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 CLT applies to a sampling distribution of the mean, where all samples have been drawn from the same parent population.
Essentially, the Central Limit Theorem (CLT), also known as the Central Theorem, states that:
Mathematically, the Central Limit Theorem is expressed as:
and
Where
= mean of the sampling distribution
= standard error of the sampling distribution
= mean of the sampling distribution
= standard deviation of the population
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.
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.
The central limit theorem states that the means of the sampling distribution will approximate a normal distribution as the number of samples increases. This gives you the license to use the normal distribution even in situations where the population distribution is not normal.
The Central Limit Theorem is a concept that arises in statistics and is utilized to analyze the behavior of the sample mean for a given random variable when the sample size is large enough. According to the theorem, the distribution of the sample mean becomes normal, regardless of the original distribution of the population that the sample is drawn from. This theorem is extensively applied in statistical hypothesis testing and inference.
Understanding the Central Limit Theorem is important since it permits making predictions and drawing inferences about a population based on a sample. By comprehending the distribution of the sample mean, we can estimate the population mean and make predictions with a certain degree of confidence. This theorem is used in several fields, including the social sciences, biology, and economics.
The Central Limit Theorem is based on several assumptions that need to be satisfied. These include random sampling, independence of observations, and a finite population variance. Additionally, the sample size must be sufficiently large, typically greater than 30, for the theorem to be applicable. When these conditions are met, the Central Limit Theorem can be applied to estimate population parameters with a high degree of accuracy.