Resarch Laboratory for Vegetable and Fruit Industry,State Research Station, Blangstedgaard,5000 Odense, Denmark

Fruit content determination in one-fruit jams and preserves

(Frugtindholdsbestemmelse i een-frugt marmelader og syltetøj)

Poul Emil Christensen

SummaryThe failure of commonly used methods of fruitcontent determination in jams and preserves isascribed to nonnormal frequency distribution ofindex-constituents in reference material.

An equation is set up which makes it possibleto discriminate between index-constituent com-binations which in all respects are equally wellsuited to the purpose.

Applying the method to two commerciallyproduced strawberry jams indicates the possibi-lity of determining the fruit content with reason-able accuracy and small security limits.

IntroductionMany countries have set up regulations governingthe quality and composition of foods. Havingestablished such regulations, it is also necessaryto lay down procedures by which they can bepoliced, usually by one of two methods, by in-spection at the point of manufacture, or by takingsamples of the final product at the point of saleand subjecting them to examination and analysis.

This investigation only deals with the lattermethod, and is restricted to a quantitative analy-sis of the amount of fruit used for that particularkind of product which is supposed to have beenmanufactured from only one kind of fruit.

Before World War 1914 Beythien (3), Baierand Hase (2), Beythien and Simmich (4), Härteland Soiling (8) and Ludwig (13) published theirmethods of fruit content determination. Thesemethods were founded on the assumption that

fixed relationsships exist among certain fruitcompounds (index-constituents).

Since then, beginning with (13) many publica-tions have appeared giving analytical data forfruit used in the preserving industry. An almostcomplete bibliography covering this matter upto 1969 has recently been published by Goodall(7).

Macara (14, 15), besides giving analytical datafor the fruits in question, also treated his datastatistically, being the first investigator to statenot only maxima and minima as well as meanvalues, but also the frequency at which a certainvalue was to bo expected.

Since then, statistical treatment of analyticaldata has been further extended, and is now com-mon practise in nearly all publications.

While, before 1940 all investigators tried tofind constant relations between several con-stituents in the fruit, Steiner (19,20) as one ofthe first discussed the possibility of using ma-thematical-statistical analysis in order to furtherincrease the analytical statement. Steiner showedthat by using an optimal linear-combinations ofseveral analytically determined index-consti-tuents, it was possible to arrive at a more precisestatement about the fruit content. This methodis now generally used to solve analytical problemsconcerning food composition. Thus, Rolle andVander cook (18) and Vander cook et al (21, 22,23) have characterized lemon juice by use ofmultiple regression analysis. Nehring and Klinger(16) and Klinger and Nehring (12), on a purely

254

theoretical basis treated the problem of fruitcontent determination using modern statistical-analytical methods. Recently Kefford (9) andAcker (1) also discussed the whole matter.

Index-constituentsAs stated by Kefford (9), the ideal constituentwould be one that was specific to a particularfruit and had the same concentration in thatfruit at all times and places, was stable to proces-sing, was amended to convenient and accuratedetermination, and so rare or expensive that itwas unlikely to be added as an adulterant.

The ideal index-constituent does not exist.Fruits are in fact very variable in compositionand the concentration of all their known consti-tuents is influenced by many factors, variety,maturity, growing area, climate, fertilization andirrigation, as shown by Kenworthy and Harris(11), Rahman et al. (17) Choureitah and Lenz(6) and several others.

Thus, the composition of a fruit can not bedefined in any absolute way, but only in termsof ranges and averages with the usual statisticalspecification to indicate the variability.

Major fruit constituents, such as sugars andacids are poor index compounds, because sugarsand acids are commonly and legitimately addedingredients of jams and preserves. Many minorconstituents have been investigated as index-constituents: inorganic elements, polyphenoliccompounds, pigments, vitamins, amino-acids etc.

Index-constituents most commonly used havebeen inorganic elements, mainly because theseelements are stable to processing and can easilybe determined with great accuracy. Among theseespecially ash, potassium and phosphate havebeen widely used. It has been common to com-bine two or three of the index-constituent in alinear equation to give the fruit content (regres-sion equation).

In spite af the fact that these regression-equa-tions normally fit very well to the results of refe-rense-material analysis, it is also recognized thatthese equations very often give poor resultswhen applied to commercially manufacturedfruit products.

