Deming's funnel

W Edwards Deming was an American statistician and management guru who used physical demonstrations to illuminate his teachings. In one demonstration Deming repeatedly dropped marbles through a funnel at a target, marking where they landed, and observing the resulting pattern. He applied a sequence of "rules" 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 after each drop. E.g. If the last drop 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 of the target.
Rule 4: The funnel is moved directly over the last place a marble landed.

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

Note that rules 2, 3, and 4 give successively worse results. Trying to deterministically compensate for a random process is counter-productive, but -- according to Deming -- quite a popular pastime: see the Further Information, below for examples.

Stretch goal 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 goal 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\n" (p (stddev rxs)) (p (stddev 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

Rule 2:
Mean x, y   : 0.0009, -0.0103
Std dev x, y: 1.0371, 0.8999

Rule 3:
Mean x, y   : 0.0439, -0.0063
Std dev x, y: 7.9871, 4.7784

Rule 4:
Mean x, y   : 3.1341, 5.4210
Std dev x, y: 1.5874, 3.9304