up:: [[Normal distributions]] Tags:: # Central Limit Theorem (CLT) ^585c40 ###### What is the central limit theorem (CTL)? For any population with mean μ and standard deviation σ (irrespective of the shape of the distribution!), the complete distribution of sample means for sample size n will have a mean of μ and a standard deviation of σ/sqrt(n) and will approach a normal distribution as n approaches infinity. But we can reasonably estimate the distribution is normal when n is 30 or greater or the population is normally distributed. The standard error of the mean is the same as σ/sqrt(n). In other words, it's the standard deviation of the sample mean distribution for a particular sample size. ^7e645c If the population distribution is normal already than the mean sampling distribution will be normal even with a n value of lower than ==30==. To calculate this we no longer compare it to the population distribution as that would give us the same answer but rather compare it to the distribution of sample means of that same size. **As the sample size increases the distribution of sample means gets more and more narrow** ![[Pasted image 20220912151254.png]] There are two powerful effects of taking samples and distributing the sample means: 1. By the law of large numbers the distribution of sample means becomes progressively ==narrower== as sample size increases. In other words the [[Variance and Standard Deviation#^36d688|standard deviation]] is ==lower== for the distribution of sample means the higher the sampling size with the replacement with the population. 2. The distribution of sample means becomes ==normal== as sample size n increases towards infinity. The ==Central Limit Theorem== states the sample size must be over n = ==30== for this to be true. Related: Created: [[30-09-2022]] ___ # Resources