Central Limit Theorem

• In probability theory, the central limit theorem (CLT) establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution(informally a “bell curve”) even if the original variables themselves are not normally distributed.
• Suppose that a sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic mean of the observed values is computed.

If this procedure is performed many times, the central limit theorem says that the distribution of the average will be closely approximated by a normal distribution. A simple example of this is that if one flips a coin many times the probability of getting a given number of heads in a series of flips will approach a normal curve, with mean equal to half the total number of flips in each series. (In the limit of an infinite number of flips, it will equal a normal curve.)

Basic Intuition

• In CLT we plot a graph with sample means VS frequency of the means
• Here sample mean is a Random variable on x axis with its respective frequency on y axis
• Sample means here is expectation value of n random samples chosen

Normal Distribution

Normal Distribution, also known as Gaussian distribution, is ubiquitous in Data Science. We will encounter it at many places especially in topics of statistical inference. It is one of the assumptions of many data science algorithms too.

Z-distribution

In statistics, the Z-distribution is used to help find probabilities and percentiles for regular normal distributions (X). It serves as the standard by which all other normal distributions are measured. The Z-distribution is a normal distribution with mean zero and standard deviation 1