Central Limit Theorem

A landmark leading to the normal distribution.

The Central Limit Theorem or CLT is considered to be the second fundamental theorem of statistics, the first being the law of large numbers. The Central Limit Theorem applies to a sampling distribution of the mean, where all sample have been drawn from the same parent population.

Statement of the CLT:

Essentially, the CLT, also known as the Central Theorem, states that:

The expected value (equivalent to the mean) of a sampling distribution of the mean is equal to the mean of the parent population.
The standard error (equivalent to the standard deviation) of a sampling distribution of the mean is equal to the standard deviation of the parent population divided by the square root of the sample size.
Irrespective of the underlying distribution of the parent population, the sampling distribution of the mean increasingly reaches the normal distribution, as the sample size increases,. This is one of the most important elements of the Central Limit Theorem and explains why so many natural phenomena can be described with the Normal distribution.

Mathematical Description of the CLT:

Mathematically, the Central Limit Theorem is expressed as:

and

Where

	= mean of the sampling distribution
	= standard error of the sampling distribution
	= mean of the population
	= standard deviation of the population

The Concept Behind the Central Limit Theorem:

The concept of the CLT is closely related to the law of large numbers, and the Chebyshev's theorem, both of which explain the behavior of stochastic probability characteristics of a population, as the sample size increases.

Law of Large Numbers:It states that, the mean value of a trial will increasingly become equal to the expected value as the number of trials increase. For example, the expected value for a roll of die is 3.5 (= ). According to the law of large numbers, as the sample size (the number of die rolls) increases, the average result increasingly approaches 3.5.
Chebyshev's Theorem:It states that, in any stochastic probability distribution (which describes the behavior of random processes), the proportion of the population that is more than k standard deviations away from the mean, is always less than 1/k2.

Example of the Central Limit Theorem:

Toss 10 coins simultaneously, once. Count the number of coins that show heads.
Toss 10 coins simultaneously, ten times. Add the number of coins that show heads from each toss. Plot a graph of number of coins that show heads (0-10) versus number of tosses (1-10).
Toss 10 coins simultaneously, 500 times. Add the number of coins that show heads from each toss. Plot a graph of number of coins that show heads (0-500) versus number of tosses (1 - 500).

According to the Central Theorem, for 10 coins, the expected value of coins which will turn up heads is 5; the probability that only 2 or 8 of the tossed coins will turn up heads is lower than the probability of the sample mean.

Below is the graph for ten coins tossed 10 times. It shows that the number of heads is evidently close to the expected value. However according to the Central Limit Theorem, since the sample size is small, the true approximation of the normal distribution is not seen.

When the sample size is increased and ten coins are tossed 500 times, the number of coins that turn up heads for the most times are 5. Right according to the Central Theorem, the sampling distribution approximates a normal distribution even though the underlying population distribution is not normal.

Central Limit Theorem Graph Large Sample

Summary

The CLT states that the means of the sampling distribution will approximate a normal distribution as the number of samples increase. This gives you license to use the normal distribution even in situations where the population distribution is not normal.

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