In remedy of these shortcomings this investiga-tions was started in 1969.

TheoryA relevant formula for a jam or preserve couldbe as follows (Ace. to Nehring and Klinger (16)).

Formula

content of index-constituent xt

Indgredient Kilogram x0 x1 x2 ~ xrFruit CO vO v-0 vO w vO

Sucrose S1 x* x* x\ » x\Other sugar... S2 x% x\ x | » xfPectin 53 xl x\ x | » *3Acid .94 v4 V 4 y4 \v v4

Preservative... S5 x$ x\ x$ » x°

Ingredient m. . Sm x™ x™ x™ » x"rl

We standardize this formula so thatm

Ii = 0

S* = 100

then we have if x0 is soluble solid as measured byrefractometry

the mean soluble solid in the formula mix andlikewise

2)

the content of the /'the index-constituent in themix.

The mean soluble solid x0 with given recipeis almost constant because the soluble solidmainly consists of sugar which has been added.Variations in the soluble solid in the fruit arenormally left aside when setting up the formula.

In the case that the concentration of all index-constituents in all formula ingredients (exceptthe fruit) is small and has small standard devia-tions, it is justified to make further simplifica-tions.

255

In that case, we can in the equation for thecontent of index-constituent Yi in the jam as afunction of the recipe content x%

3)

replace the terms xjmi and we get

xf with the means

f 1n +fi\ 4 )

In the error-term fy are included all deviationscaused by the above mentioned approximations.Setting

S2mJ H + Smmf

Smm5)

and substitute in 4) we get

S°x<>

YQfx 6)

choosing the proper index-constituents we caninterpret Yokt as a correction term and Yofi asan approximation term which as such can beneglected.

With the notations used above, the fruit con-tent t in the jam can be equated as follow.

t = 100 YS°

0 * S°x°0

and then using 6) and 7) we finally get

t

7)

8)

Now we will proceed in two different ways:

1. MethodMultiplying 8) with a factor bi we get

1009)

and then summing for all i finally

Zbi (Yi — Yoki) = Zbi Xi100

choosing the factors bi in such a way that

Zbi Xi = 100

which is possible always, we get

10)

t = — YQki) 11)

In equation 10) the term Zbi Xi is a linear com-bination of all index-constituents and can beformally regarded as a regression-equation.Hence, choosing the ^-coefficients by the me-thod of least-squares, provided that the xf aremultivariate normally distributed, the aboveequation will give us the fruit content with theleast variance. The coefficients bi will be denotedas regressions-coefficients.

The sum of squared residuals from fitting themodel by the least squared methods is

o2 _

JKXt..)

S2Xi, Xj 12)

where s2 Xi, Xj are the elements of the dispersion-matrix and f(xi ..) denote that the summingmust be done for all possible permutations ofthe Xi.

If n samples of jam are analysed, and if theindex-constituent content in sample / is denotedby Yif, the corrected content Yif — YOfki withZif, then we have the following equation for thefruit content tf

the arithmetric mean

if

tn= —Ztjn

13)

14)

is then a result where all n jam samples are takeninto consideration.

The fruit content in the jam samples with itsstatistical security limits can finally be stated asfollows.

256

n

± toc/2;O,O5 15)

1002

2. MethodTaken logarithm on both sides of 8) we get

In (Yi — Yoki) = — 4,6052 16)

If, as above, n samples of jam are analysed andthe index-constituent content in sample / isdenoted by Yif, then

ln(Yif-Yofki)= Vif 17)

and if we further assume that the index-consti-tuent Yf in the jam are normally distributed wecan regard 17) as n determinations of the nor-mally distributed variable Vif, we can computeIn tf by the equation

In tif = Vif + 4,6052 — In 18)

We can calculate a probable value In tf wherewe are taking into account the index-constituentcontent of all compounds in sample /, by linearcombination of all Vif values.We get

In tf = ECi Vif + 4,6052 — ZCt In xt 19)

with ZCi = 1

the arithmetric mean

In tn = —Zlntfn

is then a result where all n jam samples are takeninto consideration.

The sum of squared residuals from fitting thelinear model is

s% = Q Cj s2 In xi; In 21)

where again s2 In xf, In Xj denotes the element inthe logarithmic dispersion matrix and f(xi..)that the summing shall be performed for allpossible permutations of Q .

