PYTHON · NUMPY

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.

Predict firstA batch runs 5% scrap — parts outside the spec band. The fastest LEGITIMATE way to get scrap below 2% is to…

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).

▶ Live · scrub & hover — the process spread and center against the spec limits set the 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.

If your run fails, check:

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, which 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 →

PYTHON · NUMPY

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.

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