This embedded calculator will determine the minimum and maximum thickness of an assembly, given the thickness of each item. 

This calculator uses a statistical tolerancing method known as Root Sum Squared (RSS). It assumes every part is manufactured following a Gaussian distribution, i.e. parts deviate from their nominal dimension following a Gaussian curve. If every part follows a Gaussian distribution and the assembly is a linear combination of the parts' dimensions and tolerances, the assembly will also follow a Gaussian distribution (Source).
When using RSS tolerancing, there are 2 probabilistic considerations: 1) manufacturing assumption and 2) design success. 

1. Manufacturing assumption is how often the manufacturer will be within the tolerance provided. Most machine shops can hit ±3σ, meaning they are within the specified tolerance 99.73% of the time, or 27 / 10,000 parts will be out of tolerance. More over,  this implies that you're assuming ~68.26% of parts are within ±1σ: 1/3 of your specified tolerance range, and ~95.46% of parts are within ±2σ: 2/3 of your specified tolerance range.  

2. Design success is determining how often the assembly will fit together, or the success rate of the assembly. It is standard to aim for the same ±3σ, meaning the assembly will fit together 99.73% of the time. If you'd like to have more confidence in how often the assembly will fit, you can design for ±4σ, targeting a 99.9937% success rate. This means that even when setting individual part tolerances to ±3σpart, if your system can handle ±4σassembly deviation, you've effectively increased your assembly process capability, reducing likelihood of failure from ~27 / 10,000 to less than 1 / 10,000. 

To see why RSS solves system variance for independent, combined distributions, see this video. 

This calculator also accounts for mean shifts! Mean shifts account for the possibility that the manufacturing of a part is not centered around your nominal dimension, and that instead, the mean of the part's dimension is shifted. This doesn't affect the shape of the curve, but it does move where the curve lies relative to your nominal dimension and tolerances. See more about mean shifts here. 

A common alternative to this tolerancing scheme is worst-case tolerancing, which this calculator computes in parallel with RSS statistical tolerancing. Worst-case tolerancing is the process of adding all relevant tolerances together, and is very conservative, guaranteeing success in the most extreme and improbable scenarios. Its drawback is that it can lead to extremely tight tolerances and create unnecessarily expensive parts, or worse, make tight assemblies seem impossible. The logic behind RSS tolerancing is that it's extremely unlikely that all parts in an assembly will simultaneously be at their limits in a way that causes an assembly failure. If you can characterize the overall variation of an assembly to some probability, you can make tolerances on individual parts looser with confidence that the assembly will still fit together a high percentage of the time. 

It's important to note that there are limitations with this tolerancing scheme:

1. RSS assumes all the tolerances are independent from one-another. You should verify this before using RSS tolerancing.

2. RSS assumes all the parts in an assembly follow a Gaussian distribution in how they deviate from their nominal dimensions, which is a good rule of thumb, but not always true. 

   • The Central Limit Theorem states that the sum of a large number of independent random variables, regardless of their individual distribution curves, will tend to be normally distributed. This generally applies if your number of independent random variables is over 30, but if your variables are all approximately normally distributed, less variables are needed for the resultant distribution curve to be normally distributed.
     Source

3. RSS relies on dimensions and tolerances being linear combinations, meaning they are directly added or subtracted from one-another. This works great for the calculator above, looking at 1-dimension: thickness and length. However, when you get into 2 and 3 dimensions, tolerance schemes are generally not linear and need a more advanced tolerancing method (Delta Method or Monte Carlo Simulation are common).