Further, it can be shown that s^ is minimizedby choosing the weights Q according to thefollowing equation:

m

^ s2 In xf, In

m

^ s2 In xf, In

— 1

— 122)

where s2 In xf, In Xj - 1 are the elements in theinverted logarithmic dispersion matrix.

Both methods mentioned above require a cal-culation of the correction term

= Yof xSmm™

! 23)

We can split the right side into two terms and get

S2m\% =

S°m°0+—S™

Szm\ +—Sn'm™

sv0S1m1

0

24)

setting

S°m°0 -\ Smm™= Jof and

S1m) + S2m2i - KSif

S3m3 + - - Smm™ = KUif

20) we get

= Jof x KSif+ Jof x 25)

the common factor JOf is a measure of the con-centrating of soluble solids from the recipe-mixto the final jam. It will be possible to give a

17 TfP 257

max. - and a min. - value for this factor. We geta high value when the fruit content is high andsugar solutions are used for instance glucose-sirup or invert-sugar solutions. We get a lowvalue when the fruit content is low and solidsugars are used. Using the results of variousanalyses of the jam, (e.g. water insoluble solids,glucose-sirup content and invert-sugar content)it is possible to set up preliminary model formulaefor the jam in question, thus giving a high, alow and the most probable value for Jof.

Likewise by analysing various samples ofsugars, pectins, acids etc. it will be possible(taking into proper consideration factors likefirmness, acid content etc. of the jam) to state ahigh, a low and the most probable value forKSi/and KUy. Having these values, a highest, alowest and the most probable value for t can becomputed, using either equation 11) or 19).

ExperimentalIn the strawberry season 1970, samples of washed,deep frozen strawberries were collected from themain growing areas in Denmark. All sampleswere collected at random at the end of the free-zing-band at four freezing plants. Samples werecollected morning and afternoon. All samples,totalling 84, were stored at — 30° C until ana-lysed.

From two dactories manufacturing strawberryjam, samples of the finished product were takenduring the cooking season, autumn 1970. Samp-les were taken directly from the production-lineat the end of the cooling band.

Samples of raw material, other than straw-berry, used according to recipe were also collected.New samples were taken everytime new deli-veries arrived at the factory. These latter sampleswere mixed, so that only 2 to 5 samples of everyraw material were analysed.

The strawberry samples (reference material)were analysed for the following constituents,using well known and proven methods.

x1: Water insoluble substance (extraction inSoxhlet-apparatus with water).

x2'. Nitrate (Ion selective electrode).

x3: Formol value.JC4: Ash. (Dry ashing at 500° C).

Sodium. (Flame - phometry).Potassium. (Flame - photometry).Calcium. (Atomic - absorption spec-trometry).Magnesium. (Atomic - absorption spec-trometry).Phosphor. (Colorimetric as phospho-vanadomolybdate complex).

The strawberry jam samples and the rawmaterial samples were analysed for the same 9constituents mentioned above. Further, the jamswere analysed for:

a: Total soluble solide (refractometry).b: Invert sugar content,c: Total acid,d: Benzoic acid.e: Firmness (Breaking strength by Tarr-

Baker).

ResultsThe results of the analyses of the referencematerial are given in table 1. Here also are giventhe means x, the variance s%lt the standarddeviation sXi, and the coefficient of variablility

sXi X 100

In table 2, 3, 4 and 5 the calculated dispersionmatrix, the correlation matrix, the logarithmicdispersion matrix and the logarithmic correlationmatrix are tabulated.

In table 6 and 7, the results of the analyses ofthe strawberry jam and the raw materials aregiven.

In table 8, three model-formulae for the twojams in question based upon these analyses arestated. Using these formulae and the results fromthe raw material analyses, the corrected mean z$values are calculated. These values are also givenin table 6 and 7 as zf (the high value), z^ (thelow value) and zp

{ (the most probable preliminaryvalue).

