PYTHON · NUMPY

Tolerance stack-up

Predict whether a chain of toleranced parts will assemble using both worst-case and statistical (RSS) stacks, and state when each method applies.

01Challenge

A 3-part stack (plate 6.000, spacer 9.000, cap 5.000) drops into a housing pocket of depth 20.200; the retaining-clip gap must stay in 0.10–0.50 mm. With supplier tolerances (plate ±0.10, spacer ±0.12, cap ±0.08, housing ±0.05), will the clip always fit? Compute the worst-case range, then run Monte-Carlo 5000 and notice how much head-room the statistical answer gives. Tighten the cheapest tolerances until guaranteed.

Predict firstThree stacked parts each meet their own tolerance. Will the assembled gap ALWAYS stay in spec?

02Model

Three parts (six, nine, five) sum to twenty, leaving a two-tenths gap; the clip fits at nominal. But nothing is at nominal, and each part's little band stacks. The pessimist's method, worst-case: assume every part lands at its worst extreme at once and just add the tolerances. 0.10+0.12+0.08+0.05 = 0.35 mm of swing, which can eat the whole gap. Worst-case is a guarantee: fit here and it fits always.

But all four hitting their worst extreme simultaneously is wildly unlikely; they're independent draws. So the statistical stack combines them as root-sum-of-squares: √(0.10²+0.12²+0.08²+0.05²) = 0.18, half the worst-case. Both are true. Worst-case never fails but forces tight, costly tolerances; RSS is realistic and cheaper but admits a tiny reject rate.

A part must land inside its tolerance band.

03Guided practice

Step A (worked): nominal gap 0.200; worst-case Σ|tol| = 0.350 → [−0.150, 0.550] FAIL; RSS √Σtol² = 0.1825 → [0.018, 0.382], still fails low but far less.
Step B (fade): compute worst-case sum and RSS for a new supplier set; the Monte-Carlo overlay confirms RSS predicts the cloud width.
Step C (independent): make the stack worst-case-safe at lowest cost (tightening the largest contributor, the spacer at 0.12, gives the most relief per cost unit), and state one scenario to ship the RSS answer.

04Feedback

PASS WHEN Worst-case gap range ⊆ [0.10,0.50] (guaranteed fit), typed worst-case sum and RSS each within ±0.005 mm of reference, the tightening costs ≤1.6× the cheapest worst-case-safe solution, and a correct RSS-vs-worst-case statement, on MC seed 303.

If your run fails, check:

Worst-case gap is [−0.05, 0.45]: the low end is negative, so in the worst extreme the parts interfere. Tighten until the worst-case low ≥ 0.10 mm.
You entered 0.35 for RSS, but that's the linear (worst-case) sum. RSS = √(0.10²+0.12²+0.08²+0.05²) = 0.182. Re-enter the RSS value.
You tightened the cap (0.08), but the spacer (0.12) is the largest contributor; tightening it removes the most worst-case swing per cost unit.
The assembly is guaranteed, but you tightened every part to ±0.03 at cost 4.1×. You only needed the spacer and plate; loosen the cap back toward ±0.08.

05Retrieve & space

From 3.2: if a tool-wear drift shifts one part's mean off-nominal, is the RSS prediction still trustworthy? → no; RSS assumes centered/random, a systematic drift biases the stack.
From 3.1: roughly how much does halving the spacer's band cost? → ~doubles that part's cost share.
From M2: which sequencing/setup choice helps hold a tight thickness tolerance? → single-setup machining of the stack faces.

06Mastery & project

On a fresh 3-part stack seed with new nominals and gap, compute both worst-case and RSS ranges (±0.005), tighten so the stack is worst-case-guaranteed at ≤1.6× the cheapest such solution, and state when each method applies. Unlocks Module 4.

Feeds the M5 capstone: the worst-case-vs-RSS decision becomes a line-level yield lever; the same √Σtol² logic scales from a 3-part stack to the full assembly tolerance budget the capstone factory must satisfy.

← Where variation comes from Assembly sequencing →

PYTHON · NUMPY

Tolerance stack-up

Both stack methods plus a deterministic 5000-draw Monte-Carlo, exactly as the autograder uses. Edit the tolerances to guarantee the fit.

You read

the arrays and values already in scope

You change

the code you write in each cell

Fixed

the dataset and the checks that grade you