Central Limit Theorem

The central limit theorem states that if sufficiently large random samples are taken from the population, then the distribution of these sample means will be approximately normally distributed. This holds true regardless of whether the population is normal or skewed, provided the sample size is sufficiently large (usually greater or equal to 30).

Suppose X1, X2, X3,……..Xn are n random variables that are independent and identically distributed with a mean μ and a standard deviation σ then, the central limit theorem states that:

{(X−μ)/ σ /√n }→ N (0,1)

as n→∞

Example

Let’s assume that the average salary in India is INR 1,00,000 pa (population mean). If we take a random sample of sufficiently large size from this population, the mean of the sample i.e., the average salary of this sample should also be around INR 1,00,000 pa (close to population mean). The only thing to note here is that the sample should be random and should not be biased based on age, geography, demographics, etc.