The central limit theorem states that, given a distribution with a mean μ and variance σ2, the sampling distribution of the mean x-bar approaches a normal distribution with a mean (μ) and a variance σ2/N as N, the sample size, increases.
The amazing and counter-intuitive thing about the central limit theorem is that no matter the shape of the original distribution, x-bar approaches a normal distribution.
Keep in mind that N is the sample size for each mean and not the number of samples. Remember that in a sampling distribution the number of samples is assumed to be infinite. The sample size is the number of scores in each sample; it is the number of scores that goes into the computation of each mean.
Two things should be noted about the effect of increasing N:
Based on the central limit theorem, when the sample size is large, you can:
Note that the central limit theorem does not prescribe that the underlying population must be normally distributed. Therefore, the central limit theorem can be applied to a population with any probability distribution.
| jayj001: SAMPLE SIZE = no. of observations in each sample|
e.g. 100 samples with 5 observations in each
NOT normal distribution as N < 30
| Bududeen: it depends if the distribution itself is not normal...if the distribution is normal , then the sample size does not affect the normality of the distribution|
so even with a sample size of 5, if the distribution is normal then the distribution follows a normal dist.
| GouldenOne: What is variable X?|
|sgossett86: Basically we're navigating from initial statistic theory to how to take our own imperfect data and make standard probability distributions so we can standardize and make inferences comparatively as an analyst.|