Jou nal o Labo a o y Chemical Educa ion 2021, 9(2): 26-35
DOI: 10.5923/j.jlce.20210902.02
Noise-injec ion as an App oach o Gene a ing Random
Da a Se s o Online Tes s and Vi ual Labs
D. González-A jona1,*, M. M. Domínguez1, G. López-Pé ez1, W. H. Mulde 2
1Depa men o Physical Chemis y, Uni e sidad de Se illa, Se illa, Spain
2Depa men o Chemis y, The Uni e si y o he Wes Indies, Mona Campus, Jamaica
Abs ac A me hodology based on noise injec ion o gene a ion o andomized abula ed da a is p esen ed. The s a egy
can be used bo h in online eaching and o speci ic nume ical/g aphical exe cises, when indi idualized da a se s a e equi ed
simul aneously o he s uden s. Res ic ions imposed on eaching me hods du ing o he SARS-Co id-19 pandemic,
especially o labo a o y sessions in chemis y o e en o p epa ing w i en exams, ha e led o a need o app oaches based
on andomized da a se s based on li e a u e da a o heo e ical equa ions. Commonly a ailable sp eadshee so wa e has been
used o gene a ing andom da a and o analysis and calcula ions, which acili a es easy and low cos applica ion o he
me hodology p esen ed he e. Uni o m and Gaussian dis ibu ions ha e been employed o gene a e di e en ypes o noise.
S a is ical analyses on linea eg ession pa ame e s o he di e en dis ibu ion and le els o injec ed noise ha e been
pe o med. As examples, hese esul s a e employed o in oduce andomness in h ee ypical expe imen s pe o med in
Physical Chemis y labs in ol ing he modynamics, chemical kine ics and conduc i i y o elec oly e solu ions. Li e a u e
alues a e employed o he expe imen s as empla es o which di e en le els o noise a e applied. The esul s indica e ha
he applica ion o noise has o be ca e ully con olled. Uni o m noise is sugges ed o da a se s ha al eady con ain na u al
andom noise, whe eas Gaussian noise should be employed o da a se s c ea ed di ec ly om heo e ical o empi ical
equa ions, so as o p oduce da a se s wi h a mo e na u al, ealis ic appea ance.
Keywo ds Random noise-injec ion, Online lea ning, Vi ual labs
1. In oduc ion
Objec i e: To build andom da a se s om well- ounded
heo ies o expe imen al da a o be used in on-line lea ning
wi h examples om he a ea o Physical Chemis y.
Usually li e a u e da a a e employed o gene a e
nume ical o g aphical exe cises in highe ( e ia y)
educa ion. F om hese da a se s, by using he app op ia e
heo e ical backg ound, he a ge pa ame e s a e ob ained.
Some imes, i is necessa y o cons uc a plo o ob ain hese
pa ame e s o an in e media e esul .
Nowadays, online lea ning is commonly p ac iced and
in ol es he co esponding online examina ions. This mode
o cou se deli e y has now e en become he s anda d in
hese imes o lockdowns as a esul o he CoVID-19
pandemic. This has made i ine i able o pe o m
examina ions emo ely while a he same ime s uden s ha e
ull access o In e ne and social media o ob ain
in o ma ion. Unde hese condi ions, ypical es esul s do
no necessa ily e lec he s uden ’s ue knowledge abou
* Co esponding au ho :
[email p o ec ed] (D. González-A jona)
Recei ed: May 26, 2021; Accep ed: Jun. 25, 2021; Published: Jun. 30, 2021
Published online a h p://jou nal.sapub.o g/jlce
he opic being examined. Thus, i seems ad isable o use
some ools o s a egies o limi ins ances o chea ing.
In ac , some web-based lea ning pla o ms [1,2], p o ide
ools o andomize on-line es s. Fo example, ques ions can
be andomly selec ed om a ques ion bank. Fu he mo e, in
mul iple-choice ques ions he o de o he op ions can also
be sc ambled. In he case o nume ical p oblems whe e a
o mula has o be employed o pe o m calcula ions, i is
possible o gene a e a numbe o di e en se s o inpu
pa ame e s aken om a selec ed in e al, o example
empe a u e and p essu e.
Ano he s a egy ha can be employed o minimize
oppo uni ies o chea ing is o change he ph asing o a
nume ical p oblem, while main aining he s uc u e o a
ques ion, e.g. he applica ion o a pa icula o mula, ha is,
o di e si y he ex e nal appea ance o he nume ical
p oblem. Addi ional nume ical p oblem se s can be easily
cons uc ed, jus by changing he inpu da a uni s and/o
dimensions.
