Central Limit Theorem & Gaussian Distribution — A tidy Examination of their relationship
Gaussian distribution is a probability distribution that is symmetrical and bell-shaped. It is parametrized by its mean and standard deviation, and it is commonly used to model random variables that arise in natural phenomena such as the heights of individuals, the weight of objects, and many other physical phenomena.
The Central Limit Theorem (CLT) is a fundamental theorem in statistics that describes the behavior of the sample mean of a large number of independent, identically distributed random variables.
The CLT states that as the sample size increases, the distribution of the sample mean approaches a normal distribution, regardless of the shape of the original distribution.
This means that if we take a large number of samples from any distribution, and calculate the mean of each sample, the distribution of the sample means will be approximately normal.
The relation between the Normal distribution and the CLT is that the Normal distribution provides the limiting distribution of the sample mean as the sample size approaches infinity. This means that if the sample size is large enough, we can assume that the sample mean follows a Normal distribution, regardless of the shape of the original distribution.
The Normal distribution is important for statistical tests because it provides a mathematical framework for analyzing and understanding data. Many statistical tests, such as the t-test and ANOVA, assume that the data is normally distributed. Therefore, it is important to determine whether the data follows a normal distribution before applying these tests.
The Normal distribution is also important in hypothesis testing because it allows us to calculate the probability of observing a particular sample mean, given a hypothesized population mean and standard deviation.
This probability is calculated using the Z-score, which is a measure of how many standard deviations the sample mean is away from the hypothesized population mean. The Z-score is calculated as (sample mean — population mean) / (standard deviation / square root of sample size), and it follows a standard Normal distribution.