Theorem 2:
Let x¯ be the mean of a random sample X = (XI, X2, ..., Xn, ) of size n from a population with finite mean μ and finite variance σ2 . Then, for any real x,
lim P(x¯ - μ/ (σ/√n) ≤ x) = Φ (x)
n→∞
where Q, ( x ) is the cdf of the standard normal variable.
The theorem is true under more common assumptions concerning the component of the random vector X. Actually, it is not essential to assume that X is a random sample. The theorem is true even while the components Xi, i = 1, 2, ... n, of X have various marginal distributions.
One interpretation of the theorem is that for a finite n the distribution of x¯ may be approximated by the normal distribution with mean μ and variance σ2/n. The approximation is good enough, from practical points of view, if the sample size n ≥ 25 in general, however for some special population distribution a much smaller value of n may be good though.
We may have another interpretation of the theorem. Let S = X1, + X2, + ... +Xn,, then x¯ = S/n. When x¯ follows a normal random variable Z is defined as the sum of n random variables then the distribution of Z , for large n may be assumed to be normal. This is the reason why the normal distribution model is found to fit quite well in situations where the random variables, e.g., the yield per acre Z of n crop is dependent on innumerable factors influencing the quality of the seed, the conditions of the soil, the quality of the fertilizers. the weather conditions etc. and so it is expressible as Z = X1, + X2 + ... where X1, X2, X3, ... may be interpreted as contributions to Z due to the factor number one, the factor number two, the factor number three etc. respectively. Similarly, the power consumption Z in large factory can be expressed: IS sum of small contributions from many departments of the factory.