In many chemis y p oblems and exe cises i is equen ly
equi ed o ob ain he inal o a pa ial esul om he
pa ame e s ex ac ed om a, usually linea , plo . This kind
o scena io is no commonly implemen ed in web-based
lea ning pla o ms. Mo eo e , hese kind o exe cises,
cons uc ion o g aphs, a e e y common in ace- o- ace
Jou nal o Labo a o y Chemical Educa ion 2021, 9(2): 26-35 27
eaching labs, whe e a linea eg ession analysis is
usually o be pe o med. Un o una ely, du ing las yea ’s
lock-downs, no ace- o- ace labs we e held. The e a e some
so wa e pla o ms [3-6, and e e ences he ein], ha
p o ide simula ed lab expe imen s o i ual labs, ha can
be used as p epa a o y aining p io o he eal lab
expe ience. This pa ial solu ion necessi a es on he one
hand o modi y some opics and secondly, in ol es wo
ex a cos s, an economic one and a cos in e ms o ime
in es ed by he ins uc o in implemen ing he new
scena io.
Ob iously, he ace- o- ace lea ning mode is essen ial o
a lab-based p og am. Video u o ials can help, bu can ne e
subs i u e o he hands-on expe ience. In o de o educe
he loss incu ed as a esul o hese lea ning objec i es
being comp omised, a di e en app oach has been adop ed
las yea . A new amewo k was designed o each lab
expe imen , adap ed o he new lea ning en i onmen . This
included ques ions abou he s ep by s ep p ocedu e o he
expe imen as well as p oblems a ising om some unusual
ou comes ha could be ob ained on an ac ual lab day. Da a
se s we e p o ided o indi idual s uden s o wo k wi h,
wi h andomized o iginal lab da a. Thus, some noise was
supe imposed on he o iginal da a, while main aining he
signi icance and a iabili y o he esul s ha can be
ob ained om hese new andomized da a.
In his pape , he use o con olled noise injec ion o
gene a e a clus e o andomized abula ed da a, o use in
online eaching, is discussed. The p ocedu e will be
analyzed s ep-by-s ep and illus a ed wi h some examples.
Fi s , a b ie in oduc ion abou he di e en kinds o noise
o be employed, o he abo e men ioned pu pose, and i s
ela ionship wi h he basic concep s o accu acy and
p ecision, will be p esen ed. Nex , he use o sp eadshee
so wa e o cons uc di e en noise dis ibu ions will be
explained and e alua ed. And inally, he p ocedu e will be
explained and applied o i ualize some lab expe imen s
and g aphical exe cises in he ield o Physical Chemis y.
The s a egy in oduced in his pape has been success ully
employed since he las academic yea o online es s and
lab i ualiza ion. The me hodology p o ides a as and
inexpensi e way o de eloping andomized da a se s wi h
ealis ic and eliable esul s o use in online es s.
2. Gene a ing Random Digi al Noise
The undamen al limi o he esolu ion and accu acy and
model analysis om da a is hei le el o noise. The noise
can be ca ego ized acco ding o he ocus o in e es . Thus in
ins umen a ion he noise is ca ego ized by he in e e ing
sou ce: powe lines, empe a u e e ec s on he ins umen s
and senso s and andom noise om he ins umen i sel .
This las ype o noise can be analyzed using s a is ical and
p obabilis ic p inciples.
Thus, he goal in Digi al Signal P ocessing is o elimina e
he in e e ing noise wi hou al e ing he signal o in e es .
The cha ac e iza ion o noise and i s le el is a undamen al
ask. S a is ical analysis can be applied o he signal. Thus,
he mean alue and s anda d de ia ion p o ide in o ma ion
abou he signal a e age le el and sp ead. These wo
pa ame e s can u nish in o ma ion abou he accu acy and
p ecision o he signal. La ge de ia ions o he mean om
he ue alue and la ge s anda d de ia ions indica e lowe
signal quali y o high noise le el. I he sou ce o noise is
iden i ied, he quali y o he signal can be imp o ed by
emo ing he sou ce. Bu andom noise can only be pa ially
il e ed ou o minimized and special ca e has o be aken
when il e ing o a oid dis o ion o he signal unde s udy.
Using hose basics concep s om Digi al Signal P ocessing
(DSP), he in e se me hodology has been employed o
noise injec ion in o a p e iously gene a ed da a se [7]. The
amoun and ype o andomized noise ha e o be con olled
o a oid comp omising he da a analysis, keeping in mind
he a ge o o e ing ealis ic indi idualized da a o each
s uden .
The e a e many s anda d p obabili y dis ibu ions
(Binomial, Poisson, Uni o m, Chi-Squa ed, Gaussian,
Be noulli, Logno mal…) and mos o hem can be gene a ed
om andom numbe s [8]. He e, he Uni o m and Gaussian
dis ibu ions will be employed o gene a e he andom alues
o be injec ed in o da a se s.
The Uni o m dis ibu ion applies o a ini e numbe
in e al, usually any alue be ween 0 and 1. The main
cha ac e is ic is ha selec ing any numbe in his ange
has he same p obabili y, in his case wi h a mean alue
o 0.5 and a a iance o
1 12
. This kind o dis ibu ion is
also known as ‘whi e noise’, and has been employed in
C yp og aphy and in Mon e Ca lo simula ions.