258

Table 1. Reference-material

Water in-soluble

substance %

Source Anxs

Source B

n3c*

Soucre C

nxs

Soucre D

nxs

Total

nxs2

s

s x 100X

261,670,17

171,780,184

171,780,191

241,720,211

841,700,03310,1820

10,71

NO3

mgeqv/kg

X2

261,660,58

171,100,212

171,150,194

241,310,239

841,370,18860,4343

31,74

Formolvaluemgeqv/100 g

X3

261,000,19

170,960,153

171,000,161

240,800,151

840,940,03730,1931

20,59

Ash%X4

260,45

170,450,060

170,460,052

240,430,047

840,440,00290,0539

Namg/100g

X B

260,650

Kg/kg

X6

261,512

Ca Mg Pmg/10 g mg/10g mg/10g

X7 X8 X9

262,111

261,132

0,064 0,055 0,168 0,274 0,078

170,560,036

170,560,038

240,610,037

840,610,00300,0548

171,500,120

171,520,096

241,490,123

841,500,01890,1374

172,490,319

171,180,078

262,0690,203

172,070,157

17 17 172,45 1,19 2,05

0,368 0,082 0,146

242,600,381

842,360,14630,3825

12,19 9,04 9,14 16,22

241,230,101

841,180,00850,0923

7,84

242,140,167

842,090,03050,1746

8,34

Table 2. Dispersion-matrix

Xi X 2

0,0332 —0,01250,1886

x3—0,00290,04230,0373

X 4 å0,00300,00880,00320,0029

x5—0,00070,01090,00140,00010,0039

x60,01020,01230,00540,00510,00080,0189

X70,0251

—0,0191—0,0213—0,0006—0,0036—0,00610,1463

x80,0092

—0,0025—0,00220,0010

—0,00030,00520,01920,0085

X;90,01530,00620,00300,00400,00070,01520,01620,00780,0305

17 : 259

Table 3. Correlations-matrix

1 —Ix2

0,15821

x8—0,0839

0,50441

X,

0,31080,37370,3164

1

xs0,01310,42270,09340,1219

1

x60,40620,20640,20500,68440,1778

1

X7

0,3613—0,1148—0,2883—0,0271—0,1599—0,1154

1

x80,5451

—0,0622—0,1226

0,2041—0,0658

0,40800,5432

1

x90,48100,08130,08710,42910,12690,63430,24260,4808

1

Table 4. Logarithmic dispersion-matrix

x10,0111

x2—0,0048

0,0810

x8—0,0020

0,02710,0425

X4

0,00430,01110,00720,0139

x6—0,0006

0,01280,00220,00080,0096

Xg X7 Xg Xg

0,0042 0,0057 0,0045 0,00420,0050 —0,0038 —0,0005 0,00190,0036 —0,0105 —0,0024 0,00090,0077 —0,0012 0,0020 0,00430,0008 —0,0026 —0,0005 0,00060,0087 —0,0020 0,0031 0,0048

0,0256 0,0065 0,00300,0061 0,0031

0,0070

Table 5. Logarithmic correlation-matrix

xa—0,1592

1

xs—0,0940

0,46201

x40,34420,33120,3025

1

x5—0,0165

0,46720,08500,14211

x60,42120,18890,18440,69540,1679

1

X7

0,3386—0,0826—0,3187—0,0599—0,1776—0,1367

1

x80,5419

—0,0241—0,1469

0,2153—0,0712

0,41850,5246

1

x90,47340,07980,04970,43890,13290,61850,22140,4760

1

260

Table 6. Content of index-constituent in Strawberry-jam

Factory

E y-

yHi

yLi

zfzf~z\Szsz

sz x 100if

Sucrose m1

Invert-sugar m11

Pectin mni

Acid mIV

Preserva-tives mv

n

2020202020202020

20

4

433

1

X i

0,620,710,550,620,710,550,00140,0376

6,09

0

000

0

x2

0,810,910,690,440,660,320,00310,0556

12,67

0,27

0,80531,227,58

0

x8

0,320,380,260,320,380,260,00120,0348

10,94

0

000

o :

