# The Performance Curve of a Device

May 20, 2011

A little-know yet important subject, the performance curve of a device is a concept that allows us to evaluate a device’s behavior in several points along the engineering specification range. Through the results of this field of study, it is possible to determine the probability of acceptance of each measured part of the studied device.

This study will be explained from the example below.

Bias = - 0,1. The bias is calculated by the difference between the reference value and the average measured value observed.

RR = 0,25 (25%). As 5,15 sigma means that 99% of the measurements are within the normal distribution, we can calculate sigma, where:

5,15 sigma = RR Þ 5,15 = 0,25 Þ sigma = 0,04854

Engineering limits: ILE = 80 and SLE = 82

Reference values: 79,95; 81; 82; 82,04; and 82,12 (random values, although, near the engineering specification).

Example of hom to create the performance curve of a device:

Z = (ILE – X) / sigma = (80 – 79,85) / 0,04854 Þ Z = 3,09 Þ P(Z) = 0,10%

Z = (SLE – X) / sigma = (82 – 79,85) / 0,04854 Þ Z = 44,29 Þ P(Z) = 0,00%

ps.: P(Z) = 0.10% is extracted from the normal distribution table for Z = 3,09

Z = (X – ILE) / sigma = (80,9 – 80) / 0,04854 Þ Z = 18,54 Þ P(Z) = 0,00%

Z = (SLE – X) / sigma = (82 – 80,9) / 0,04854 Þ Z = 22,66 Þ P(Z) = 0,00%

Z = (X – ILE) / sigma = (81,9 – 80) / 0,04854 Þ Z = 39,14 Þ P(Z) = 0,00%

Z = (SLE – X) / sigma = (82 – 81,9) / 0,04854 Þ Z = 2,06 Þ P(Z) = 1,97%

Z = (X – ILE) / sigma = (81,94 – 80) / 0,04854 Þ Z = 39,96 Þ P(Z) = 0,00%

Z = (SLE – X) / sigma = (82 – 81,94) / 0,04854 Þ Z = 1,24 Þ P(Z) = 10,82%

Z = (X – ILE) / sigma = (82,02 – 80) / 0,04854 Þ Z = 41,61 Þ P(Z) = 0,00%

Z = (X - SLE) / sigma = (82,02 – 82) / 0,04854 Þ Z = 0,41 Þ P(Z) = 34,02%

After accomplishing all the calculations and tracing the curve, for the resulting value of each part, the mode graph shall be used in order to know the probability of acceptance for each related part. In the X axis of the graph, the value found for the part that is being analysed has to be entered and an imaginary line has to be created an then, from the found value, until this imaginary line reaches the graph and, at this moment, it is read in the Y”axis the probability of acceptance.

The technician responsible for the analysis ha to evaluate the importance of the characteristic to be controlled to define whether the study should be accomplished or not, due to its complexity.

The performance curve of a device’s objective is solely and exclusively to evaluate the probability of acceptance of each measured part. This study allows us a higher safety margin in decisions such as: to liberate or scrap a batch of manufactured parts as well as to invest in financial resources to improve the RR values or even to reduce the device’s tendency value. The smaller the RR values and the bias are the higher is the probability of acceptance for the values near the engineering limits.

This study will be explained from the example below.

5,15 sigma = RR Þ 5,15 = 0,25 Þ sigma = 0,04854

Example of hom to create the performance curve of a device:

## First Device Performance Curve Point (79,95)

X = 79,95 – 0,1 = 79,85Z = (ILE – X) / sigma = (80 – 79,85) / 0,04854 Þ Z = 3,09 Þ P(Z) = 0,10%

Z = (SLE – X) / sigma = (82 – 79,85) / 0,04854 Þ Z = 44,29 Þ P(Z) = 0,00%

ps.: P(Z) = 0.10% is extracted from the normal distribution table for Z = 3,09

## Second Device Performance Curve Point (81)

X = 81 – 0,1 = 80,9Z = (X – ILE) / sigma = (80,9 – 80) / 0,04854 Þ Z = 18,54 Þ P(Z) = 0,00%

Z = (SLE – X) / sigma = (82 – 80,9) / 0,04854 Þ Z = 22,66 Þ P(Z) = 0,00%

## Third Device Performance Curve Point (82)

X = 82 – 0,1 = 81,9Z = (X – ILE) / sigma = (81,9 – 80) / 0,04854 Þ Z = 39,14 Þ P(Z) = 0,00%

Z = (SLE – X) / sigma = (82 – 81,9) / 0,04854 Þ Z = 2,06 Þ P(Z) = 1,97%

## Fourth Device Performance Curve Point (82,04)

X = 82,04 – 0,1 = 81,94Z = (X – ILE) / sigma = (81,94 – 80) / 0,04854 Þ Z = 39,96 Þ P(Z) = 0,00%

Z = (SLE – X) / sigma = (82 – 81,94) / 0,04854 Þ Z = 1,24 Þ P(Z) = 10,82%

## Fifth Device Performance Curve Point (82,12)

X = 82,12 – 0,1 = 82,02Z = (X – ILE) / sigma = (82,02 – 80) / 0,04854 Þ Z = 41,61 Þ P(Z) = 0,00%

Z = (X - SLE) / sigma = (82,02 – 82) / 0,04854 Þ Z = 0,41 Þ P(Z) = 34,02%

After accomplishing all the calculations and tracing the curve, for the resulting value of each part, the mode graph shall be used in order to know the probability of acceptance for each related part. In the X axis of the graph, the value found for the part that is being analysed has to be entered and an imaginary line has to be created an then, from the found value, until this imaginary line reaches the graph and, at this moment, it is read in the Y”axis the probability of acceptance.

The technician responsible for the analysis ha to evaluate the importance of the characteristic to be controlled to define whether the study should be accomplished or not, due to its complexity.

The performance curve of a device’s objective is solely and exclusively to evaluate the probability of acceptance of each measured part. This study allows us a higher safety margin in decisions such as: to liberate or scrap a batch of manufactured parts as well as to invest in financial resources to improve the RR values or even to reduce the device’s tendency value. The smaller the RR values and the bias are the higher is the probability of acceptance for the values near the engineering limits.