The o he well-known dis ibu ion is he Gaussian o
no mal dis ibu ion. In his case he alues a e dis ibu ed
so as o c ea e a bell-shape a ound a mean alue ha
coincides wi h he mode, and p obabili ies alling o
exponen ially o alues away om he mean. This ype
o dis ibu ion is commonly ound in many na u al
p ocesses, e.g. i s exponen ial cha ac e is encoun e ed
in he Maxwell-Bol zmann dis ibu ion, applicable o
physicochemical p ocesses.
These wo kinds o dis ibu ion can be easily gene a ed
by using sp ead-shee p og ams. The well-known Excel o
Lib eO ice p og ams can be used in e changeably. The
andom numbe algo i hm employed he e has been imp o ed
o he la es e sion o Excel wo kshee [9]. Despi e he
imp o emen o he algo i hm, his is no ye ecommended
o use in p o essional c yp og aphy no o Mon e Ca lo
simula ions, bu i is pe ec ly adequa e o mee ing he
objec i es o his pape .
Excel uses he RAND() unc ion o gene a e a eal numbe
be ween 0 and 1. E e y ime he wo k shee is modi ied, a
new andom numbe is gene a ed o all he cells which
con ain ha unc ion. The numbe gene a ed has a uni o m
dis ibu ion wi h a mean alue o 0.5 and a s anda d
de ia ion o
1 12
[10].
28 D. González-A jona e al.: Noise-injec ion as an App oach o Gene a ing
Random Da a Se s o Online Tes s and Vi ual Labs
Ano he popula way o gene a ing uni o m dis ibu ions
is by using he modulus ope a ion (MOD) [11]. This me hod
has some weaknesses when i comes o gene a ing uly
andom sequences, bu o he objec i es o his pape i is
s ill use ul.
The Cen al Limi Theo em can be in oked o gene a e he
no mal dis ibu ion wi hou using any new Excel unc ion.
Simply adding wel e andom numbe s be ween 0-1, an
excellen app oxima ion o a no mal dis ibu ion is ob ained
wi h mean alue o six (sum o each indi idual uni o m
mean) and one as s anda d de ia ion (squa e oo o sum o
he a iances) [12]. This dis ibu ion can be easily modi ied
o p oduce selec ed mean and s anda d de ia ion alues.
The e a e some o he algo i hms, based in he Box-Mulle
T ans o ma ion, capable o gene a ing s anda d no mal
dis ibu ions om jus wo uni o m dis ibu ions:
2ln(RAND()) cos(2 RAND())x


[13].
This algo i hm has he ad an age o a sho e de ini ion o
he cons uc ion unc ion ha simpli ies w i ing i in
sp eadshee , hence minimizing he isk o ypos.
Figu e 1. Sp eadshee algo i hms employed o he ype o da a se
dis ibu ion gene a ion wi h a selec ed mean and s anda d de ia ion
Da a se s, ha ing mo e han 5000 poin s, ollowing bo h
dis ibu ions, Uni o m (U) and No mal (N), ha e been
gene a ed by he di e en equa ions displayed in Figu e 1.
Al hough he mean and s anda d de ia ion can be selec ed
a bi a ily, in his wo k ze o and one, espec i ely, we e
always chosen.
Table 1. Selec ed s a is ics pa ame e s o he di e en dis ibu ions
gene a ed
Table 1 collec s some s a is ical pa ame e s (mean,
s anda d de ia ion and max and min alues) o he di e en
da a se s gene a ed wi h he algo i hms desc ibed abo e. I
can be seen ha bo h algo i hms o uni o m dis ibu ion
pe o m iden ically and p oduce he expec ed esul s,
especially he s anda d de ia ions ob ained. Bo h algo i hms
using he no mal dis ibu ion also p o ide he expec ed
esul s. The numbe o RAND imes employed in he unc ion
is also shown in Table 1, showing hei in luence on he
s a is ical pa ame e s o he dis ibu ions. Thus, an inc ease
in he use o he RAND unc ion implies a highe alue o he
s anda d de ia ion and b oade ange o he ex eme alues
max and min.
Figu e 2 shows ha i is ha d o dis inguish be ween he
wo ypes o dis ibu ions, uni o m o no mal, jus by isual
inspec ion o he sequence o alues. The cons uc ion o a
his og am c ea ing a se ies o bins (class in e als) wi h an
inc emen o 0.02 each and so ing alues acco ding o he
in e als (bins) in which hey occu , leads o he dis ibu ions
shown in Figu e 3. This ope a ion can be easily pe o med in
Excel by using he Da a Analysis ool.