x4

0,180,230,140,130,190,080,00050,0223

17,72

0,003

0,031,740,18

21.5

x8

2,002,251,75

0,49

0,2611,380,60

1597

x6

0,4890,5280,4400,4590,5070,4100,0007

X7

1,5751,3700,9900,9191,2700,8100,0080

0,026970,0892

5,88

0,036

0,0321,9480,026

0

9,71

0,054

0,60030,730,435

0

x8

0,5210,5900,4900,5010,5700,4650,00060,0241

4,82

0,007

0,1161,6200,242

0

x9

0,7230,7800,6050,7040,7600,5850,00240,0490

6,96

0,016

0,0551,1280,528

0

Benzoic-RT Inv. acid

0/ 0/ 0/

/o /o /oo65,17 26,83 0,7565,81 33,964,42 21,95

Breakingstrength:59 g/cm2

75%

Table 7. Content of index-constituent in Strawberry-jam

Factory

F

s.Z

SucroseInvert-sögarPectinAcidPreserva-

y-i

y"i

y'i

*f*7*L,*Isz

x 100

zL

m1

m11

mm

mlv

n

55

555555

5

1

011

Xx

0,460,570,360,460,570,360,00640,080

17,1

0

-00

x2

0,890,990,830,520,690,370,00430,0659

12,7

0,25

-

30,9419,49

x3

0,280,340,230,280,340,230,00180,0424

15,2

0

-00

x4

0,180,190,160,130,150,100,00020,0130

10,2

0,05

-

1,610,05

x6

2,512,852,20

0,02

-

10,400,08

X6

0,3560,3860,3250,330,360,300,00060,0236

7,2

0,034

-

1,8550,076

X7

0,8120,8450,7900,580,680,4900,00050,0214

3,7

0,185

-

29,000,790

x8

0,3530,3900,3100,330,3700,2900,00090,0303

9,1

0,020

-

1,4400,050

x9

0,5390,6230,4900,520,6030,4700,00270,0524

10,1

0,020

-

1,1950,010

RT0/

/o64,5164,9564,08

Bonzoic-Inv. acid

% 7oo18,29 0,8522,2414,64

Breakingstrength:61 g/cm2

tives 1 0 21,5 1597 0

261

Table 8. Model recipes

Factory E Factory FHighest Lowest Probable Highest Lowest Probable

value value

Fruit 40% 20%Sucrose 45 % 75 %Invert-sugar 15 % 5 %Pectin. 5°/00 2°/00

A c i d . . . 100/oo 6,6°/00

Preservative 1,5°/Oo l°/oo

Based upon the analysis results of the refe-rence material, all regression-coefficients b$ andall weights Q were computed, together withthe corresponding variance s^ for all possibleindex-constituents combinations (totalling 502comb.).

value

35%55 %10%

3,5°/00

8,3°/00

l°/oo

value

40%60 %10%5°/00

5°/0 0

value

20%80 %0%2°/00

3°/00

l°/oo

value

30%70 %0%

3,5°/00

4°/00

l°/oo

compounds in the combination. Also, in diagram1 the standard deviation s* as well as the coef-

• s ! /

ficient of variability

4 x 100

A reason for tabulating all these numbers can are given.

Diagram 1. Diagram 2.

Number of index-constituents in combination.

hardly be found, since they can be displayedmore easily and illustratively otherwise. There-fore, in diagram 1 (method 1) I have displayedthe dependence of the mean ~s* of all s2

y - num-bers within each group containing either 2, 3 etc.index-constituents in relation to numbers of

Number of index-constituents in combination.

Likewise, in diagram 2 you will find the valuescalculated according to method 2.

In diagram 3, the dependence of the means ofthe regression-coefficients b% multiplied with x\(reference-material) within each group contain-ing 2, 3 etc. indexconstituents is displayed.

262

Diagram 3. Diagram 4.

Number of index-constituents in combination.

Finally, in diagram 4 the means of the weightsQ are illustrated in the same way.

Doing so, we get an idea of how much (inpercentage) every index-constituent contributeson the average to 100% fruit content, as well ashow much this contribution varies according tovariation among the index-constituents usedwithin each group.

DiscussionBoth diagrams 1 and 2, show that the meanssj, after an abrupt fall in the beginning, verysoon become almost constant.

This behaviour indicates that in practice itwill not be advisable or economical to use morethan 5 to 8 index-constituents.

We can get an impression of the main differencebetween method 1 and 2 by looking at the curvesfor the coefficient of variability of above men-tioned means i.e.

Number of index-constituents in combination.

ss*g

For method 1, these coefficients show the sametendency as J | , i.e. an approach to an almostconstant value after an abrupt fall in the begin-ning. In contrast with this, method 2 shows noabrupt fall but an almost constant value overthe whole range.

This behaviour indicates that in our case me-thod 1 would be preferable to method 2. Com-paring the elements in the correlation matricesalso indicates this.

