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# Central Limit Theorem, Law of Large Numbers

Asked by
Vishwadeep Khatri

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

**Law of Large Numbers** - states that as sample size grows, the sample mean gets closer to the population mean irrespective whether the data set is normal or non-normal e.g. consider the roll of a single dice. If you roll the dice sufficiently large number of times, the average would tend to be close to 3.5.

An application oriented question on the topic along with responses can be seen below. The best answer was provided by Atul Dev on 15th September 2017.

Q9. What is the difference between Central Limit Theorem and Law of Large Numbers? If central limit theorem can help us achieve a normal distribution, why should we not always make use of it and get rid of Non-Normal data?

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