Central Limit Theorem[CLT]

Central Limit Theorem is the most important theorem of Statistics.

According to the central limit theorem, if X₁, X₂, …, X_n is a random sample of size n taken from a population with mean µ and variance σ² then the sampling distribution of the sample mean tends to normal distribution with mean µ and variance σ²/n as sample size tends to large.

This formula indicates that as the sample size increases, the spread of the sample means around the population mean decreases, with the standard deviation of the sample means shrinking proportionally to the square root of the sample size, and the variate Z,

Z = (x – μ)/(σ/√n)

where,

• z is z-score
• x is Value being Standardized (either an individual data point or the sample mean)
• μ is Population Mean
• σ is Population Standard Deviation
• n is Sample Size

This formula quantifies how many standard deviations a data point (or sample mean) is away from the population mean. Positive z-scores indicate values above the mean, while negative z-scores indicate values below the mean. Follows the normal distribution with mean 0 and variance unity, that is, the variate Z follows standard normal distribution.

According to the central limit theorem, the sampling distribution of the sample means tends to normal distribution as sample size tends to large (n > 30).

Sampling distribution is essential in various aspects of real life. Sampling distributions are important for inferential statistics. A sampling distribution represents the distribution of a statistic, like the mean or standard deviation, which is calculated from multiple samples of a population. It shows how these statistics vary across different samples drawn from the same population.

In this article, we will discuss the Sampling Distribution in detail and its types along with examples and go through some practice questions too.