component dimensions.
One drawback of the worst-case analysis is that it is unlikely that the actual measured assembly dimension will take on the calculated minimum or maximum value. It is more likely to be near the middle of the range, as component dimensions produced near their upper limits cancel out the effect of dimensions produced near their lower limits. This drawback becomes more pronounced when more dimensions are included in the stack-up chain.
The stack-up equation can also be used to solve a tolerance assignment problem containing a single unknown. When the assembly dimension is known, and only one dimension needs to have its dimensional limits assigned, the separated worst-case equations can be re-arranged to solve for the final tolerance to assign. For example, when the limits of dimension B are desired,
Bmax = Amax + Cmin + Dmin − Emax − Fmax − Gmax + Hmin
Bmin = Amin + Cmax + Dmax − Emin − Fmin − Gmin + Hmax
The stack-up equation used for analysis must be separated before it can be re-arranged for use in assignment. This is because of the way the analysis equation is derived: dimension A reflects the total variation of all dimensions. Using this equation for assignment may reveal that there is not enough tolerance left to give to dimension B and other assignment changes must be made. This is indicated by a meaningless result showing a minimum value for dimension B greater than its maximum value. Often, an assignment problem requires assigning tolerances to several dimensions, and the analysis equation cannot be re-arranged to solve for the missing dimension. In this case, an iterative approach is used.
Statistical Method
The statistical stack-up method shifts the focus from the possible to the probable. The resulting predictions of assembly dimensions are more realistic provided the data and/or assumptions are realistic. Statistical stack-ups also leave larger tolerances available for assignment. The analysis gives likely values and ranges depending on confidence intervals. The analysis is not a guarantee of conformance to specs because some outliers may exist. Assembly yield rates or defect rates can be predicted.
When a large number of dimensions contribute to a stack-up, the result is typically distributed according to a normal distribution. Even though individual components will be fully screened at inspection, statistical methods predict the likely assembly dimensions more accurately than the more conservative worst-case method.
Statistical stack-up calculations are appropriate when it is known or can be assumed that the contributing dimensions follow a normal distribution. When a dimension follows a normal distribution, most of the values are clustered around the mean value. It is also possible for a dimension to attain a value outside its specification limits, though with low probability. If no quality inspection reports exist for a component being designed, but data for analogous components and manufacturing processes shows a normal distribution, statistical methods may also be applied.
For large production runs, Statistical Process Control (SPC) is typically used to control the manufacturing process and ensure dimensions are under control. SPC controls the process, not the component. Sample components are inspected at intervals and used to determine whether the process is in control. Inspection is not used to determine which components fall within specifications and which do not on a part-by-part basis. Under these conditions, a normal distribution of dimensions typically results. This will typically be true of commodity hardware components like screws.
Using Variables with a Normal Statistical Distribution
The normal distribution for a given dimension is described by two parameters: the mean (μ) and standard deviation (σ). The mean indicates the average value, and the standard deviation is a measure of the variation within the sample. A larger standard deviation indicates more dispersed values of the dimension. Stacking up dimensions described by a normal distribution results in an assembly dimension with a normal distribution. This distribution can then be compared to assembly spec limits to determine the likelihood of a given assembly being out-of-spec, or the percentage of assemblies that will be produced out-of-spec (defect rate).
For the stack-up equation derived in the previous section,
Assembly mean: μA = μB − μC − μD + μE + μF + μG − μH
Assembly standard deviation:
Note that the mean values sum algebraically to the assembly mean whereas the standard deviations combine as the square root of the sum of the squares of the standard deviations. Note also that standard deviations all add as positive terms under the radical. A statistical stack-up using all normally-distributed dimensions is often called the Root Sum of Squares (RSS) method.
For a tolerance assignment problem where an assembly dimension is known, and all dimensions are prescribed but one, the analysis stack-up equation can be rearranged as it was in the worst-case example. If dimension B is desired,
Dimension mean: μB = μA + μC + μD − μE − μF − μG + μH
Dimension standard deviation:
As with the worst-cast stack-up, most assignment problems will require simultaneous assignment of tolerances to several dimensions and an iterative approach. If the calculation of the dimension standard deviation results in a negative number under the square root sign, the assignment problem can not be solved with the tolerances currently chosen. There is insufficient variation remaining to be assigned to dimension B.
Using the Standard 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)
