Machine Designers Reference. J. Marrs

Normal

      The property of the normal curve that is most useful to machine designers is that the area under the curve, bounded between lower and upper points ZL and ZU on the Z-axis, represents the percentage of all Z’s that will be between ZL and ZU. The total area under the normal distribution curve is always equal to 1, or 100%. Calculating the area under a given normal distribution curve can be tedious, so a transformation of variables is used to take advantage of tabulated values.

      The standard normal is simply a normal distribution curve with μ = 0 and σ = 1. The area under the standard normal curve is pictured in Figure 3-20 and tabulated in Table 3-14. The values indicate the area under the curve to the left of Z. The table is read by finding the value of Z by summing the column and row headers and locating the area at the intersection. The area under the curve to the left of Z = −1.25 is 0.10565, or 10.565% of the population. This percentage is found at the intersections of row “−1.2” and column “−0.05” corresponding to the value of −1.25.

Figure 3-20: Standard Normal Curve

      Because the curve is symmetric, the table only gives the area under the curve for half the curve: from the left up through Z = 0. To calculate the area for positive values of Z, use the identity:

      Area(Z) = 1 − Area(−Z)

      The following example demonstrates how to use the table to determine what percentage of assemblies fall between the upper and lower spec limits of AUSL = 0.35 and ALSL = 0. First, calculate the assembly mean and standard deviation:

μA	=	μB − μC − μD + μE + μF + μG − μH
	=	1.735 − 8.5 − 3.175 + 2.66 + 0.51 + 8.5 − 1.5625
μA	=	0.17

Table 3-14: Area Under the Standard Normal Curve

      To calculate the values for Z we will look up in the table:

      The area under the normal curve between ALSL and AUSL is the area under the curve to the left of ZUSL minus the area under the curve to the left of ZLSL, ^or:

=	Area(3.911) − Area(−3.694)
=	[1 − Area(−3.911)] − Area(−3.694)
=	(1−0.00005)−0.00011
=	0.99984

      According to this sample calculation, 99.984% of assemblies will be in-spec, and 0.016% out-of-spec. The standard normal table can also be used to determine spec limits for a given desired defect rate.

       Modeling the Distribution

      Performing an accurate analysis using the statistical method requires knowledge of the dimensions of the produced parts. This is not always available, and assumptions must be made. Given upper and lower specification limits (USL, LSL), what will be the mean (μ) and standard deviation (σ) of the population? These values can be assumed outright when their meaning is fully understood.

      Often the mean and standard deviation are assumed by way of other measures. The process capability index (Cpk) is used as an indicator of how well a manufacturing process is capable of producing dimensions on-target between the spec limits. The terms Cpl and Cpu are the lower and upper process capability indices, respectively.

      A Cpk of 1.0 with a mean (μ) on target (halfway between the specification limits) corresponds to a standard deviation (σ) of one-sixth of the tolerance range (USL − LSL), or 99.73% of dimensions in-spec. Larger values of Cpk indicate greater control over the process and incur greater manufacturing expense: smaller values, less control at less cost. A centered process with Cpk of 2.0 represents a Six Sigma process, a goal of many corporate quality programs using SPC, and may be assumed by the design engineer purchasing components from reputable and reliable companies. A smaller assumption on Cpf would be more conservative.

      However, knowing Cpk alone is not enough to select both μ and σ When the terms Cpl and Cpu are equal, the μ is on target and centered between the spec limits. They need not be equal. A larger upper process capability index occurs when the mean is shifted closer to the LSL; a larger lower process capability index means the mean is shifted toward the USL. When Cpl and Cpu are known or can be estimated, μ and σ can be calculated: