4Schepers.qxd


In the construction of psychometric tests it is normally assumed

that the items included in a test form a linear scale. To test this

assumption the dimensionality of the vector space of test items

must first be determined. Should it turn out to be

multidimensional, the test items must first be categorised

according to the construct measured. The categorisation of the

test items can be done with the aid of factor analysis, but the

procedure is not free of problems.

The gist of the problem concerns the fact that test items usually

vary in terms of their degree of skewness (i.e. are differentially

skew), and this affects their mutual intercorrelations. If test

items that are differentially skew are subjected to factor analysis,

a multiplicity of factors is usually obtained. This tends to obscure

the true structure of the intercorrelation matrix of test items (cf.

Ferguson, 1941).

To illustrate the problem, binary items (i.e. items scored

dichotomously), with different marginal splits, will be used.

Thereafter the principle will be generalised to items where the

responses are endorsed on continuous scales (usually five or

seven-point scales) which are often differentially skew. But to

start off with the concept ‘marginal split’ will first be clarified:

If test items are scored dichotomously, the proportion of

respondents endorsing an item according to the scoring key (pg)

and the proportion of respondents not endorsing the item

according to the scoring key (qg), will vary from item to item.

The ratio of pg to qg is usually referred to as the marginal split of

item g. In the general literature pg is usually referred to as the

‘difficulty value’ of item g. For convenience sake, the expression

‘p-value’ will be used in this paper.

In Table 1 the maximum intercorrelations of 10 test items with

different marginal splits are given. The items have been arranged

according to their marginal distributions from 0,05:0,95 to

0,95:0,05.

From Table 1 it is evident that there is a clear gradient underlying

the intercorrelations: adjacent items correlate the highest with

one another, for example items 1 and 2, whilst items that are far

apart, for example items 1 and 10, correlate the lowest. The

highest intercorrelations are adjacent to the principal diagonal

and systematically taper off as you move away from the principal

diagonal. This is true for both the rows and the columns of the

intercorrelation matrix. Such intercorrelation matrices are

known as simplexes, and must not be confused with ordinary

unidimensional intercorrelation matrices.

The inverse of a simplex has rather special properties: the

principal diagonal is positive and the two adjacent diagonals are

negative. All the other elements are equal to zero (Guttman,

1954, 1955, 1957; Schepers, 1962; Jöreskog, 1970).

The inverse of the intercorrelation matrix in Table 1 is given in

Table 2.

From Table 2 it is clear that the inverse of the intercorrelation

matrix, in Table 1, reveals all the properties of a simplex. The

implications of this will now be carefully scrutinised.

The significance of a simplex structure is clearest if the

standardised multiple regression coefficients for predicting each

variable (item) by the rest, are viewed. In the case of a perfect

simplex only adjacent variables can be used to predict a

particular variable. The regression coefficients of all the other

variables are equal to zero.

The regression coefficients for predicting each variable by the

rest, in respect of Table 1, are given in Table 3 (cf. Schepers, 1962,

p. 301).

From Table 3 it can be seen that the regression coefficients in the

two diagonals adjacent to the principal diagonal vary from

0,3052 to 0,6063. All the other coefficients are equal to zero.

As mentioned earlier, the intercorrelation matrix in Table 1

represents the maximum intercorrelations of 10 test items with

specified marginal distributions. Furthermore, the assumption

was made that all the items relate to a single central construct

(Bohrnstedt & Knoke, 1988; Gorsuch, 1974; Guilford, 1950;

Magnusson, 1967). It would therefore be logical to expect a

single factor underlying the matrix of intercorrelations. The

structure of the intercorrelation matrix was accordingly

investigated.

The eigenvalues of the unreduced intercorrelation matrix are

given in Table 4.

JOHANN M SCHEPERS
Department of Human Resource Management

Rand Afrikaans University

ABSTRACT
The principal objective of the study was to develop a procedure for overcoming the effects of differential skewness

of test items in scale construction. It was shown that the degree of skewness of test items places an upper limit on

the correlations between the items, regardless of the contents of the items. If the items are ordered in terms of

skewness the resulting intercorrelation matrix forms a simplex or a pseudo simplex. Factoring such a matrix results in

a multiplicity of factors, most of which are artefacts. A procedure for overcoming this problem was demonstrated

with items from the Locus of Control Inventory (Schepers, 1995). The analyses were based on a sample of 1662 first-

year university students.

OPSOMMING
Die hoofdoel van die studie was om ’n prosedure te ontwikkel om die gevolge van differensiële skeefheid van

toetsitems, in skaalkonstruksie, teen te werk. Daar is getoon dat die graad van skeefheid van toetsitems ’n boonste

grens plaas op die korrelasies tussen die items ongeag die inhoud daarvan. Indien die items gerangskik word volgens

graad van skeefheid, sal die interkorrelasiematriks van die items ’n simpleks of pseudosimpleks vorm. Indien so ’n

matriks aan faktorontleding onderwerp word, lei dit tot ’n veelheid van faktore waarvan die meerderheid artefakte is.

’n Prosedure om hierdie probleem te bowe te kom, is gedemonstreer met behulp van die items van die Lokus van

Beheer-vraelys (Schepers, 1995). Die ontledings is op ’n steekproef van 1662 eerstejaaruniversiteitstudente gebaseer.

OVERCOMING THE EFFECTS OF DIFFERENTIAL SKEWNESS 

OF TEST ITEMS IN SCALE CONSTRUCTION

Requests for copies should be addressed to: JM Schepers, Department of Human

Resource Management, RAU, PO Box 524, Auckland Park, 2006

27

SA Journal of Industrial Psychology, 2004, 30 (4), 27-43

SA Tydskrif vir Bedryfsielkunde, 2004, 30 (4), 27-43



SCHEPERS28

TABLE 1

IINTERCORRELATION MATRIX OF TEST ITEMS WITH DIFFERENT DEGREES OF SKEWNESS

1 2 3 4 5 6 7 8 9 10

1 1,0000 0,5461 0,3974 0,3126 0,2536 0,2075 0,1683 0,1325 0,0964 0,0526

2 0,5461 1,0000 0,7276 0,5725 0,4644 0,3800 0,3083 0,2425 0,1765 0,0964

3 0,3974 0,7276 1,0000 0,7868 0,6383 0,5222 0,4237 0,3333 0,2425 0,1325

4 0,3126 0,5725 0,7868 1,0000 0,8112 0,6637 0,5385 0,4237 0,3083 0,1683

5 0,2536 0,4644 0,6383 0,8112 1,0000 0,8182 0,6637 0,5222 0,3800 0,2075

6 0,2075 0,3800 0,5222 0,6637 0,8182 1,0000 0,8112 0,6383 0,4644 0,2536

7 0,1683 0,3083 0,4237 0,5385 0,6637 0,8112 1,0000 0,7868 0,5725 0,3126

8 0,1325 0,2425 0,3333 0,4237 0,5222 0,6383 0,7868 1,0000 0,7276 0,3974

9 0,0964 0,1765 0,2425 0,3083 0,3800 0,4644 0,5725 0,7276 1,0000 0,5461

10 0,0526 0,0964 0,1325 0,1683 0,2075 0,2536 0,3126 0,3974 0,5461 1,0000

sg 0,2179 0,3571 0,4330 0,4770 0,4975 0,4975 0,4770 0,4330 0,3571 0,2179

pg 0,05 0,15 0,25 0,35 0,45 0,55 0,65 0,75 0,85 0,95

qg 0,95 0,85 0,75 0,65 0,55 0,45 0,35 0,25 0,15 0,05

Note:
–

,
gk g k

g g g phi
g k

p p p
s p q r

s s
= =

TABLE 2

INVERSE OF INTERCORRELATION MATRIX

1 2 3 4 5 6 7 8 9 10

1 1,4250 -0,7782

2 -0,7782 2,5500 -1,5462

3 -1,5462 3,7500 -2,0653

4 -2,0653 4,5500 -2,3729

5 -2,3729 4,9500 -2,4750

6 -2,4750 4,9500 -2,3729

7 -2,3729 4,5500 -2,0653

8 -2,0653 3,7500 -1,5462

9 -1,5462 2,5500 -0,7782

10 -0,7782 1,4250

TABLE 3

STANDARDISED MULTIPLE REGRESSION COEFFICIENTS FOR PREDICTING EACH ITEM BY THE REST

1 2 3 4 5 6 7 8 9 10

1 - 0,5461

2 0,3052 - 0,6063

3 0,4123 - 0,5508

4 0,4539 - 0,5215

5 0,4794 - 0,5000

6 0,5000 - 0,4794

7 0,5215 - 0,4539

8 0,5508 - 0,4123

9 0,6063 - 0,3052

10 0,5461



TABLE 4

EIGENVALUES OF INTERCORRELATION MATRIX

Root Eigenvalue 

1 5,0752

2 1,7929

3 1,0005

4 0,6704

5 0,4671

6 0,3281

7 0,2374

8 0,1785

9 0,1389

10 0,1111 

Trace 10,000 

From Table 4 it is clear that there are three eigenvalues greater

than unity. Accordingly three factors were postulated (Kaiser,

1961).

Next, the intercorrelation matrix was subjected to a principal

factor analysis. Three factors were extracted and rotated to

simple structure by means of a Varimax rotation. The rotated

factor matrix is given in Table 5.

TABLE 5

ROTATED FACTOR MATRIX (VARIMAX)

Variable Factor 1 Factor 2 Factor 3 h2j

1 0,071 0,545 0,060 0,3056

2 0,192 0,874 0,092 0,8088

3 0,464 0,730 0,083 0,7553

4 0,676 0,542 0,114 0,7643

5 0,822 0,347 0,192 0,8322

6 0,822 0,192 0,347 0,8322

7 0,676 0,114 0,542 0,7643

8 0,464 0,083 0,730 0,7553

9 0,192 0,092 0,874 0,8088

10 0,071 0,060 0,545 0,3056

Table 5 shows that items 4, 5, 6 and 7 have high loadings on

Factor 1, whilst items 1, 2 and 3 have high loadings on Factor 2.

Items 8, 9 and 10 have high loadings on Factor 3. Items 1, 2 and

3 have p-values between 0,05 and 0,25; items 4, 5, 6 and 7 have

p-values between 0,35 and 0,65, and items 8, 9 and 10 have p-

values between 0,75 and 0,95. The items therefore cluster

according to their marginal distributions.

It is also striking that the communalities of items 1 and 10 are

considerably lower than the rest. This phenomenon is typical of

simplexes, because in a simplex it is only adjacent items that

share common variance, and the first and the last items have only

one adjacent item each.

All the limitations that have been referred to above are the direct

consequence of the simplex structure of the particular

intercorrelation matrix. But the most troublesome aspect is the

artefactual factors that emerge if a simplex is subjected to a

principal factor analysis. In the present case there were only

three factors, but it is not uncommon to obtain a multiplicity of

factors, particularly in respect of large matrices.

Different solutions have already been proposed to overcome the

problem of artefactual factors. Gorsuch (1974, p. 262), for

instance, maintains that the best solution is to avoid working

with variables that are too skew. His suggestion is highly

acceptable, provided the variables used derive from well

constructed measuring instruments. Where necessary the

obtained scores can even be normalised before doing a factor

analysis. But when working with test items his suggestion is not

acceptable. The discrimination power and reliability of a test will

be lowered if all the items have equal p-values. A wide

distribution of p-values (0,10 to 0,90) is necessary to ensure

good discrimination power and reliability. The rationale for this

should be apparent from an analysis of Kuder-Richardson

Formula 20 (Kuder & Richardson, 1937). The derivation of the

formula is given in Appendix 1.

, where  (1)

K = number of test items

m = mean of test

pg = proportion of subjects endorsing item g according to the key

=  variance of pg’s

The larger K�p2, the larger the test reliability would be (Horst,
1953).

Tucker (1949, p. 119) has shown that the variance of the p-values

for a rectangular distribution of the pg’s, varying from 0,00 to

1,00, is equal to 0,083. For a normal distribution with p = 0,00 at

-3� and p = 1,00 at +3�, the variance is equal to 0,028.

From the foregoing it is clear that for a longer test or scale, with

a wide distribution of pg’s, the value of K�p2 can be quite
substantial.

Formula 1 applies only to binary data.

For continuous data, Kuder-Richardson Formula 20 and

Cronbach’s coefficient alpha (Cronbach, 1951), can be used. The

transformed KR20 formula can be written as follows:

,                          (2)

where

K = number of test items

= sum of item variances

�x2 = test variance

[The derivation of Formula 2 is given in Appendix 1].

The total test variance can never exceed (cf. Schepers,

1992, p. 33), therefore, the greater the test variance, the higher the

reliability will be.

From the foregoing it is clear that Gorsuch’s suggestion of

avoiding items that are too skew, will lead to a reduction in test

reliability.

2

1

K

g

g

K
=

σ∑

2

1

K

g

g =
σ∑

( )

2 2

1
20 2

–

1–
– 1

k

g x

g

x

K

KR
K

=

 
σ σ 

 =  σ 
  

∑

2
pσ

2
2

20 2

– –
1–

– 1

p

x

K
K KKR

K

 µ
µ σ 

 =
σ 

  

DIFFERENTIAL SKEWNESS OF TEST TERMS 29



Horst (1965, p. 516) maintained that one way of getting rid of the

artefactual factors is to fit an appropriate simplex to the binary

data matrix and to separate it from the true structure. This is

equivalent to throwing the baby away with the bath water.

The fitting of a simplex poses problems of its own. Firstly, an

appropriate model must be decided upon, and then an

appropriate method must be found to fit the model to the data.

If the measuring units of all the variables are the same, a simplex

can be fitted to the variance-covariance matrix, but if the units of

measurement differ, the intercorrelation matrix must be used

(Jöreskog, 1970).

Jöreskog (1970, p. 122) distinguishes six different simplex

models. These models can broadly be divided into Markov

simplexes and Wiener simplexes. The Markov simplexes can be

subdivided into perfect simplexes, quasi-simplexes and restricted

quasi-simplexes. Similarly, Wiener simplexes can be subdivided

into perfect simplexes, quasi-simplexes and quasi-simplexes with

equal error variances. Markov simplexes are scale free, i.e. the

measuring units of the variables do not have to be the same.

Wiener simplexes, on the other hand, are scale dependent, ie. they

can only be used if the units of measurement of all the variables

are the same (Jöreskog, 1970, p. 128).

Perfect simplexes only arise if true scores are used. The moment

there is measurement error it becomes a quasi-simplex. According

to Jöreskog (1970, p. 130) the fitting of a perfect simplex is

straightforward, but the fitting of quasi-simplexes is complex

and requires iterative procedures. Reasonable fits can be

obtained with the LISREL-program (Jörgeskog & Sorbom, 1982).

There are many intercorrelation matrices that superficially

resemble simplexes, but that are really pseudo-simplexes. This

complicates the issue of fitting simplexes even further. It is also

doubtful whether the fitting of a simplex overcomes the

problem of differential skewness, and the generation of

‘difficulty factors’.

So far the research problem has been developed in terms of

binary items that are differentially skew. It will now be

broadened to continuous items that are differentially skew.

The principal objectives of the present study were to:

1. Examine the gradients of the correlations (by row and by

column) in an intercorrelation matrix based on continuous

variables, arranged in terms of degree of skewness.

2. Determine the factor structure of an intercorrelation matrix

based on continuous variables, arranged in terms of degree

of skewness.

3. Develop and evaluate a procedure for overcoming the effects

of differential skewness on the factor structure of test items,

and to examine the properties of the resulting scales.

METHOD

Sample

The full complement of first-year university students at the

Rand Afrikaans University, during 1995, was subjected to an

extensive psychometric test programme. The programme

stretched over four days, and to ensure complete records,

students who did not attend all the test sessions were excluded

from the sample. The final sample consisted of 1662 students,

and can be considered representative of the population of first-

year university students at the Rand Afrikaans University,

during 1995. The ages of the students varied from 26 to 54

years, with a mean of 27,30 years and a standard deviation of

1,842 years. As far as gender is concerned 49,8% were female

and 47,2% were male. Missing information accounted for 3,0%.

The majority of the students were Afrikaans-speaking (969).

Three hundred and seventy nine were English-speaking, and

195 spoke both English and Afrikaans. Only 27 had an African

language as vernacular. Thirty-nine spoke other languages, and

53 did not indicate their home language. As far as ethnic group

is concerned 88,7% were White, 1,4% were Indian, 4,7% were

Coloured and 2,2% were African.

Measuring instrument

For the purpose of this study the items of the Locus of Control

Inventory (Schepers, 1995), were used.

The construct of ‘locus of control’ was created by Rotter

(1966) and pertains to a person’s expectat ion of

reinforcement of his/her behaviour, arising from the social

environment. Therefore, it is theoretically based on social

learning theory (Mischel, 1979). Rotter (1966) distinguished

bet ween t wo different orientations in people, namely an

inter nal control or ientation and an external control

orientation. People with an internal control orientation are

convinced that their behaviour depends on their own

achievements, abilities and dedication, whereas people with

an external control orientation believe that random or

fortuitous events, fate, Lady Luck and certain influential

people are responsible for their behaviour.

Concept ually, the Locus of Control Inventory (LCI) is 

based on attribution theory and social learning theory. People

are constantly att uned to finding the causes of their

behaviour and those of others. The attributing of causes to

specific behaviour is called attr ibutions. The causative

attributions that people make and their interpretation thereof

determine their perceptions of the social world to a large

extent. Is it a friendly or a hostile world? Is it a just or unjust

world? Is it a predictable or an unpredictable world? Can we

exercise control over certain events through our own abilities

or are our lives controlled by certain influential people?

According to at tribut ion theory the causes of human

behaviour can be divided into t wo broad categories, namely

dispositional causes and situational causes. Dispositional

causes pertain to one’s nat ural disposition and include the

organismic attributes of people. Situational causes pertain to

the external world and include all environmental factors

(Roediger III et al., 1991).

Social learning theory links up with attribution theory: whereas

social learning theory deals with the nature of reinforcements

arising from the social behaviour of the learner, attribution

theory pertains to the way in which a person gathers information

about the stable or invariant characteristics of others – their

motives, intentions and traits – as well as the external world

(Baron, Byrne & Kantowitz, 1980).

A construct closely related to internal control is autonomy.

Autonomy can be defined as ‘the tendency to attempt to master

or be effective in the environment, to impose one’s wishes and

designs on it’ (Wolman, 1973, p. 37). It is expected that persons

high on autonomy would seek control of situations that offer

possibilities of change, would readily accept the challenge of

solving complex problems, would take the initiative in

situations requiring leadership, would prefer to work on their

own and to structure their own work programme.

With attribution theory and social learning theory as frames of

reference, the domain of locus of control was extensively

sampled. Altogether 80 items were written, representing the

constructs of internal control, external control and autonomy.

Roughly equal numbers of items were written in respect of each

of the constructs.

The items of the LCI are all in the form of questions and

the responses are endorsed on a seven-point scale. Only 

the end-points of the scales are verbally anchored. Separate

answer sheets that can be read by an optical page reader are

used, and the responses can be read directly onto a stiff y 

or compact disk.

SCHEPERS30



Procedure

The LCI was applied to the full complement of first-

year university students at the Rand Afrikaans University

during 1995. Thereafter the answer sheets were carefully

scrutinised for double markings or incompleteness. Where

necessary the markings were made clearer and stray marks

were erased. If more than two items were left blank or 

spoilt, the person’s record was not used. If one or two 

items were left blank or spoilt, the person’s mean for that

construct (according to the a priori key) was estimated, and

his/her item mean was substituted for the particular item.

Care was taken not to estimate item means if several

respondents skipped the same item. In this way 1662 complete

records were obtained.

Statistical analysis

In order to examine the gradients of the correlations in 

an intercorrelation matrix of continuous variables that have

been arranged according to their degree of skewness, the

means, standard deviations, coefficients of skewness 

and kurtosis of the 80 items of the LCI were computed. 

Next, the items were arranged according to their degree of

skewness. To ensure stabilit y of the data, composite 

scores were formed by adding the scores of eight adjacent

items together. In this way 10 new variables were formed 

The 10 new variables were then intercorrelated. Following 

this the inverse of the obtained intercorrelation matrix 

was computed.

To determine the factor structure of an intercorrelation matrix

based on continuous variables arranged in terms of degree of

skewness, the same data as above were used, except that the

subscores were based on parcels of four adjacent items instead of

eight. The resulting 20 subscores were then intercorrelated and

subjected to a principal factor analysis. The obtained factor

matrix was rotated to simple structure by means of a Direct

Oblimin rotation.

To overcome the effects of differential skewness of the items of

the LCI, the following procedure (as suggested in Schepers, 1992)

was followed:

1. The 80 items were intercorrelated.

2. The eigenvalues of the unreduced intercorrelation matrix

were calculated.

3. As many factors as there were eigenvalues greater than unity

were postulated.

4. An iterative principal factor analysis was done.

5. Iteration was done on the number of factors as determined

at step 3.

6. The obtained factor matrix was rotated to simple structure by

means of a Varimax rotation.

7. All the items with high negative loadings were reflected.

8. All the items with high loadings on a specific factor were

added together and a subscore for each factor was computed.

Every item was used only once.

9. The obtained subscores were intercorrelated and steps 2 to 4

were repeated.

10. The obtained factor matrix was rotated to simple structure by

means of a Direct Oblimin rotation.

11. All subscores with negative loadings on the first principal axis

were reflected.

12. Separate scales were formed, corresponding to each of the

factors, by grouping all the items together that had

substantial loadings on a factor, ie. all the items in the

relevant subscores (cf. step 8).

13. Separate item analyses (NP50) were done for each of the

scales formed.

14. Iteration was done in terms of the indices of reliability of the

test items.

15. The reliability of the scales were determined by means of

Cronbach’s coefficient alpha.

RESULTS

Objective 1: Gradients of correlations in an intercorrelation

matrix based on continuous variables, arranged in terms of

skewness

As a first step, the means, standard deviations, and coefficients

of skewness of the 80 items of the LCI were computed. Next, the

items were arranged according to their degree of skewness. The

descriptive statistics are given in Table 6.

TABLE 6

MEANS, STANDARD DEVIATIONS AND COEFFICIENTS OF

SKEWNESS OF THE TEST ITEMS

Variable Mean Standard Coefficient 

deviation of skewness

New variable 1 Q19 6,3285 0,8537 -1,7592

Q60 6,1330 1,1200 -1,6226

Q61 5,9675 1,1309 -1,6141

Q42 6,1005 1,0588 -1,5781

Q37 5,9537 1,0426 -1,4967

Q18 6,0632 1,0177 -1,4407

Q49 6,2202 0,8993 -14065

Q10 6,1258 0,9300 -1,2905

New variable 2 Q63 6,1943 0,9003 -1,2590

Q75 5,9693 1,0867 -1,2468

Q59 5,7443 1,3507 1,2163

Q31 5,8207 1,1219 -1,1491

Q13 5,7966 0,9513 -1,0591

Q62 5,6227 1,2052 -1,0363

Q28 5,4633 1,3655 -1,0208

Q8 5,6071 1,1489 -1,0048

New variable 3 Q22 5,8039 1,1023 -0,9947

Q16 5,1799 1,6189 -0,9934

Q67 5,7635 1,0554 -0,9602

Q66 6,0897 0,9021 -0,9457

Q69 5,7419 1,1072 -0,9350

Q48 5,3538 1,3030 -0,9026

Q24 5,4398 1,2491 -0,8827

Q54 5,4266 1,1774 -0,8541

New variable 4 Q40 5,3057 1,3288 -0,8485

Q2 5,4940 1,2772 -0,8381

Q73 5,1986 1,2821 -0,8258

Q70 5,6492 1,2166 -0,8215

Q6 5,6859 1,0957 -0,8036

Q68 5,7389 1,0071 -0,7734

Q46 5,3995 1,1133 -0,7371

Q76 5,0235 1,5189 -0,7322

New variable 5 Q55 5,5890 1,0004 -0,7082

Q27 5,6227 1,0881 -0,6982

Q7 5,6637 1,0453 -0,6947

Q17 5,3153 1,2883 -0,6697

Q29 5,2882 1,2113 -0,6390

Q32 5,2515 1,2339 -0,6214

Q30 5,2058 1,3249 -0,6133

Q25 5,2744 1,3027 -0,6092

New variable 6 Q14 5,2010 1,1651 -0,6063

Q74 5,2100 1,1291 -0,6023

Q5 5,2557 1,0171 -0,5963

Q44 5,1871 1,0226 -0,5259

Q39 4,7786 1,3998 -0,5103

Q9 5,0391 1,2906 -0,5086

Q26 4,6546 1,5714 -0,4864

Q1 4,7960 1,3723 -0,4761

DIFFERENTIAL SKEWNESS OF TEST TERMS 31



New variable 7 Q64 4,3454 1,2960 -0,4112

Q50 4,7569 1,3762 -0,3913

Q47 4,6191 1,4449 -0,3822

Q15 4,6907 1,5028 -0,3266

Q3 4,5000 1,3634 -0,3011

Q77 4,3063 1,7838 -0,2711

Q71 4,4302 1,2917 -0,1796

Q51 3,6516 1,3752 -0,0815

New variable 8 Q72 4,2942 1,3370 -0,0414

Q65 3,7100 1,6761 -0,0330

Q80 3,6342 1,5409 0,0431

Q38 3,5812 1,4367 0,1515

Q20 33965 1,3670 0,1660

Q4 3,6203 1,8354 0,1662

Q21 3,4260 1,3251 0,1872

Q57 3,2816 1,4653 0,2859

New variable 9 Q36 3,1949 1,4547 0,2941

Q43 3,1270 1,6261 0,4077

Q56 3,0355 1,4983 0,5286

Q35 2,8670 1,5248 0,5352

Q33 2,8008 1,4597 0,5819

Q12 2,7882 1,5176 0,5919

Q23 2,9777 1,4437 0,6314

Q79 2,6901 1,4626 0,6807

New variable 10 Q34 2,6444 1,5183 0,7148

Q78 2,9362 1,6902 0,7495

Q41 2,5487 1,4407 0,8386

Q45 2,5048 1,3590 0,9770

Q11 2,4711 1,3919 0,9944

Q58 2,3424 1,5642 1,1727

Q53 2,1721 1,3036 1,2429

Q52 2,1420 1,4670 1,4317

Note:  Items arranged according to degree of skewness.

From Table 6 it can be seen that the coefficients of skewness vary

from –1,7592 to 1,4317.

Next, new variables were formed by adding the scores of eight

adjacent items together. The new variables were then arranged in

terms of their degree of skewness. The descriptive statistics of

the new variables are given in Table 7.

TABLE 7

MMEANS, STANDARD DEVIATIONS AND COEFFICIENTS OF SKEWNESS

AND KURTOSIS OF THE NEW VARIABLES

Variable Standard Coefficient Coefficient 

deviation of skewness of kurtosis

New variable 1 48,892 4,750 -1,006 1,605

New variable 2 46,218 4,677 -0,421 0,109

New variable 3 44,799 5,102 -0,352 0,397

New variable 5 43,211 5,029 -0,343 0,046

New variable 4 43,495 4,962 -0,233 0,072

New variable 6 40,122 4,724 -0,065 -0,019

New variable 7 35,300 4,534 -0,054 0,189

New variable 8 28,944 5,453 0,012 -0,176

New variable 9 23,481 6,441 0,276 -0,254

New variable 10 19,762 6,219 0,478 0,125

Note: Standard error of coeffient of skewness = 0,06

Standard error of coefficient of kurtosis = 0,12

From Table 7 it can be seen that the coefficients of skewness

range from –1,006 to 0,478. The new variables are therefore less

skew than the original items.

Next, the new variables were intercorrelated. The matrix of

intercorrelations is given in Table 8.

Table 8 shows that the highest correlations are adjacent to the

principal diagonal and taper off as you move from left to right

and from top to bottom. However, small departures from this

trend is also visible. These gradients are typical of a simplex.

Furthermore, the column totals also reveal the typical pattern

of a simplex: the successive totals increase in size until a

maximum is reached, and then systematically decrease in size.

However, a more objective test is to inspect the inverse of the

correlation matrix.

The inverse of the intercorrelation matrix was accordingly

calculated and is given in Table 9.

It is evident from Table 9 that the principal diagonal is positive

and the two adjacent diagonals are negative. However, the off-

diagonals are not equal to zero. The intercorrelation matrix is

therefore a pseudo-simplex.

From the foregoing it is clear that if continuous variables are

arranged according to their degree of skewness, the correlations

between the variables show gradients similar to that of a simplex.

It is therefore expected that such matrices will generate factors of

skewness, if factor analysed.

Objective 2: Factor structure of an intercorrelation matrix

based on continuous variables, arranged in terms of degree of

skewness

To determine the factor structure of an intercorrelation matrix

based on continuous variables, arranged according to degree of

skewness, the LCI data were used. Parcels of four adjacent items

in terms of skewness, were formed. The descriptive statistics are

given in Table 10.

From Table 10 it is clear that the coefficients of skewness vary

from –1,16 to 0,77.

Next, the new variables were intercorrelated. The matrix of

intercorrelations is given in Table 11.

Although there is not a clear gradient visible in Table 11, the

correlations nevertheless become smaller as you move from left

to right and from top to bottom.

The eigenvalues of the unreduced intercorrelation matrix are

given in Table 12.

From Table 12 it is evident that there are three eigenvalues greater

than unity, suggesting three factors.

Three factors were extracted and rotated to simple structure by

means of a Direct Oblimin rotation. The rotated factor matrix is

given in Table 13.

From Table 13 it is clear that parcels 1, 2, 3 and 9 have high

loadings on Factor III. Parcels 4 to 14 (excluding parcel 9) load

on Factor I, and parcels 15 to 20 load on  Factor II. Each of the

factors have high loadings on parcels with fairly similar

coefficients of skewness. The only exception being parcel 9. It

is therefore reasonable to identif y the three factors as factors

of skewness.

In order to examine the contents of the three factors obtained,

the items represented by the subscores were listed. The listing of

the items is given in Table 14.

SCHEPERS32



TABLE 10

MEANS, STANDARD DEVIATIONS AND COEFFICIENTS OF SKEWNESS

AND KURTOSIS OF THE NEW VARIABLES

Variable Standard Coefficient Coefficient 

deviation of skewness of kurtosis

New variable 2 24,36 2,61 -1,16 2,54

New variable 1 24,53 2,74 -0,92 0,83

New variable 3 23,73 2,82 -0,70 0,86

New variable 5 22,84 3,12 -0,54 0,29

New variable 10 21,02 3,14 -0,45 0,22

New variable 4 22,49 2,79 -0,43 -0,03

New variable 7 21,65 3,05 -0,38 0,15

New variable 6 21,96 2,89 -0,34 0,23

New variable 9 22,19 2,78 -0,33 -0,09

New variable 11 20,85 3,03 -0,33 0,10

New variable 8 21,85 2,81 -0,20 -0,14

New variable 14 16,89 2,87 -0,12 0,13

New variable 13 18,41 2,98 -0,10 0,01

New variable 12 19,27 2,81 0,01 -0,05

New variable 15 15,22 3,04 0,01 0,16

New variable 16 13,72 3,72 0,04 -0,48

New variable 17 12,22 3,94 0,21 -0,48

New variable 19 10,63 3,84 0,36 -0,19

New variable 18 11,26 3,57 0,37 0,03

New variable 20 9,13 3,65 0,77 0,38 

Note: Standard error of coefficient of skewness = 0,06

Standard error of coefficient of kurtosis = 0,12

From Table 14 it is clear that 24 of the 40 items loading on Factor

I, are classified as Autonomy according to the scoring key. Eleven

of the items are classified as Internal Control and five are

classified as External Control.

As far as Factor II is concerned 23 of the 24 items are classified

as External Control and one as Autonomy.

Fifteen of the 16 items loading on Factor III are classified as

Internal Control and one as Autonomy.

From the foregoing it is clear that 24 of the 26 Autonomy

items (according to the key) were correctly classified, 23 

of the 28 External Control items were correctly classified, 

and 15 of the 26 Internal Control items were correctly

classified. Eleven of the Internal Control items were

misclassified as Autonomy. This is also evident in the 

matrix of intercorrelations of the factors (see Table 13): 

Factor I (Autonomy) correlates 0,496 with Factor III 

(Internal Control).

According to Table 10 the coefficients of skewness of 

the parcels loading on the first factor, range from –1,16 to

–0,70, and those loading on the third factor, range from 

–0,54 to 0,01. The coefficients of skewness of the parcels

loading on the second factor range from 0,01 to 0,77. The

coefficients of skewness of the parcels loading on Factors I 

and III are thus essentially negative, whereas those of Factor II

are positive.

DIFFERENTIAL SKEWNESS OF TEST TERMS 33

TABLE 8

MATRIX OF INTERCORRELATIONS OF THE NEW VARIABLES ARRANGED ACCORDING TO THEIR DEGREE OF SKEWNESS

New var 1 New var 2 New var 3 New var 5 New var 4 New var 6 New var 7 New var 8 New var 9 New var 10 Total

New var 1 1,000 0,640 0,539 0,548 0,535 0,386 0,174 -0,152 -0,157 -0,303 3,211

New var 2 0,640 1,000 0,606 0,602 0,600 0,533 0,224 -0,214 -0,237 -0,379 3,375

New var 3 0,539 0,606 1,000 0,670 0,660 0,554 0,280 -0,158 -0,150 -0,270 3,732

New var 5 0,548 0,602 0,670 1,000 0,615 0,565 0,266 -0,134 -0,078 -0,253 3,801

New var 4 0,535 0,600 0,660 0,615 1,000 0,556 0,258 -0,210 -0,193 -0,332 3,488

New var 6 0,386 0,533 0,554 0,565 0,556 1,000 0,351 -0,121 -0,130 -0,233 3,461

New var 7 0,174 0,224 0,280 0,266 0,258 0,351 1,000 0,230 0,158 0,151 3,093

New var 8 -0,152 -0,214 -0,158 -0,134 -0,210 -0,121 0,230 1,000 0,444 0,515 1,201

