Central Limit Theorem

Central Limit Theorem (CLT) - states that for multiple samples taken from a population (with known mean and variance), if the sample size is large, then the distribution of the sample mean, or sum, will converge to a normal distribution even though the random variable x (individual data points within a sample) may be non-normal. This proves to be a key concept in probability theory as it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. It usually gives the below conditions

1. Sample means always follow normal distribution irrespective of distribution of individual data in population
2. Mean of sample means tends to population mean as the number of samples tend to infinity
3. Variance of sample means is 'n' times less than the variance of population, where 'n' is size of sample
E.g. Consider the roll of 2 dice. If this is done multiple times and the average or the sum of the rolls is plotted, then this plot will converge to a normal distribution

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Review the answer provided by Mr Venugopal R, Benchmark Six Sigma's in-house expert.

Question

Q﻿﻿. 155  Explain the practical application of Central Limit Theorem while collecting and analyzing data. Provide a few practical examples to support your responses.

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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.

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What i learn from the central limit theorem and am keen to apply to my work is that, It may not be correct to overly invest in problems, what is more important is to be consistent with actions. Even if there are mistakes being made in the process, it is important to be consistent and keep actions alive, the end result will come to normal. As there will be both negative and positive results from the actions produced, towards the end the extreme results will neutralise and demonstrate normalcy.

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The central limit theorem is used to define the location of data distribution, there are three measures of central tendency Mean, Median and Mode.

Mean is simply the arithmetic average of data.

Median is a middle number in an ordered dataset.

Mode represents the frequently occurred data in a dataset, a dataset can have no mode or bi-mode (two-mode), if the data set has more than 2 modes then we have to consider as the data set has no mode. In Fig.1 the data is normally distributed so all three measures are the same, sometime depends on the data there is a better way to represent the location/middle of the data.

In Fig.2&3 the data is skewed mean & mode are same but these are not representing the actual location of distribution, but the median represents the location more accurately, here the difference in mean and median is because of the nature of mean, mean is easily affected by the outlier.

When facing the skewed distribution, it is advisable to look for the median to find the location of distribution, for right-skewed distribution mean will always be higher than median whereas in left-skewed distribution median will always be higher than mean.

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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.

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Please refer to the Benchmark Expert view to get a clear picture of usage of Central Limit Theorem.

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