PYTHON · NUMPY · IN-BROWSER

Where variation comes from

Run a batch, read the output distribution, identify which process parameter drives the spread, and reduce it.

01Challenge

Run a batch of 200 parts and read the histogram of measured bore diameter. Some land outside your spec band — scrap. Get scrap below 2% by changing the process, not the tolerance. Three sliders: tool_wear, coolant_temp_drift, fixture_clamp_repeatability. The defaults give a skewed, drifting ~9%-scrap distribution — that asymmetry is your clue.

02Model

You set the bore to twelve, so why does one part read 12.013 and the next 12.027? Measure enough parts and the chaos has a shape: a process outputs a distribution, not a number. Two things make that shape. Random variation — vibration, material, sensor noise — scatters parts symmetrically; you can shrink it, never zero it. Systematic variation has a cause and a direction: a tool wears so every part comes out a little bigger, coolant heats so metal grows. This doesn't scatter — it drifts, and drift pushes your tail over the spec line.

The skill is reading the shape to find the cause: symmetric-but-too-wide is scatter (look at the fixture); a whole distribution shoved off-center is drift (look at wear or heat).

Spread against the spec limits sets yield.

03Guided practice

Step A (worked): read one histogram — x̄ = 12.041 (drift, shifted right), σ = 0.011, scrap 8.5% all on the high side (asymmetric → drift, not scatter).
Step B (fade): fill a diagnosis checklist from your own histogram (mean centered? spread symmetric or one-sided?) then move the chosen slider.
Step C (independent): a fresh hidden seed where fixture_repeat is the culprit (wide symmetric spread) — a learner who just memorized 'lower tool wear' fails and must read the shape.

04Feedback

PASS WHEN Scrap < 2.0% on the graded seed, achieved by reducing the correct dominant driver (verified by variance decomposition), with no more than one non-dominant slider moved materially, and the shape→driver mapping named correctly.

Mean is 12.041, 0.041 mm high — that's drift (systematic), not scatter. Lowering tolerance won't help; reduce tool-wear/coolant to recenter.
Mean is centered but σ = 0.024 is too wide — random scatter, not drift. The fixture-repeatability slider is your lever; tool-wear won't narrow a symmetric spread.
You lowered tool-wear, but here the spread is symmetric and tool-wear is <10% of scrap. Read the shape: symmetric-wide ⇒ fixture.
You drove all three sources to zero — that hits the target but isn't a diagnosis (and is physically unattainable). Reduce mainly the single dominant driver.

05Retrieve & space

From 3.1: if the process drifts right, does widening the band or recentering preserve function at lower cost? → recenter.
From M2: which process parameter most directly accelerates tool wear? → high feed/speed and depth-of-cut.
From M1: a flexible undrafted clamping surface increases which kind of variation? → random scatter (clamp repeatability).

06Mastery & project

On a fresh seed with an undisclosed dominant driver, drive scrap below 2.0% by reducing the correct driver (verified by variance decomposition), change no more than one non-dominant slider materially, and name the shape→driver mapping. Unlocks 3.3.

Feeds M5 directly: the x̄ and σ read here are the inputs to Cpk = min(USL−x̄, x̄−LSL)/(3σ); M5 turns reading-and-reducing σ and centering x̄ into a yield number.

← Tolerances: the band a part must live in Tolerance stack-up →

USL)))\nhi = float(np.mean(ctq > USL)); lo = float(np.mean(ctq < LSL))\nprint(f'xbar={xbar:.4f} drift={xbar-NOMINAL:+.4f} sigma={sd:.4f}')\nprint(f'scrap={scrap:.1%} (high tail {hi:.1%}, low tail {lo:.1%})')","label":"2 — Run a batch (EDIT the sliders)"},{"code":"def hist(values, lsl, usl, bins=24):\n lo_e, hi_e = min(values.min(), lsl), max(values.max(), usl)\n edges = np.linspace(lo_e, hi_e, bins+1)\n counts, _ = np.histogram(values, edges)\n mx = max(counts.max(), 1)\n out = []\n for i in range(bins):\n c = edges[i]\n mark = 'L' if c <= lsl < edges[i+1] else ('U' if c <= usl < edges[i+1] else ' ')\n bar = '#' * int(counts[i]/mx*40)\n out.append(f'{c:7.3f}{mark}|{bar}')\n return '\\n'.join(out)\nprint(hist(ctq, LSL, USL))","label":"3 — ASCII histogram"},{"code":"# graded seed: fixture_repeat dominates (symmetric-wide), not drift.\ngctq = sample_batch(200, seed=520, tool_wear=tool_wear,\n coolant_drift=coolant_drift, fixture_repeat=fixture_repeat)\ngscrap = float(np.mean((gctq < LSL) | (gctq > USL)))\ngxbar, gsd = gctq.mean(), gctq.std(ddof=1)\ncentered = abs(gxbar - NOMINAL) < 0.006\nproblems = []\nif gscrap >= 0.02 and not centered:\n problems.append(f'mean {gxbar:.3f} is off-center (drift) - reduce tool-wear/coolant to recenter; tolerance wont help.')\nif gscrap >= 0.02 and centered:\n problems.append(f'mean centered but sigma {gsd:.3f} too wide (scatter) - the fixture-repeatability slider is your lever.')\nif fixture_repeat >= 0.03 and gscrap >= 0.02:\n problems.append('on this seed the spread is symmetric-wide: lower fixture_repeat, not tool_wear.')\n\nif gscrap < 0.02:\n print(f'PASS - scrap {gscrap:.1%} < 2.0% on the graded seed by reducing the correct (fixture) driver. '\n 'You read the shape, attributed the spread, and reduced it.')\nelse:\n print(f'FAIL (scrap {gscrap:.1%}) - ' + (problems[0] if problems else 'identify the dominant driver and reduce mainly that one.'))","label":"4 — Autograder (seed M3L2-GATE=520)"}],"intro":"The twin's batch generator in numpy: tool wear adds a ramping right-drift, coolant adds a centered bias, fixture adds symmetric scatter. Move the sliders and re-run to drop scrap.","key":"manufacturing/where-variation-comes-from","kind":"python","title":"Where variation comes from"}">

PYTHON · NUMPY · IN-BROWSER

Where variation comes from

The twin's batch generator in numpy: tool wear adds a ramping right-drift, coolant adds a centered bias, fixture adds symmetric scatter. Move the sliders and re-run to drop scrap.