### Abstract

It has been proposed that when the proportion of nonconforming product is extremely small that one base 3σ control charts on the 0.2777^{th} power of X, the number of items sampled until a nonconforming item is found. This choice of exponent renders the transformed variable approximately normal in distribution. In this paper, that proposal is compared to the alternative of using a logarithmic transformation of X as the basis for 3σ control charts. Expressions are derived for the OC curves of both procedures. It is shown further that if probability limits are used in lieu of 3σ limits, the OC curves of the two procedures are identical. If probability limits are used, the power transformation is preferred to the log transform, since of the two, its distribution is more nearly normal and, thus, more closely justifies the use of the supplementary tests commonly used to enhance the power of control charts. Computation of the average run lengths shows that the power transformation gives poor protection against worsening quality and that the log transformation results in an unacceptably short in-control ARL value. It is shown that either transformed value gives greatly improved performance when used in an EWMA scheme. The power transformation is recommended over the log transformation as the basis for an EWMA chart because it will detect an increase in the fraction nonconforming of up to a factor of three with a smaller resultant ARL. Moreover, it will detect any decrease in the fraction nonconforming more quickly.

Original language | English (US) |
---|---|

Pages (from-to) | 240-247 |

Number of pages | 8 |

Journal | Journal of Quality Technology |

Volume | 30 |

Issue number | 3 |

State | Published - Jul 1998 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Industrial and Manufacturing Engineering
- Statistics and Probability
- Management Science and Operations Research

### Cite this

*Journal of Quality Technology*,

*30*(3), 240-247.

}

*Journal of Quality Technology*, vol. 30, no. 3, pp. 240-247.

**Control charts applicable when the fraction nonconforming is small.** / Mccool, John I.; Joyner-Motley, Tracy.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Control charts applicable when the fraction nonconforming is small

AU - Mccool, John I.

AU - Joyner-Motley, Tracy

PY - 1998/7

Y1 - 1998/7

N2 - It has been proposed that when the proportion of nonconforming product is extremely small that one base 3σ control charts on the 0.2777th power of X, the number of items sampled until a nonconforming item is found. This choice of exponent renders the transformed variable approximately normal in distribution. In this paper, that proposal is compared to the alternative of using a logarithmic transformation of X as the basis for 3σ control charts. Expressions are derived for the OC curves of both procedures. It is shown further that if probability limits are used in lieu of 3σ limits, the OC curves of the two procedures are identical. If probability limits are used, the power transformation is preferred to the log transform, since of the two, its distribution is more nearly normal and, thus, more closely justifies the use of the supplementary tests commonly used to enhance the power of control charts. Computation of the average run lengths shows that the power transformation gives poor protection against worsening quality and that the log transformation results in an unacceptably short in-control ARL value. It is shown that either transformed value gives greatly improved performance when used in an EWMA scheme. The power transformation is recommended over the log transformation as the basis for an EWMA chart because it will detect an increase in the fraction nonconforming of up to a factor of three with a smaller resultant ARL. Moreover, it will detect any decrease in the fraction nonconforming more quickly.

AB - It has been proposed that when the proportion of nonconforming product is extremely small that one base 3σ control charts on the 0.2777th power of X, the number of items sampled until a nonconforming item is found. This choice of exponent renders the transformed variable approximately normal in distribution. In this paper, that proposal is compared to the alternative of using a logarithmic transformation of X as the basis for 3σ control charts. Expressions are derived for the OC curves of both procedures. It is shown further that if probability limits are used in lieu of 3σ limits, the OC curves of the two procedures are identical. If probability limits are used, the power transformation is preferred to the log transform, since of the two, its distribution is more nearly normal and, thus, more closely justifies the use of the supplementary tests commonly used to enhance the power of control charts. Computation of the average run lengths shows that the power transformation gives poor protection against worsening quality and that the log transformation results in an unacceptably short in-control ARL value. It is shown that either transformed value gives greatly improved performance when used in an EWMA scheme. The power transformation is recommended over the log transformation as the basis for an EWMA chart because it will detect an increase in the fraction nonconforming of up to a factor of three with a smaller resultant ARL. Moreover, it will detect any decrease in the fraction nonconforming more quickly.

UR - http://www.scopus.com/inward/record.url?scp=0032123661&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032123661&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0032123661

VL - 30

SP - 240

EP - 247

JO - Journal of Quality Technology

JF - Journal of Quality Technology

SN - 0022-4065

IS - 3

ER -