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What is the Central Limit Theorem?

Central Limit Theorem is one of the most important and most utilized theorems in Statistics. Many theories are based on the Central Limit Theorem. It is the foundation to determine the mean, variance and Standard deviations. This theorem is considered as the heart of probability theory.

The central limit theorem states that when an infinite number of successive random samples are taken from a population, the distribution of sample means calculated for each sample will become approximately normally distributed with mean μ and standard deviation σ/√N (-N (μ, σ)/ √N) when the sample size is N becomes larger, irrespective of the shape of the shape of population.

The different components of Central Limit Theorem are as follows.

  • Successive sampling from a population.
  • Increasing the sample size.
  • Population distribution.

The theorem suggests that the approximation steadily improves as the number of observations increases.

For Ex: suppose an ordinary coin is tossed 100 times and the number of heads is counted. This is equivalent to scoring 1 for a head and 0 for a tail and computing the score. Thus the total number of heads is the sum of 100 independent, identically distributed random variables. By the central limit theorem, the distribution of total number of heads will be, to a very high degree of approximation, normal. After a large number of repetitions the curve appears like a normal curve. It has been empirically observed that various natural phenomena, such as the heights of individuals, follow approximately a normal distribution. A suggested explanation is that these phenomena are sums of a large number of independent random effects and hence are approximately normally distributed by the central limit theorem.

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