Deming's funnel

Deming's funnel is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

W Edwards Deming was an american statistician and management guru who used physical demonstrations to illuminate his management teachings. In one such demonstration he dropped multiple marbles through a funnel at a target, marking where they landed, and observing the resulting pattern. A sequence of "rules" are tested to try to improve performance. In each case, the experiment begins with the funnel positioned directly over the target.

• Rule 1: The funnel remains directly above the target.
• Rule 2: Adjust the funnel position by shifting the target to compensate for the miss. If the last shot missed 1 cm east, move the funnel 1 cm to the west of its current position.
• Rule 3: As rule 2, but first move the funnel back over the target, before making the adjustment. E.g. If the funnel is 2 cm north, and the marble lands 3 cm north, move the funnel 3 cm south.
• Rule 4: The funnel moves directly over the last place a marble landed.

Task: Simulate 50 pseudorandom drops. The radial displacement of the drop from the funnel position is given by a Gaussian distribution (standard deviation is 1.0) and the angle of displacement is uniformly distributed. Apply each rule to the set of 50 drops, and show scatter plots of all four results.

Further information

• Further explanation and interpretation
• Video demonstration of the funnel experiment at the Mayo Clinic.

Racket

<lang racket>#lang racket (require math/distributions plot)

(define radii (map abs (sample (normal-dist 0 1) 100))) (define angles (sample (uniform-dist (- pi) pi) 100))

(define dxs (map (λ (r theta) (* r (cos theta))) radii angles)) (define dys (map (λ (r theta) (* r (sin theta))) radii angles))

(define (funnel dxs dys rule)

 (let ([x 0] [y 0])
   (for/list ([dx dxs] [dy dys])
     (let ([rx (+ x dx)]
           [ry (+ y dy)])
       (set! x (rule x dx))
       (set! y (rule y dy))
       (vector rx ry)))))

(define (experiment label rule)

 (displayln label)
 (plot (points (funnel dxs dys rule))
       #:x-min -15 #:x-max 15 #:y-min -15 #:y-max 15))

(experiment "Rule 1" (λ (z dz) 0)) (experiment "Rule 2" (λ (z dz) (- dz))) (experiment "Rule 3" (λ (z dz) (- (+ z dz)))) (experiment "Rule 4" (λ (z dz) (+ z dz)))</lang>