From diagram 3 and 4 it's seen that the curvesfor some bixi values and Q 100 values very soonbecome almost parallel with the abscissae e.g.bz, 64, b5, and b7 as well as C3, C4, C5, and C7,whereas the others constantly diminish with in-creasing number of index-constituents in thecombination. Many interesting speculations canarise from regarding this curious behaviour of

263

certain index-constituents, but these speculationsare purely mathematical-statistical in nature andas such outside the scope of this publication.

With the present material and using eithermethod 1 or 2, we find considerable deviationsbetween the result of fruit content determinationin the two jam samples, depending on the index-constituent combination used. This is so, dispitethe fact that many of these combinations givethe same security limits on the calculated t-values.

The result thus confirms earlier findings, thatthe methods often fail when applied to commer-cially manufactured products.

The question now arises. Are our chemicalanalysis results wrong, or are the mathematical-statistical methods used not valid in our case ?

As all analysis results are means of two ormore determinations with small deviations andthe analysis methods used have been well prac-tised, the conclusion is that the fault must bewithin the assumptions.

These assumptions are mainly two.1. The concentration of all index-constituents

in all recipe-ingredients, except the fruit, issmall and has small standard deviation.

2. The index-constituents x$ are multivariate -normally distributed.

Ré 1. As seen from table 6 and 7 we must con-clude, especially when excluding sodium, thatassumption 1 holds in our case. We are then for-ced to accept that assumption 2 is not justified.

In order to get an impression of the deviationsfrom normality of the reference material the twoquantities yt, and y2 given by the equations,

as well as Pearson's skewness factor given by theequation

Yx =

72 =

and

Sk= -+ 3/*2

2 IH

are calculated where /u2, JU3, and /^ denote thesecond, third, and fourth factorial moments. Thecalculated moments used are corrected for group-ing using Sheppard's corrections.

Although the reference material is too smallto justify a precise assertion of the exact shapeof the distribution curves, they nevertheless givesom indication of how these curves deviate fromnormal ones. From table 9, where these valuesare tabulated, we find that y2 for the variablesx2, x7 are > 0 and for all others < 0. As Pear-son's skewness factor also departs from 0 to aless or greater extent, we must conclude that thedistribution curves deviate considerably fromnormal ones.

In the case of the variable x6, the above men-tioned calculations have been omitted, becausethis variable is apparently bimodal-distributed.

In order to further investigate the nature ofthe reference material, I have tested whether themeans and the variances calculated for samplingplace are possibly alike. The results of thesetests is specified in table 10 and 11.

The result indicates that there exist variance-homogeniety in the referencematerial betweensampling places, but that the means in severalcases are different.

Tests whether variance-homogeneity betweenthe jam-samples and the reference material exist,using their dispersion-matrices for comparisonhas also been performed. The test method usedis described by Box (5), see also Kendall (10).

The result is that homogeneity does not exist.All these tests have convinced me that the

assumption 2, presupposed by the theory is not

Table 9. Measures ofkurtosis and skewness

Yi

Y*SK

xx0,2722

—0,95410,8749

x22,57847,58284,3493

x30,1311

—2,87120,0242

x40,9424

—0,07679,7102

x50,2660

—0,39780,2078

x6———

x70,38070,5895

—0,1553

x80,2348

—0,9586—0,6755

x91,2495

—1,65090,2338

264

Table 10. Test for variance homogenity

Sources X t X2 X3 X4 X5 X6 X7 X8 X9

A : B x x x x x x x xA:C x x x x x x xA : D x x x x x x x xB : C x x x x x x x x xB : D x x x x x x x x xC : D x x x x x x x x x

x indicates that the hypothesis hold at 95% level.

Table 11. Test for equal means

Sources X t X2 X3 X4 X5 X6 X7 X8 X9

A : B x x x xA: C x xx xA : D x x xB : C x x x x x x x x xB : D x xx xC: D x x x x

x indicates that the hypothesis about equal meanshold at 95% level.

justified in the present case. The question nowarises, on what considerations our selection ofthe most suitable index-constituent combinationshould be based.

One method seems natural.As we have found that the index-constituent

is not normally distributed, we could possiblytransform the Xj-values so that the new variableswere more normal-distributed, that is, we couldfit a type of Pearson curve to the analysis resultsand then transform the values.

