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Used in ANSI Y14.5 measurements, this selection will use the Rayleigh distribution to model measured characteristics resulting from two Normal variables. To fit a Rayleigh distribution, only the specified sigma (process, sample, or population) is needed. Note that the curve generated will be used for K-S values and process capability, but that process sigma is always used to define the distribution limits on the Individual-X chart.

Pyzdek (1992) recommends assuming a Rayleigh distribution for true-position data:

The Rayleigh distribution results when x and y measurements are converted to radial measurements. The Rayleigh (distribution) assumes that both x and y are normally distributed with zero mean and equal variances. An approximation to this situation occurs frequently with true position measurements. True position is calculated using Equation 13.11:

True Position = 2*SQRT(XIn Equation 13.11 X is the deviation from nominal in the x direction and Y is the deviation from nominal in the y direction. With true position, obviously, no negative values are possible. thus, there is no lower specification. Table 13.5 shows the percentage above the upper specification based on the value of Z^{2}+Y^{2})_{t}where:Z_{t}=True Position Specification / Sigma

(Editor's note: Sigma is the process standard deviation of the calculated true-position values (as defined in a control chart) Also note that Pyzdek subsequently advised using SPC-PC software to predict the percent defective based on a fitted distribution for the calculated true position values. (Quality Engineering 4(2), 235-241 (1991-1992)).

Table 13.5 (abridged)

Z |
% Out of Spec |

2.0 |
60.65 |

2.5 |
45.78 |

3.0 |
32.47 |

4.0 |
13.53 |

4.5 |
7.96 |

5.0 |
4.39 |

5.5 |
2.28 |

6.0 |
1.11 |

6.5 |
0.51 |

7.0 |
0.22 |

7.5 |
0.09 |

8.0 |
0.03 |

8.5 |
0.01 |

9.0 |
0.00 |

Learn more about the Statistical Inference tools for understanding statistics in Six Sigma Demystified (2011, McGraw-Hill) by Paul Keller, or his online Black Belt certification course.

Tools

Intervals & Tests

Hypothesis Test Of Sample Mean Example

Hypothesis Test Of Two Sample Variances Example

Hypothesis Test Of A Standard Deviation Compared To A Standard Value Example

Distributions

Area Under the Standard Normal Curve

Non-Normal Distributions in the Real World

Rayleigh Distribution for True Position