Question

Why is a normal distribution converted to a standard normal distribution?

Answer 1

Answer

For X ~ N(µ, σ²), to obtain any probability on X, say P(X ≤ t), one needs to evaluate

∫_-∞^t [{1/σ√(2π)}exp-[(1/2){(x - µ)/σ}²], which cannot be done by the normal process of integration. The evaluation involves a very tedious process of numerical method. But, the probability values of N(0, 1), i.e., standard normal distribution, are readily available in Standard Statistical Tables. So, for any Normal distribution probability calculations, the given variable is converted to the Standard Normal Variable by Z = {(X - - µ)/σ}      

DONE

[Going beyond,

The above not withstanding, in the present times, Excel Function: Statistical NORMDIST and NORMINV can readily give probability for any N(µ, σ²), obviating the necessity to convert to Standard Normal Variable]