However, the present material does not war-rant this immense computational task. In fact,more than 500 samples would be necessary tojustify such a transformation, without any gua-rantee that the final result would be better,especially considering the lack of variance-homo-geneity between jam-samples and reference-material.

Therefore, considering the above mentioneddeviations from normality and the lack of va-riance-homogenity, it seems reasonable that acombination where one index-constituent ac-counts for a considerable percentage of the cal-culated fruit content would be inferior to a com-bination where all index-constituents are alike.

It would also seem likely, that an index-consti-tuent with very great correction-terms would beinferior to one where these correction-terms arediminutive.

Based on these considerations I have set uptwo equations which allow us to discriminate be-tween the various combinations.

These equations are:

1. Method

bj S2 Xi, Xj

1 +

with

II Too

= i

and for

2. Method

= -X y CiCj s2 In xi,\nn — 1 ^^^

1 +I Yp

~ I withith ^V d =

where n is the number of index-constituents inthe combination.

The value of £/& and Uc are esaily computedtogether with the regression coefficients bt andthe weights Q.

By choosing the combination for which eitherU\) or Uc is at the lowest, we are assured thatthe influence of all deviations from the assump-tions made in the theory are diminished.

For both jam-samples under consideration,the above stated equations have given that thebest suited combination would be the one con-taining the following compounds, xx, x2, x3, xé,x6, x7, and x9 and using method 1.

In table 12 the fruit contents in the jamsamples are tabulated, using this combinationand method 1. Also given are the figures for thefruit content as received from the two factoriesafter this investigation had been finished.

In our case where we are assured that the twofactories have not altered their recipes during

265

Table 12. Combination 1-2-3-4-6-7-9

Source Sample

E 90919293949596979899

100101102103104105106107108109

7

»True value« 1

Source Sample

F 8586878889

1

MethodFruit-

contenthigh

33,2033,6030,8432,9338,0933,4934,1533,1135,1936,1635,6635,5535,1635,0733,9635,4233,3635,2335,3437,38

34,5

52,5% ± 0 , 5 %

Fruit-content

high

25,8426,5422,6423,6826,82

25,1

1

Fruit-content

low

32,1932,6729,9831,7436,9132,7933,1432,1034,4635,1534,6532,5434,1534,0632,9534,4032,3434,2234,3336,29

33,6

Fruit-content

low

24,6125,3121,4122,4525,59

23,9

Fruit-content

probable

32,6633,0630,6932,3937,5533,2633,6132,5934,6836,6235,1233,0134,6234,5333,4234,8732,8134,6734,7736,73

34,1 ± 1,4

Fruit-content

probable

25,1525,8521,9522,9925,1324,2 ± 3,4

»True value« 23,7 % ± 0,5 %

Using the results from reference-material ana-lysis and the combination found most suitable,two jam populations can be constructed, onepopulation containing 20% fruit in the jam, andthe other 35%. Then an discriminant equationcan be set up in the usual way, which by insertingthe corrected Zif values assign the jam-sampleto either the population containing 20 to 27,5%fruit or to the population containing 27,5-35%fruit.

The result for factory 2 samples is, that all 5samples belong to population 20-27,5% fruitcontent with the probalility of misclassificationin the order of 0,4%.

ConclusionThe shortcomings of many methods of fruit con-tent determination in jams and preserves are ac-cording to this investigation, due to the fact,that the frequency-distribution of the index-constituents used, deviates considerably from anormal distribution. Hence the statistic used(regression-analysis) is not valid. A method isdeveloped which to a certain extent remediesthese difficulties. Using the methods, we havesucceeded in determining the fruit content intwo strawberry jams with reasonable accuracy,despite the fact, that the index-constituent con-tent in these jams in several ways must be re-garded as outliers compared with the same in thereference material.

The method developed can eliminate consider-able deviations from normality as well as uncer-tainties regarding the cooking-recipe used andthe quality of the other raw materials, besidesfruit, used in the production.

the sampling period, the method used above forcalculating the fruit content and the statisticalsecurity limits can be regarded as valid. In othercases, especially when recipe-alteration can besuspected, it will be more appropriate to performa discriminant analysis.

This has been done for purely illustrative pur-poses, using the 5 jam-samples from factory 2.