New var 9 -0,157 -0,237 -0,150 -0,078 -0,193 -0,130 0,158 0,444 1,000 0,663 1,319

New var 10 -0,303 -0,379 -0,270 -0,253 -0,332 -0,233 0,151 0,515 0,663 1,000 0,559

Total 3,211 3,375 3,732 3,801 3,488 3,461 3,093 1,201 1,319 0,559

TABLE 9

INVERSE OF INTERCORRELATION MATRIX

New var 1 New var 2 New var 3 New var 5 New var 4 New var 6 New var 7 New var 8 New var 9 New var 10

New var 1 1,909 -0,783 -0,230 -0,322 -0,260 0,156 0,001 -0,045 -0,050 0,144

New var 2 -0,783 2,390 -0,316 -0,331 -0,271 -0,348 -0,095 0,046 0,099 0,253

New var 3 -0,230 -0,316 2,414 -0,719 -0,652 -0,263 -0,127 0,036 0,064 -0,039

New var 5 -0,322 -0,331 -0,719 2,333 -0,357 -0,411 -0,034 0,010 -0,265 0,135

New var 4 -0,260 -0,271 -0,652 -0,357 2,266 -0,349 -0,133 0,120 0,016 0,171

New var 6 0,156 -0,348 -0,263 -0,411 -0,349 1,842 -0,346 0,030 0,044 0,060

New var 7 0,001 -0,095 -0,127 -0,034 -0,133 -0,346 1,328 -0,281 -0,051 -0,225

New var 8 -0,045 0,046 0,036 0,010 0,120 0,030 -0,281 1,463 -0,247 -0,484

New var 9 -0,050 0,099 0,064 -0,265 0,016 0,044 -0,051 -0,247 1,871 -1,117

New var 10 0,144 0,253 -0,039 0,135 0,171 0,060 -0,225 -0,484 -1,117 2,257 



TABLE 12

EIGENVALUES OF INTERCORRELATION MATRIX

Root Eigenvalue 

1 6,251

2 2,765

3 1,202

4 0,949

5 0,807

6 0,793

7 0,774

8 0,686

9 0,611

10 0,589

11 0,552

12 0,544

13 0,513

14 0,487

15 0,456

16 0,442

17 0,435

18 0,401

19 0,388

20 0,376 

Trace 20,000 

SCHEPERS34

TABLE 11

MATRIX OF INTERCORRELATIONS OF THE NEW VARIABLES ARRANGED ACCORDING TO THEIR DEGREE OF SKEWNESS

Variables New var 2 New var 1 New var 3 New var 5 New var 10 New var 4 New var 7 New var 6 New var 9 New var 11

New var 2 1,000 0,577 0,546 0,390 0,330 0,383 0,326 0,392 0,500 0,360

New var 1 0,577 1,000 0,538 0,428 0,365 0,428 0,401 0,416 0,476 0,394

New var 3 0,546 0,538 1,000 0,397 0,363 0,394 0,372 0,397 0,464 0,386

New var 5 0,390 0,428 0,397 1,000 0,508 0,465 0,454 0,442 0,469 0,520

New var 10 0,330 0,365 0,363 0,508 1,000 0,435 0,458 0,471 0,443 0,526

New var 4 0,383 0,428 0,394 0,465 0,435 1,000 0,453 0,462 0,454 0,563

New var 7 0,326 0,401 0,372 0,454 0,458 0,453 1,000 0,491 0,442 0,545

New var 6 0,392 0,416 0,397 0,442 0,471 0,462 0,491 1,000 0,485 0,501

New var 9 0,500 0,476 0,464 0,469 0,443 0,454 0,442 0,485 1,000 0,486

New var 11 0,360 0,394 0,386 0,520 0,526 0,563 0,545 0,501 0,486 1,000

New var 8 0,423 0,461 0,427 0,461 0,392 0,445 0,429 0,493 0,480 0,488

New var 14 0,001 0,026 -0,055 0,095 0,140 0,108 0,115 0,113 0,046 0,143

New var 13 0,234 0,212 0,224 0,247 0,276 0,296 0,257 0,279 0,225 0,362

New var 12 0,174 0,162 0,209 0,284 0,286 0,264 0,256 0,190 0,230 0,306

New var 15 -0,080 -0,122 -0,137 -0,077 0,001 -0,084 -0,095 -0,063 -0,111 -0,075

New var 16 -0,077 -0,149 -0,192 -0,146 -0,090 -0,149 -0,202 -0,133 -0,164 -0,172

New var 17 -0,038 -0,116 -0,173 -0,040 0,048 -0,113 -0,090 -0,055 -0,066 -0,034

New var 19 -0,160 -0,247 -0,284 -0,206 -0,109 -0,216 -0,230 -0,197 -0,193 -0,159

New var 18 -0,130 -0,201 -0,230 -0,208 -0,091 -0,168 -0,163 -0,145 -0,140 -0,119

New var 20 -0,216 -0,269 -0,316 -0,190 -0,139 -0,236 -0,249 -0,167 -0,292 -0,216

Total 4,936 4,781 4,328 5,294 5,612 5,186 4,969 5,373 5,235 5,804

New var 8 New var 14 New var 13 New var 12 New var 15 New var 16 New var 17 New var 19 New var 18 New var 20

