STANDARD NORMAL DISTRIBUTION AND APPROXIMATIONS

ABSTRACT

The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, become normally distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal. Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed. This paper presents five new formulas for approximation of cumulative standard normal probabilities. Two of these approximations are polynomial based and are only accurate for 0 ≤ z ≤ 1; the other three formulas are accurate on the interval − 3.4 ≤ z ≤ 3.4 which is the domain often used in normal tables. We recommend the last of these new formulas. 

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