Simple, Closed-Form Expressions Relating Long-Term (Z score) and Short-Term (Defects per Opportunity) Variability
A simple and accurate analytical expression relating the expected process (long term) and sampling (short-term) product variability is developed using a variational mathematical principle. Of the several complex functional forms discovered, simplicity and ease of use are used to select an expression providing the most reliable estimation for and convenient expression of Z score (σ level) as a function of defects per opportunity (DPO) or per million opportunities (DPMO). In the absence of scientific calculators or computers, this expression allows engineers to accurately estimate long term process variability to within 0.01 of its true value without resulting to (laborious) tables or a computer. Also, a high precision approximation is provided for cases when DPO is less than 1% which estimates Z-score to within 0.003 of the actual value (at 6σ).