Sums of Random Variables and Tolerance Stack-Up

Tolerance stack-up is simply working out how much a gap or a total length varies when it's built from several parts that each vary a little. The formulas look heavy, but there are only two ideas. Averages add: $E(aX + bY + c)$ is just the parts' average sizes added up, with a minus sign for parts that close the gap. Variances add: $a^2\sigma_X^2 + b^2\sigma_Y^2$ (each part's variance, the minus signs squared away), plus a covariance term that's zero when parts vary independently. The worst-case and RSS formulas, and the fit probability, are those two ideas put to work.

A gap built from many parts. A gearbox shaft carries two bearings, a spacer and a gear hub between two shoulders. Every part comes out a little different, so the end play, the gap left over, changes from gearbox to gearbox. Too tight and the shaft binds when it warms up; too loose and it rattles. The design question: given how much each part varies, how much does the gap vary, and what fraction of gearboxes will be in spec?

Write the assembly as a sum. Every one-dimensional stack is a linear combination: gap = shoulder distance − bearing − bearing − spacer − hub, so $G = \sum a_iX_i$ with $a_i = +1$ for dimensions that open the gap and $a_i = -1$ for those that close it. Drawing this loop of dimensions, with a sign on each, is the first step of every stack-up.

Means add. Expectation is a sum or an integral, so it's linear: $E(G) = \sum a_i\mu_i$. Nominal sizes stack exactly the way you'd expect.

Variances add, even for differences. In Measurement Error you saw two independent pieces add their variances. Here is the general rule. Write each part as its mean plus a random deviation, $X = \mu_X + \delta_X$ and $Y = \mu_Y + \delta_Y$, with $E(\delta) = 0$. Then

The first two terms are $a^2\sigma_X^2$ and $b^2\sigma_Y^2$, and the last is the covariance term, which is 0 for independent parts. The coefficients get squared, so a part that subtracts ($a = -1$) still adds its variance. A hole's variation and a pin's variation both make the clearance less predictable.

Why standard deviations add like a right triangle. Independent deviations behave like perpendicular vectors: their sizes combine by Pythagoras, $\sigma_G = \sqrt{\sigma_1^2 + \sigma_2^2 + \cdots}$. Ten parts with σ = 10 µm each give $\sigma_G = 10\sqrt{10} \approx 32$ µm, not 100 µm, because the deviations partly cancel: some parts run large while others run small.

Two ways to tolerance a stack.

• Worst case: every part sits at its limit, all pushing the same way: $T_{\text{wc}} = \sum T_i$. Every assembly fits, but for $n$ equal parts the stack grows like $n$, so each part gets only $T/n$.
• Root-sum-square (RSS): if each tolerance is $\pm3\sigma_i$, the stack's $\pm3\sigma$ is $\sqrt{\sum T_i^2}$, which grows like $\sqrt n$. Each part can be $\sqrt n$ times looser, at the price of about 0.27% of assemblies landing beyond $\pm3\sigma$.

The probability an assembly fits. A sum of independent normal parts is exactly normal, and a sum of several independent parts of any shape is close to normal by the central limit theorem. So $G \approx N(\mu_G, \sigma_G)$, and the fraction in spec is a difference of two normal areas, just as in Using the Normal Distribution.

When RSS is too optimistic. It assumes independent, centered parts. Parts cut by the same worn tool or taken from the same batch are correlated, so the $2ab\operatorname{Cov}$ terms come back. A process mean that has drifted shifts $\mu_G$. And a stack of just two or three flat (uniform) distributions isn't normal yet. Many design guides inflate the RSS answer by a factor of about 1.5 to cover all this, and the Monte Carlo lesson checks it directly.

A common mistake. When a dimension subtracts from the gap, it's tempting to subtract its variance too. It isn't subtracted: the coefficient $-1$ gets squared, so every part adds its variance, whichever way it pushes. Two cousins of this slip: adding standard deviations instead of variances (that's the worst case in disguise), and treating one part counted twice as two independent parts, when $\operatorname{Var}(2X) = 4\sigma^2$ but $\operatorname{Var}(X_1 + X_2) = 2\sigma^2$.

The same algebra runs the rest of this unit: Propagation of Error applies it to curved formulas, and process capability later asks whether each part's σ is small enough to spend.

Tolerance stack-up is everyday work in mechanical design: gearboxes, engine blocks, phone housings and aircraft panels all have critical gaps built from dozens of parts. Electrical engineers stack resistor and voltage-reference tolerances the same way, and chip fabs budget the misalignment between layers as a root-sum-square of tool, mask and wafer contributions. Choosing RSS over worst case where it's safe is one of the biggest cost levers in manufacturing, and knowing when it isn't safe (correlated or drifting parts) is what keeps it honest.