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: Apply the four rules to the set of 50 pseudorandom displacements provided as dxs and dys (e.g in the Racket solution). Output: calculate the means, standard-deviations, min and max values of x and y for each rule.

Stretch 1: Generate fresh pseudorandom data. 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.

Stretch 2: Show scatter plots of all four results.

Further information

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

Racket

The stretch solutions can be obtained by uncommenting radii etc. (delete the 4 semi-colons) to generate fresh data, and scatter-plots can be obtained by deleting the #; . <lang racket>#lang racket (require math/distributions math/statistics plot)

(define dxs '(-0.533 0.270 0.859 -0.043 -0.205 -0.127 -0.071 0.275 1.251 -0.231

             -0.401 0.269 0.491 0.951 1.150 0.001 -0.382 0.161 0.915 2.080 -2.337 
             0.034 -0.126 0.014 0.709 0.129 -1.093 -0.483 -1.193 0.020 -0.051
             0.047 -0.095 0.695 0.340 -0.182 0.287 0.213 -0.423 -0.021 -0.134 1.798
             0.021 -1.099 -0.361 1.636 -1.134 1.315 0.201 0.034 0.097 -0.170 0.054 
             -0.553 -0.024 -0.181 -0.700 -0.361 -0.789 0.279 -0.174 -0.009 -0.323
             -0.658 0.348 -0.528 0.881 0.021 -0.853 0.157 0.648 1.774 -1.043 0.051 
             0.021 0.247 -0.310 0.171 0.000 0.106 0.024 -0.386 0.962 0.765 -0.125 
             -0.289 0.521 0.017 0.281 -0.749 -0.149 -2.436 -0.909 0.394 -0.113 -0.598
             0.443 -0.521 -0.799 0.087))

(define dys '(0.136 0.717 0.459 -0.225 1.392 0.385 0.121 -0.395 0.490 -0.682 -0.065

             0.242 -0.288 0.658 0.459 0.000 0.426 0.205 -0.765 -2.188 -0.742 -0.010 
             0.089 0.208 0.585 0.633 -0.444 -0.351 -1.087 0.199 0.701 0.096 -0.025 
             -0.868 1.051 0.157 0.216 0.162 0.249 -0.007 0.009 0.508 -0.790 0.723
             0.881 -0.508 0.393 -0.226 0.710 0.038 -0.217 0.831 0.480 0.407 0.447
             -0.295 1.126 0.380 0.549 -0.445 -0.046 0.428 -0.074 0.217 -0.822 0.491 
             1.347 -0.141 1.230 -0.044 0.079 0.219 0.698 0.275 0.056 0.031 0.421 0.064
             0.721 0.104 -0.729 0.650 -1.103 0.154 -1.720 0.051 -0.385 0.477 1.537 
             -0.901 0.939 -0.411 0.341 -0.411 0.106 0.224 -0.947 -1.424 -0.542 -1.032))

(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/fold ([rxs null] [rys null])
     ([dx dxs] [dy dys])
     (let ([rx (+ x dx)]
           [ry (+ y dy)])
       (set! x (rule x dx))
       (set! y (rule y dy))
       (values (cons rx rxs) (cons ry rys))))))

(define (experiment label rule)

 (define (p s) (real->decimal-string s 4))
 (let-values ([(rxs rys) (funnel dxs dys rule)])
   (displayln label)
   (printf "Mean x, y    : ~a, ~a\n" (p (mean rxs)) (p (mean rys)))
   (printf "Std dev x, y : ~a, ~a\n" (p (stddev rxs)) (p (stddev rys)))
   (printf "x: min .. max: ~a .. ~a\n" (p (apply min rxs)) (p (apply max rxs)))
   (printf "y: min .. max: ~a .. ~a\n\n" (p (apply min rys)) (p (apply max rys)))
   #;(plot (points (map vector rxs rys)
         #: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>

Output:

Rule 1:
Mean x, y    : 0.0004, 0.0702
Std dev x, y : 0.7153, 0.6462
x: min .. max: -2.4360 .. 2.0800
y: min .. max: -2.1880 .. 1.5370

Rule 2:
Mean x, y    : 0.0009, -0.0103
Std dev x, y : 1.0371, 0.8999
x: min .. max: -4.4170 .. 2.4490
y: min .. max: -2.4380 .. 1.9190

Rule 3:
Mean x, y    : 0.0439, -0.0063
Std dev x, y : 7.9871, 4.7784
x: min .. max: -12.5600 .. 12.5840
y: min .. max: -8.2830 .. 8.2670

Rule 4:
Mean x, y    : 3.1341, 5.4210
Std dev x, y : 1.5874, 3.9304
x: min .. max: -0.5330 .. 6.6800
y: min .. max: -0.0370 .. 12.1490