Deming's funnel

Template:Draft Task American statistician and management guru W Edwards Deming 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 a scatter plot of the result.

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>