AcknowledgementsThe author thanks lic.agro. Poul Hansen forvaluable aid concerning the many analyses, lie.agro. Karl Sandvad for the regression analysesand lic.agro. Karl Kaack for many valuablediscussions throughout this work, also thanks tostud. mag. Sten Schousboe for aid with the Eng-lish translation.

266

References1) Acker, L. (1969). Über Verfälschungen von Le-

bensmitteln. Die Naturwissenschaften 56. 72-77.2) Baier, E. & Hse, P. (1908). Zeitschrift für Lebens-

mittel-Untersuchung und-Forschung 15. (140-00).3) Beythien, A. (1903). Zeitschrift für Lebensmittel-

Untersuchung und-Forschung 6. (1095-0000).4) Beythien, A. und P. Simmich (1910). Zeitschrift

für Lebensmittel-Untersuchung und-Forschung20. (241-000).

5) Box, G. E. P. (1949). A general distribution theoryfor a class of likelihood criteria. Biometrica. 36.317.

6) Choureitah, A. & Lenz, F. (1971) Inhaltstoffe beiErdbeeren (Senga Sengana) in abhängigkeit vonErnährung und Fruchtbetrag. Der Erwerbs Ost-bau 13. 133-136.

7) Goodall, H. (1969) The Composition of Fruits.B.F.M.I.R.A. Scientific and Technical Surveyes59. 1-101.

8) Härtel, F. & Soiling, J. (1911) Zeitschrift fürLebensmitteluntersuchung und - Forschung 21(168, 553).

9) Kefford, J. F. (1969) Analytical Problems withFruit Products. Food Preservation Quarterly 29.65-71.

10) Kendall, M. G. & Stuart, A. The AdvancedTheory of Statistics. London: Ch. Griffin. Vol. 13'ed 1969, vol. 2 2'ed 1967, vol. 3 2'ed 1968.

11) Kenworthy, A. L. & Harris, N. (1963) Compositionof Mclntosh, Red Delicious and Golden Deli-cious apples as related to environment and season.

12) Klinger, H. & Nehring, P. (1965) Zur Beurteilungvon Obstkonfitüren. Zeitschrift für Lebensmittel-Untersuchung und - Forschung 129, 76-83.

13) Ludwig, W. (1907). Zeitschrift für Lebensmittel-Untersuchung und - Forschung 13. (5-0).

14) Macara, T. (1931) The composition of fruits asused for jam manufacture in Great Britain.Analyst 56. 35-43.

15) Macara, T. (1935) The composition of rasp-berries. Analyst 60. 592-595.

16) Nehring, P. & Klinger, H. (1965) Zur beurteilungvon Obstkonfifüren. Zeitschrift für Lebensmittel-Untersuchung und - Forschung 129. 1-9.

17) Rahman, A. A., Shalaby, A. F. & Monayeri, M.O.El. (1971) Effect of Moisture Stress on MetabolicProducts and Ion Accumulation. Plant and Soil34. 65-90.

18) Rolle, L. A. & Vandercook, C. E. (1963) LemonJuice Composition. III. Characterization of Cali-fornia-Arizona Lemon Juice by Use of a MultipleRegression Analysis. J. Ass.off.agric. Chem. 46.362-365.

19) Steiner, E. H. (1948) Application of Statical Me-thods in Calculating Proportions of Ingredientsin certain Food Products. Analyst 73. 15-30.

20) Steiner, E. H. (1949) The Statical Use of SeveralAnalytical Constituents for Calculating Propor-tions of Ingredients in Certain Food Products.Analyst 74. 429-438.

21) Vandercook, C. E., Rolle, L. A. & Ikeda, R. M.(1963) Lemon Juice Composition. I. Characteri-zation of California-Arizona Lemon Juice byIt's Total Acid and 1-Malic Acid Content J.Ass. off. agric. Chem. 46. 353-358.

22) Vandercook, C. E. & Rolle, L. A. (1963) LemonJuice Compostion. II. Characterization of Cali-fornia-Arizona Lemon Juice by its PolyphenolicContent. J. Ass. off. agric. Chem. 46. 359-362.

23) Vandercook, C. E. & Guerrero, H. C. (1969)Citrus juice characterization. Analysis of thephosphorus fraction. J. agric. Fd. Chem. 17.626-628.

267