Figu e 2. Sca e ed alues a ound ze o ob ained by using he uni o m dis ibu ion unc ion scaled be ween -0.5 and 0.5, U, and a no mal dis ibu ion
algo i hm, N. Bo h ha e he same  = 1/√12
Jou nal o Labo a o y Chemical Educa ion 2021, 9(2): 26-35 29
Figu e 3. Compa a i e his og ams o di e en dis ibu ions, wi h ze o as
mean alue. Red: Uni o m,  = 1/√12; Blue: No mal,  = 1/√12; Viole :
No mal,  = 1/(2√12) and G een: No mal log based:  = 1/(2√12)
Figu e 3 shows he his og ams o di e en ypes o
dis ibu ions. The alues o he uni o m dis ibu ion ( ed
ba s) a e andom dis ibu ed showing no pa e n, wi h
hei alues pe ec ly con ined be ween -0.5 and +0.5.
Ne e heless he no mal dis ibu ion (blue ba s) wi h he
same s anda d de ia ion,  = 1/√12, p oduces he
bell-shaped pa e n, bu he e a e alues g ea e han 0.5 and
below -0.5. These la e alues accoun o app oxima ely
5% o he en i e se . The iole and g een his og am a e
ob ained o a no mal dis ibu ion using RAND unc ion and
he Box-Mulle ans o ma ion, bo h scaled o a s anda d
de ia ion 1/(2√12), hal o he blue his og am. Bo h
his og ams a e independen o he algo i hm employed.
Mo eo e , wi h s anda d de ia ion alue selec ed, less han
0.5% o he alues lays ou side o he -0.5/+0.5 in e al. The
selec ion o he dis ibu ion ype and i s scaling is ou nex
ask.
3. Analysis o he In luence o Noise
Type on Linea Reg ession
A common ask pe o med by s uden s is o c ea e a
plo om he da a o ex ac in o ma ion abou a physical
phenomenon. Among he di e en kinds o g aphical
analyses, linea eg ession is he mos equen ly
encoun e ed p ocedu e. In o de o analyze he in luence o
noise on he linea pa ame e s, a con olled le el o andom
noise has been in oduced in o he da a along bo h coo dina e
axes. The amoun o noise in oduced a each da a poin is se
o a selec ed pe cen age o he ac ual alue, hence he new
alue luc ua es andomly a ound he o iginal alue.
The uni o m dis ibu ion p o ides easy con ol o he
ange o ou pu alues, and has he e been se o a y be ween
-0.5 and 0.5, so wi h a mean alue o ze o. Thus o each da a
poin , a andom pe cen age o luc ua ion a ound i s ac ual
alue can be added, employing he equa ion:
 
New da a Ac ual da a 1 (RAND (-0.5 0.5)) % noise/100    
whe e, RAND(-0.5↔0.5), ep esen s he algo i hm o ob ain
a eal numbe be ween -0.5 o 0.5 wi h a selec ed ype o
dis ibu ion. Thus, he pe cen age o noise is dis ibu ed
andomly (unde a uni o m o no mal dis ibu ion), a ound
he ac ual alue, adding o sub ac ing hal he pe cen age o
he noise selec ed.
Howe e , his p ocedu e implies ha he noise- ee alue
ze o is singula , i.e. i has no noise added using he abo e
equa ion. Ne e heless, his can be emedied i an ex a
pe cen age o andom noise is included o each da a poin as
an o se , posi i e o nega i e, educing his singula i y. The
amoun o noise used o p oduce he o se is chosen a a
le el ha is less by a ac o o en compa ed o he le el o
noise selec ed. As s a ed be o e, a no mal dis ibu ion wi h
he same s anda d de ia ion as he uni o m one, p oduces
alues ou side o hose selec ed o he uni o m dis ibu ion.
Thus, o main ain app oxima ely he same in e al o alues,
he s anda d de ia ion o he no mal dis ibu ion should be
scaled, di iding by 1/(2√12) ≈1/7, as can be seen in igu e 3,
p oducing alues ha a e mo e concen a ed a ound he
mean alue. In his sense, when noise is injec ed using
scaling o he no mal dis ibu ion he ac ual alues a e less
biased.
All hese di e en app oxima ions o injec ing noise
in he da a se ha e been es ed and analyzed: uni o m
dis ibu ion, no mal dis ibu ions wi h di e en s anda d
de ia ion scales, and o each case, wi h and wi hou
addi ional o se . The analyses ha e been pe o med o e a
da a se con aining en poin s ha we e gene a ed using a
simple linea ela ionship o he o m y = x + 1. Fo each ype
o noise dis ibu ion, i e di e en pe cen ages o noise
le els we e injec ed: 0.5%, 1%, 2%, 5% and 10% o he
o dina e and 0.2%, 0.5%, 1%, 2% and 5% o he abscissa.
The sum o e squa ed de ia ions a ies app oxima ely om
2·10-4, o 0.2% o 0.1 o 5% in he case o he uni o m
dis ibu ion applied o he abscissa and om 1·10-3 o 0.5%
o 0.5 o 10% in he o dina e. When he co ec ion o he
singula noise- ee alue as men ioned abo e is applied, he
alues o he sum o squa ed de ia ions a e e y simila .
