TY - BOOK

T1 - Normalizing Transformations for Shewhart-type Control Schemes

T2 - Report No.118

AU - Shore, Haim

PY - 1994

Y1 - 1994

N2 - Traditional Shewhart-type process control schemes and process capability analyses assume that the process distribution (or distributions of statistics derived thereof) are approximately normally distributed. When this assumption fails to materialize, normalizing transformations are sometimes needed. However, many of the currently used transformations are difficult to apply and at times require expertise that the common practitioner does not possess. In this paper, a new set of normalizing transformations are suggested that are simple to carry out, may be generally applied (since they are distribution-free), and are associated with a non-iterative standard procedure that is easy to program. The new transformations are based on linear transformations of a symmetrically disributed random variable, Z, where different transformations are applied to each half of the distribution. A three-moment or a two-moment (partial and complete) distribution-matching procedure is used to identify the parameters of the transformations. Applying an inverse transformation, a symmetrically distributed variable is obtained. If Z is the standard normal variable, a normalizing transformation results. The proposed transformations are numerically demonstrated for the p-control chart, for an X&R control scheme and for capability analysis. The latter two examples use Monte-Carlo simulated data from a highly skewed gamma distribution to demonstrate the effectiveness of the new transformations. For the binomial and the Poisson distributions, a comparison is made with current normalizing transformations used in applications of the p chart and the c chart, respectively. The new transformations are shown to result in better normalization. Furthermore, unlike some "distribution-specific" transformations, they utilize a standard procedure. Implications for current practices regarding Shewhart-type control schemes and for capability analyses are discussed.

AB - Traditional Shewhart-type process control schemes and process capability analyses assume that the process distribution (or distributions of statistics derived thereof) are approximately normally distributed. When this assumption fails to materialize, normalizing transformations are sometimes needed. However, many of the currently used transformations are difficult to apply and at times require expertise that the common practitioner does not possess. In this paper, a new set of normalizing transformations are suggested that are simple to carry out, may be generally applied (since they are distribution-free), and are associated with a non-iterative standard procedure that is easy to program. The new transformations are based on linear transformations of a symmetrically disributed random variable, Z, where different transformations are applied to each half of the distribution. A three-moment or a two-moment (partial and complete) distribution-matching procedure is used to identify the parameters of the transformations. Applying an inverse transformation, a symmetrically distributed variable is obtained. If Z is the standard normal variable, a normalizing transformation results. The proposed transformations are numerically demonstrated for the p-control chart, for an X&R control scheme and for capability analysis. The latter two examples use Monte-Carlo simulated data from a highly skewed gamma distribution to demonstrate the effectiveness of the new transformations. For the binomial and the Poisson distributions, a comparison is made with current normalizing transformations used in applications of the p chart and the c chart, respectively. The new transformations are shown to result in better normalization. Furthermore, unlike some "distribution-specific" transformations, they utilize a standard procedure. Implications for current practices regarding Shewhart-type control schemes and for capability analyses are discussed.

M3 - דוח

BT - Normalizing Transformations for Shewhart-type Control Schemes

ER -