CLT, Sample means tend to follow normal distribution, irrespective to the population distribution, and hence the properties of Normal distribution apply for the sample means. The normality gets better with higher sample size. In today’s world with so many user-friendly statistical software, the analysis and even the choice of the tools to be applied, (for instance the type of test of hypothesis to be used for comparative analysis) could be left to the software. Hence the practical application of CLT would be happening inadvertently while using these tools. Control charts that use mean value of subgroups have their limits and rules based on the CLT. The significance tests where mean values of samples are compared, have the acceptance conditions based on CLT. If these tools had been used as part of the Six Sigma projects, the CLT has been put to use as part of the inbuilt working of these statistical softwares.