Applying he scaled no mal dis ibu ion, he alues o he
sum o e squa ed de ia ions a e app oxima ely om 3·10-5,
o 0.2% o 0.04 o 5% o he abscissa and om 2·10-4 o
0.5% o 0.1 o 10% o he o dina e. No signi ican changes
a e obse ed when a small o se is added o minimize he
singula ze o alue. These alues o he sum o e squa ed
de ia ions ag ee wi h he expec ed beha io , namely ha
noise gene a ed by he uni o m dis ibu ion will p oduce
s onge luc ua ions wi h poo e eg ession s a is ics han
he scaled no mal dis ibu ion.
In o de o analyze he in luence o he kind o noise
dis ibu ion and i s le el in bo h a iables, linea eg ession
s a is ics ha e been pe o med by means o he LINTEST
ou ine in Excel.
30 D. González-A jona e al.: Noise-injec ion as an App oach o Gene a ing
Random Da a Se s o Online Tes s and Vi ual Labs
Table 2. Linea eg ession s a is ics by using he LINTEST unc ion in Excel o di e en uni o m noise le els added o he o dina e only and o bo h
coo dina es (x,y), espec i ely
Uni o m noise a o dina e y%
Reg . Pa am.
2%
5%
10%
slpe/in cp
1.00195512
0.9936106
0.9995866
0.9946863
1.0102083
0.967860847
s de
slpe/in cp
1.8104E-03
1.2279E-02
4.6607E-03
3.1611E-02
9.2095E-03
6.2462E-02
2/ s de y
9.9997E-01
1.8988E-02
9.9980E-01
4.8882E-02
9.9925E-01
9.6590E-02
F/ d
3.0630E+05
9
4.5997E+04
9
1.2032E+04
9
ss eg/ss esid
1.1043E+02
3.2448E-03
1.0991E+02
2.1505E-02
1.1226E+02
8.3966E-02
Uni o m noise a bo h coo dina es x%/y%
Reg . Pa am.
1%/2%
2%/5%
5%/10%
slpe/in cp
1.00184749
0.9968192
1.0005897
0.9899704
1.0050629
0.993968265
s de
slpe/in cp
2.4575E-03
1.6662E-02
5.9783E-03
4.0531E-02
9.8773E-03
6.7106E-02
2/ s de y
9.9995E-01
2.5777E-02
9.9968E-01
6.2635E-02
9.9913E-01
1.0412E-01
F/ d
1.6619E+05
9
2.8012E+04
9
1.0354E+04
9
ss eg/ss esid
1.1043E+02
5.9801E-03
1.0990E+02
3.5308E-02
1.1224E+02
9.7565E-02
Legend: slpe: slope; in cp: in e cep ; s de slpe/in cp: s anda d e o ; 2: coe icien o de e mina ion; s de y: s anda d
e o y es ima e; F: F s a is ic; d : deg ees o eedom; ss eg: eg ession sum o squa es; ss esid: esidual sum o squa es.
Table 2 summa izes, as an example, he esul s o a
eg ession s a is ical analysis o he injec ion o di e en
le els o uni o m noise. The sub- able a he op epo s
esul s ob ained when noise is added only o he o dina e
alues, while he lowe sub- able gi es LINTEST ou pu
when noise is added o bo h coo dina es. The % le el o
noise injec ed is indica ed in he column heade s. Clea ly,
om he da a in Table 2, lowe noise added implies be e
s a is ics o he linea eg ession pa ame e s ( 2, s d e ,
esidual, and % ela i e e o ) indica i e o a g ea e
eliabili y o he model equa ion.
Figu e 4. Rela i e e o associa ed wi h di e en pe cen ages o uni o m
noise le els injec ed along he o dina e
Figu es 4 and 5 show he ela i e e o in he o dina e, and
a close-up iew o one poin o he da a se , (2, 3),
espec i ely. Di e en le els o uni o m dis ibu ion noise
ha e been employed, in igu e 4 only in o dina e and in
igu e 5 bo h coo dina es. The shaded a ea in igu e 5
app oxima ely delinea es he limi s o he (x, y) luc ua ion
ange.
An analysis o he % ela i e e o in slopes, in e cep s and
eg ession esiduals o he di e en le els o noise and o
he di e en kinds o dis ibu ions has been pe o med. The
esul s a e summa ized in 3D plo o ma o ease o
compa ison. Many plo s o his ype ha e been gene a ed,
and he ones shown he e can be conside ed as ypical o he
gene al pa e n.
Figu e 5. Close-up iew o he app oxima e a ea o luc ua ion when
di e en pe cen ages o uni o m noise le els a e injec ed along bo h
coo dina es x, y
Figu es 6, 7 and 8, show column diag ams o he ela i e
e o s in he slope, in e cep and he squa ed sum o he
eg ession esiduals o Uni o m ( ed) and No mal scaled
(blue) dis ibu ions a di e en le els o noise o he
o dina e only and o bo h coo dina es. The % o noise is
indica ed in he plo s. In gene al, he ela i e e o in he
in e cep is app oxima ely i e imes highe han ha in he
slope, no ing he di e en scales in igu es 6 and 7.
