A specific elaboration of Test Phase-Space: A Rudimentary Mathematical Model of Testing Phenomenon in General

UNDER CONSTRUCTION

Note1: the equations presented herein are rudimentary; further inputs / corrections / suggestions are welcome.

Note2: although this article was written from the perspective of software testing, we believe this can be expanded to encompass all testing phenomena in general.

Summary: this article tries to describe testing using a Confidence Scale, with the various stages interrelated through mathematical equations. As a prerequisite, we base on the following assumptions:

Assumption#1.

confidence/level/quality scale: from 1(low) to 10(perfect), effectively divided into three ranges:

Low-confidence(1-3),

Medium-confidence(4-6),

High-confidence(7-9),

5 = median-confidence/level/quality

10 = "perfect-confidence" which will be generally accepted as highly unlikely to be attainable.

(To those familiar with the Kepner-Tregoe method of problem / decision analysis, this scale may sound familiar.)

Assumption#2.

the general relationship from "idea" to "end-user-acceptance" is:

idea > coding.willpower > product > test.iterations > acceptance

we would like to somehow provide a mathematical relationship between these stages.

Equation-A. (idea > coding.willpower > product)

w = coding.Willpower-level

i = idea robustness-level

P = productConfidence

(w*i)/((w+i)/2)=P

this would...

a) put median product confidence at 5, perfect product confidence at 10, and so on:

{1 =< low <=3}, {4 =< med <=6}, {7 =< high <=9} ;

b) forecast that a low idea robustness (level 2) combined with a high codingWillpower (level 8), will still result in a low productConfidence (3) ; with the resolution being to increase idea robustness to increase the confidence level of the output product.

Equation-B. relationship of: {coding.Willpower, coder.skill-set, coder.intention }

w = coding.Willpower

s = coder.Skill-set

N = coder.iNtent}

d = coding duration, similar to test duration -- see below Equation-D

w = ((0.5*s)+ln(N))*(0.649*ln(d))

= ((0.5*s)+ln(N))*(0.649*ln(C/c))

Example B.
coding.willpower	skill	iNtent	d=C/c
10.91282	10	10	10
9.55023	9	9	9
7.31638	9	6	6
7.07046	9	3	7	a low intent, given enough time, may generate enough codingwillpower, provided there is an ultra-high level of skill involved
6.85500	6	8	8
6.38146	9	5	5	a high skillset, coupled with only median intent and median duration is here predicted to score less than a upper-medium skillset with higher intent and duration
6.24617	6	7	7	high intent (and an increase in coding duration) can actually grow a skill set, accelerating an increase in codingwillpower
5.86005	5	5	9
5.68303	9	1	7	an ultra low intent, even when given a reasonably sufficient coding duration might yield just enough as a 5-7-7 scenario
5.61472	5	7	7	median skill-level, with high intent, coupled with a substantial coding duration will provide just enough push for codingwillpower
5.57211	6	6	6	an upper medium skillset, with an upper medium intent, will be limited by the coding duration in which they would operate
5.32523	4	7	8	if a low skillset grows, coding willpower may increase given enough time
4.99068	5	6	6
4.81467	6	5	5
4.54467	5	3	7	high skill-level or median skill level with low intent, will not produce enough codingwillpower
4.35183	3	7	7	fairly-high intent, with low skill-level, will suffer from the constraints of the limited skillset available, even with a prolonged coding period
4.29241	5	5	5	median skill-level, with median intent, and operating on quite a limited time will not provide enough push for codingwillpower (a half-hearted job will not be very productive)
3.82783	3	6	6
3.24789	3	5	5
2.92231	6	3	3
2.56581	5	3	3
1.85281	3	3	3

from this we would concede that satisfactory product outputs mostly are created/extracted from the upper-mid to high range of this scale.

Equation-C. correlation of: (product > test.iterations > acceptance)

r = testing.iterations

T = TestLevel applied in the iteration(see below equation-E)

P = productConfidence

A = AcceptanceConfidence

r(T(P)/(T+P))=A

can be rewritten as,

0.84*(ln(r+5)*(T(P)/(T+P)))=A

this last equation introduces a logarithmic “saturation” limit for successive iterations “0.84*(ln(r+5)…” for the exact same unchanging contributors -- meaning if all other factors (T,i,w) remain unchanged, there would be an increasing level of AcceptanceConfidence as iterations increase, but eventually reach a saturation limit (where increasing iterations provide marginally less info than the previous runs.)

note that every time an iteration is run, if there are changes in any of the contributing factors, the r-value reverts to 1.

So, for the following values,

r = test-iterations = 1

T = TestLevel applied in the iteration = 7

P = productConfidence = 6.46154 {I=6 , w=7}

A = AcceptanceConfidence would be logarithmic, demonstrating increasing confidence as test iteration increases, but each successive iteration adds marginally less than the test results of previous runs (so long as the contributing factors remain the same).

Examples:

Example Ca.

AcceptanceConfidence of a fairly-high testlevel effort, applied to an upper-medium thought-out-idea, and coded with a fairly-high level of coding willpower. (This would suggest that software products with high AcceptanceConfidence scores are achievable only with fairly high levels of coding effort and test effort.)