An assembly dimension often depends on several manufactured dimensions acting in series. Their variations therefore stack up.

For a simple chain

$$Y=A+B-C,$$

the nominal assembly dimension is calculated with the same signed relationship.

If each component has independent bilateral worst-case tolerance $\pm t_A$, $\pm t_B$ and $\pm t_C$, the worst-case tolerance is

$$t_Y=t_A+t_B+t_C.$$

Suppose

$$A=20.00\pm0.05,\quad B=10.00\pm0.03,\quad C=5.00\pm0.02\ \mathrm{mm}.$$

Then

$$Y=25.00\ \mathrm{mm}$$

and

$$t_Y=0.05+0.03+0.02=0.10\ \mathrm{mm}.$$

So worst-case assembly range is $25.00\pm0.10$ mm.

If component variations are statistically independent and approximately centred, a root-sum-square estimate may be used for variation rather than guaranteed worst-case bounds:

$$t_{RSS}=\sqrt{t_A^2+t_B^2+t_C^2}.$$

This gives a smaller number but represents a probabilistic assumption, not a guaranteed limit.

Tolerance allocation is therefore a design-manufacturing trade-off. Tightening every component tolerance raises process and inspection cost; allowing too much variation can make assembly function unreliable. Stack-up analysis identifies which dimensions actually control the functional requirement.