Addi ionally, he e o in he in e cep inc eases when he
ex apola ion is made a om he da a se in e al. Mo e
o en han no , when hese calcula ions a e epea ed many
imes, i is mo e likely o ob ain highe ela i e e o s and
eg ession esiduals o he uni o m han o he scaled
no mal dis ibu ion.
Gene ally, hese conclusions a e s ill alid when he
compa ison is made using he no mal dis ibu ion wi hou

Jou nal o Labo a o y Chemical Educa ion 2021, 9(2): 26-35 31
scaling wi h he uni o m dis ibu ion. Howe e , he e o
le el using he no mal dis ibu ion wi hou scaling is highe
and close o ha ob ained using he uni o m dis ibu ion.
This beha io can be expec ed as ex eme alues unde he
no mal dis ibu ion ha e ela i ely low p obabili y.
In conclusion, he injec ion o andom noise in o a da a se
can be use ul o gene a ing di e en da a se s o use in
online es s and i ual labs.
Figu e 6. Rela i e % e o in he slope pa ame e o di e en noise le els
and o bo h dis ibu ions, uni o m (U) and no mal (N)
Figu e 7. Rela i e % o e o in he in e cep pa ame e o di e en noise
le els and o bo h dis ibu ions, uni o m (U) and no mal (N)
Figu e 8. Squa ed sum o he eg ession esiduals o di e en noise le els
and o bo h dis ibu ions, uni o m (U) and no mal (N)
4. Examples o Injec ion o Con olled
Random Noise
In his sec ion, h ee kinds o expe imen s om he
unde g adua e Physical Chemis y lab will be employed as
examples o he applica ion o he s a egy p esen ed abo e.
4.1. Chemical Kine ics o he Fading o Phenolph halein
in S ong Alkaline Media unde Pseudo-Fi s O de
Condi ions
This is a classical expe imen used o eaching chemical
kine ics, in ol ing he analysis o he change wi h ime o
he abso bance o a phenolph halein solu ion a di e en
high alkaline concen a ions. The ading is second o de
o e all, pa ial o de s equal o one o each componen ,
OH- and he Phenolph halein anion. The goal is he
de e mina ion o pa ial o de s and he second o de a e
cons an . This kine ic expe imen has been pa o ou lab
p og am o decades. I is ad isable o pe o m he
expe imen a cons an ionic s eng h by adding di e en
concen a ions o an ine sal . The gene al expe imen al
condi ions ha e been ecen ly desc ibed [14], and epo ed
esul s will be employed as ini ial da a se .
Table 3. Excel able con aining he ini ial expe imen al condi ions and
li e a u e da a o he kine ics o phenolph halein ading. The ionic s eng h
is main ained a 0.435M
A new able is c ea ed in Excel om he da a in able 3, by
andomizing OH- concen a ions, ex inc ion coe icien and
concen a ion o phenolph halein and he second o de a e
cons an . The kind and he % le el o noise can be selec ed
indi idually o each pa ame e . The goal is o add jus
enough noise o andomize he ini ial da a se , while keeping
he esul s ob ained close o li e a u e alues, wi h a
ole ance le el o 20% being ad isable. Taking in o accoun
ha Excel andomizes he cell con en s each ime a cell is
modi ied a any posi ion on he wo kshee , i is equi ed o
copy hese gene a ed alues elsewhe e on he wo kshee . In
his way he new da a emain ixed o he es o he
calcula ions. This inal able can be conside ed o be he da a
se co esponding o he esul s o a lab expe imen .
A new wo kshee is gene a ed con aining he abso bance
alues o he ou alkaline concen a ions o di e en
sampling imes, by using he equa ions depic ed in igu e 9.
These abso bance alues, in ou case, a e andomized
wi h a new pe cen age o noise, o mimic he possible
luc ua ions o he abso bance du ing he ac ual
measu emen p ocess. In he o iginal lab expe imen , he
abso bance is au oma ically ead e e y second, and he
eac ion is moni o ed o 5 min.
Li e a u e Exp. Se C OH /M C Phen. /M Ex _Coe (M·cm)-1 ksd (Ms)-1
O iginal 1 0.4350 1.25E-05 30000 2.00E-02
Da a 2 0.3125
Sou ce 3 0.1875
Re . [14] 4 0.0325
32 D. González-A jona e al.: Noise-injec ion as an App oach o Gene a ing
Random Da a Se s o Online Tes s and Vi ual Labs
Figu e 9. Equa ions employed in he kine ic s udy o he ading o
phenolph halein. The symbols ha e hei usual meanings
This wo kshee , oge he wi h he li e a u e da a o he
ini ial concen a ions and mola ex inc ion coe icien , a e
he andomized da a o be supplied online o s uden s
indi idually as an Excel ile. Al e na i ely, i he
examina ion is ca ied ou ace- o- ace, as in a semina ,
hose abso bance alues can be sampled, ia Excel Da a
Analysis, o ob ain a andomized small se o abso bance
da a ha can be eadily p o ided o he s uden s.
Figu e 10. Sc een sho displaying he kine ics analysis esul s. 20%
noise e o added abso bance da a
Figu e 10 shows a sc eensho o he Excel ou pu when
20% Gaussian scaled noise is injec ed in o he abso bance
alues. Ini ially he abso bance is gene a ed using p ima y
da a wi h 10% noise le el injec ed. As can be seen, e en
wi h a high le el o noise injec ed, i is s ill possible o
ob ain alues close o hose epo ed in li e a u e. Thus, his
expe imen al design is e y obus and use ul o eaching
chemical kine ics.
4.2. Es ima ion o he Limi ing Mola Conduc i i y o an
Elec oly e Based on Kohl ausch’s Law
Ob aining he in ini e dilu ion mola conduc i i y o an
elec oly e is ano he classical expe imen in he Physical
Chemis y eaching lab. The expe imen is easily ca ied ou
wi h a cheap po able conduc i i y me e . In his case he
s uden s ha e o ob ain a linea Kohl ausch’s ela ionship
be ween he mola conduc i i y and he squa e oo o
he elec oly e concen a ion and om he in e cep a ze o
concen a ion, he limi ing mola conduc i i y o he
elec oly e is ob ained. The compa ison wi h li e a u e
da a o e s in o ma ion abou s uden ’s skills in p epa ing
solu ions and hei glasswa e-cleaning p o ocols. The
expe imen al da a a e he speci ic conduc i i ies o he
elec oly e solu ion. The speci ic conduc i i ies o he mos
dilu e solu ions ha e o be co ec ed o he con ibu ion
om he sol en , usually wa e . The ini ial da a se o
di e en ue elec oly es a e culled om he li e a u e, [15].
Ano he possible app oach is o gene a e he da a om he
Debye-Hückel-Onsage equa ion, knowing he limi ing
mola concen a ion and he cons an s A and B, see igu e 11.
Figu e 11. Debye-Hückel-Onsage equa ion and pa ame e s A, B, o
wa e a 25°C
A se o ini ial alues o mola conduc i i ies a di e en
concen a ion is selec ed, in ou case, KNO3, K2SO4,
KIO3 and sodium oxala e. These concen a ions can be
andomized o ob ain he di e en mola conduc i i ies.
F om hem, a se o andomized speci ic conduc i i ies can
be gene a ed. Fo his case, uni o m dis ibu ion- ype noise
has been selec ed. The expe imen al concen a ion ange is
usually om 10-4 M o 10-2 M, bu can be modi ied. The
speci ic conduc i i y o low elec oly e concen a ions has
o be co ec ed o he wa e con ibu ion, and his can be
ano he sou ce o andomiza ion.
The andomiza ion p ocess is analogous o ha used in he
kine ics expe imen . A wo kshee con ains he ini ial da a,
and om hese he andomized se is gene a ed. Figu es 12
and 13 show he li e a u e da a o Kohl ausch’s plo s o he
selec ed sal s and hose ecalcula ed a e andomiza ion,
espec i ely.
Figu e 12. Li e a u e-gene a ed Kohl ausch plo s o di e en selec ed
sal s
Jou nal o Labo a o y Chemical Educa ion 2021, 9(2): 26-35 33
Figu e 13. Es ima ed Kohl ausch plo s o di e en sal s a e
andomiza ion p oduced by adding 5% o uni o m noise o he li e a u e da a
In his case, a low le el o noise should be injec ed
because he pa ame e o in e es is ob ained om he
in e cep , he one mos sensi i e o highe noise le els, as
was illus a ed ea lie . Ne e heless, wi h his noise le el in
combina ion wi h li e a u e in o ma ion, many andomized
da a se s can be cons uc ed. This s a egy can be applied
also o ob ain he limi ing conduc i i y o a weak elec oly e
and i s dissocia ion equilib ium cons an . In his case, he
pa ame e s a e ob ained by ex apola ing a om he da a
se in e al. Consequen ly, special ca e should be aken in
choosing he le el and ype o noise o be injec ed.
4.3. Es ima ion o he En halpy o Vapo iza ion Based on
he Clausius-Clapey on Equa ion
The de e mina ion o he en halpy o apo iza ion o wa e
cons i u es ano he s anda d lab expe imen in he Physical
Chemis y lab. The ypical expe imen is based on he
measu emen o he change o he wa e apo p essu e wi h
empe a u e when he apo and he liquid wa e phases a e
in equilib ium. The esul s a e analyzed by using he
app oxima e Clausius-Clapey on (CC) exp ession, whe e he
mola olume o wa e is neglec ed wi h espec o ha o he
liquid, he en halpy o apo iza ion is independen o
empe a u e and ideal gas beha io is assumed o he wa e
apo [16].
The CC equa ion can be o mula ed o he liquid/ apo
equilib ium:
1
Ln A
Vap
ap
H
PRT




whe e P ap is he wa e apo p essu e, A is a cons an , ΔH ap
is he mola en halpy o apo iza ion o wa e , assumed o be
cons an in he empe a u e ange conside ed, R he gas
cons an and T he empe a u e in K.
In ou lab, a dis illa ion glasswa e sys em con aining
wa e which can be hea ed and is connec ed o a acuum
pump is employed. A simple h ee-way manually ope a ed
al e allows easy con ol o he p essu e in he sys em. The
a angemen pe mi s he measu emen o apo p essu e
(P ap) and he empe a u e (T) da a a equilib ium, om oom
empe a u e o he no mal boiling poin .
Figu e 14 shows he linea plo s (ln P ap s. 1/T) acco ding
o he CC equa ion ob ained by h ee s uden s wi h di e en
le els o labo a o y skills. As can be no ed, he deg ee o da a
sca e is highe o lowe empe a u es, whe e equilib ium
condi ions a e less easy o a ain.
Figu e 14. Expe imen al Clausius-Clapey on plo s based on da a,
independen ly ob ained by h ee unde g adua e s uden s
Li e a u e da a [15], ob ained as P ap s /°C, o he phase
equilib ium be ween wa e apo /liquid a he condi ions
p e ailing in he lab (25°C-100°C) a e plo ed as a CC linea
plo in Fig. 15.
Figu e 15. Wa e apo /liquid equilib ium Clausius-Clapey on plo om
li e a u e da a [15] o he empe a u e ange employed in he lab
No wi hs anding he good linea ela ionship, sligh
di e ences be ween he da a poin s and he s aigh line can
be no ed specially o he wo empe a u es limi s. This is
ano he sign o he accu acy o he assump ions used in
applying he CC equa ion.
Taking in o accoun all hese conside a ions, he
con olled injec ion o noise has been applied o gene a e
34 D. González-A jona e al.: Noise-injec ion as an App oach o Gene a ing
Random Da a Se s o Online Tes s and Vi ual Labs
andomized P ap/T da a om li e a u e sou ces. Bo h ypes
o noise ha e been injec ed, he Uni o m and he No mal
(log based, scaled).
Di e en le els o noise ha e been applied o he
expe imen al a iables: apo p essu e in kPa and
empe a u e in Celsius scale, anging om 10% o 40%
o P ap and 1% o 5% o empe a u e. The loga i hmic
cha ac e o he o dina e a iable in he CC equa ion, in
addi ion o he ac ha he pa ame e o in e es is ob ained
om he slope, pe mi high le els o noise o he apo
p essu e. This ac ensu es he success ul pe o mance o
hese expe imen s in he unde g adua e lab. The hea o
apo iza ion ob ained by a mode a ely skilled s uden is
always close o he li e a u e alue, boos ing s uden ’s
sel -con idence. None heless, he le el o noise applied o
empe a u e alues has o be con olled ca e ully o ob ain
signi ican esul s, and a maximum luc ua ion ange o 5%
is ecommended.
Figu e 16. Clausius-Clapey on plo s ob ained by injec ing o li e a u e
da a Uni o m- and No mal- (log based, scaled) ype noise, wi h a 40% ange
o P ap in kPa and 5% o empe a u e in °C
Figu e 16, shows he in luence o noise injec ion in he
sampled li e a u e da a om Fig. 15. The same conclusions
eached in sec ion 3 apply he e. Injec ion o uni o m noise
p oduces poo e linea eg ession pa ame e s han he no mal
scaled noise a he same pe cen age le el. E en hough
andomized Gaussian noise p oduces an ou lie , he slope
ob ained is close o he li e a u e da a. The occu ence o
hese ou lie alues in a da a se is no e y likely. The slopes
ob ained om plo s wi h injec ed Gaussian noise a e likely o
end up close o he li e a u e alue han hose ob ained wi h
uni o m noise.
5. Conclusions
In conclusion, some ecommenda ions can be
implemen ed o andomized da a se s by using injec ion o
noise. Fi s , he applica ion o noise has o be ca e ully
con olled, ha is, he esul s ob ained om he analysis o
he noisy da a se ha e o be ealis ic and eliable. Fo each
speci ic si ua ion and se o condi ions, a check should be
pe o med p io o selec ing he app op ia e amoun o noise.
Secondly, i is ad isable o use uni o m (whi e) noise o
da a se s ha al eady con ain na u al andom noise, such as
he da a se s culled om li e a u e. And inally, no mal
dis ibu ion (Gaussian) noise should be employed o
da a se s gene a ed di ec ly om heo e ical o simula ed
equa ions, p oducing noisy da a se s ha ha e a mo e na u al
appea ance.
ACKNOWLEDGEMENTS
The au ho s a e indeb ed o M. A. Gue a Aguila -Galindo
o his con inuous, deep in ol emen wi h and dedica ion o
he Labs o he Physical Chemis y Dp . o he Uni e si y o
Se ille.
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