0,423 0,001 0,234 0,174 -0,080 -0,077 -0,038 -0,160 -0,130 -0,216

0,461 0,026 0,212 0,162 -0,122 -0,149 -0,116 -0,247 -0,201 -0,269

0,427 -0,055 0,224 0,209 -0,137 -0,192 -0,173 -0,284 -0,230 -0,316

0,461 0,095 0,247 0,284 -0,077 -0,146 -0,040 -0,206 -0,208 -0,190

0,392 0,140 0,276 0,286 0,001 -0,090 0,048 -0,109 -0,091 -0,139

0,445 0,108 0,296 0,264 -0,084 -0,149 -0,113 -0,216 -0,168 -0,236

0,429 0,115 0,257 0,256 -0,095 -0,202 -0,090 -0,230 -0,163 -0,249

0,493 0,113 0,279 0,190 -0,063 -0,133 -0,055 -0,197 -0,145 -0,167

0,480 0,046 0,225 0,230 -0,111 -0,164 -0,066 -0,193 -0,140 -0,292

0,488 0,143 0,362 0,306 -0,075 -0,172 -0,034 -0,159 -0,119 -0,216

1,000 0,045 0,249 0,206 -0,118 -0,142 -0,119 -0,228 -0,199 -0,226

0,045 1,000 0,202 0,170 0,254 0,225 0,194 0,188 0,130 0,207

0,249 0,202 1,000 0,195 0,066 0,042 0,081 0,014 0,014 -0,013

0,206 0,170 0,195 1,000 0,004 -0,047 -0,071 -0,114 -0,147 -0,133

-0,118 0,254 0,066 0,004 1,000 0,291 0,266 0,316 0,252 0,251

-0,142 0,225 0,042 -0,047 0,291 1,000 0,374 0,375 0,316 0,415

-0,119 0,194 0,081 -0,071 0,266 0,374 1,000 0,533 0,470 0,445

-0,228 0,188 0,014 -0,114 0,316 0,375 0,533 1,000 0,555 0,381

-0,199 0,130 0,014 -0,147 0,252 0,316 0,470 0,555 1,000 0,346

-0,226 0,207 -0,013 -0,133 0,251 0,415 0,445 0,381 0,346 1,000

4,965 3,348 4,462 3,423 1,737 1,375 2,494 1,017 1,143 0,382 



TABLE 13

ROTATED FACTOR MATRIX (DIRECT OBLIMIN)

VARIABLES K FACTOR I FACTOR II FACTOR III h2j

New var 11: Items 5, 14, 4 0,737 -0,077 0,064 0,611

44 and 74

New var 10: Items 25, 29, 4 0,616 0,026 0,125 0,468

30 and 32

New var 7: Items 2, 40, 4 0,599 -0,155 0,091 0,470

70 and 73

New var 4: Items 8, 13, 4 0,557 -0,116 0,170 0,472

28 and 62

New var 5: Items 16, 22, 4 0,548 -0,085 0,195 0,471

66 and 67

New var 6: Items 24, 48, 4 0,519 -0,046 0,237 0,461

54 and 69

New var 12: Items 1, 9, 4 0,430 -0,088 -0,065 0,172

26 and 39

New var 13: Items 15, 47, 4 0,416 0,149 0,082 0,217

50 and 64

New var 8: Items 6, 46, 4 0,406 -0,103 0,326 0,444

68 and 76

New var 14: Items 3, 51, 4 0,392 0,310 -0,164 0,226

71 and 77

New var 17: Items 35, 36, 4 0,028 0,747 0,110 0,514

43 and 56

New var 19: Items 34, 41, 4 -0,100 0,726 0,017 0,540

45 and 78

New var 18: Items 12, 23, 4 -0,107 0,643 0,057 0,410

33 and 79

New var 16: Items 4, 20, 4 -0,066 0,570 0,025 0,325

21 and 57

New var 20: Items 11, 52, 4 -0,059 0,551 -0,138 0,391

53 and 58

New var 15: Items 38, 65, 4 0,086 0,422 -0,083 0,203

72 and 80

New var 2: Items 10, 18, 4 0,029 0,107 0,795 0,809

37 and 49

New var 1: Items 19, 42, 4 0,150 -0,039 0,638 0,544

60 and 61

New var 3: Items 31, 59, 4 0,136 -0,124 0,591 0,515

63 and 75

New var 9: Items 7, 17, 4 0,361 -0,046 0,438 0,498

27 and 55

Number of items per factor 40 24 16

INTERCORRELATIONS OF FACTORS

VARIABLES FACTOR I FACTOR II FACTOR III

FACTOR I 1,000 -0,086 0,496

FACTOR II -0,086 1,000 -0,339

FACTOR III 0,496 -0,339 1,000

Note: Factor I = Autonomy

Factor II = External Control

Factor III = Internal Control

TABLE 14

ITEMS CORRECTLY AND INCORRECTLY CLASSIFIED

ACCORDING TO CONTENT

AUTONOMY EXTERNAL CONTROL INTERNAL  CONTROL

Q1 A Q4 E Q7 I

Q2 A Q11 E Q10 I

Q3 A Q12 E Q17 A *

Q5 A Q20 E Q18 I

Q6 I * Q21 E Q19 I

Q8 I * Q23 E Q27 I

Q9 E ** Q33 E Q31 I

Q13 A Q34 E Q37 I

Q14 A Q35 E Q42 I

Q15 A Q36 E Q49 I

Q16 I * Q38 E Q55 I

Q22 A Q41 E Q59 I

Q24 A Q43 E Q60 I

Q25 I * Q45 E Q61 I

Q26 I * Q52 E Q63 I

Q28 A Q53 E Q75 I

Q29 A Q56 E

Q30 A Q57 E

Q32 I * Q58 E

Q39 A Q65 E

Q40 I * Q72 A *

Q44 A Q78 E

Q46 A Q79 E

Q47 E ** Q80 E

Q48 I *

Q50 E **

Q51 E **

Q54 I *

Q62 A

Q64 A

Q66 A

Q67 A

Q68 A

Q69 I *

Q70 A

Q71 A

Q73 A

Q74 A

Q76 I *

Q77 E ** 

Note: A = Autonomy; E = External Control; I = Internal Control

Items marked with * and ** have been misclassified according to the scoring key

DIFFERENTIAL SKEWNESS OF TEST TERMS 35



TABLE 16

EIGENVALUES OF UNREDUCED INTERCORRELATION MATRIX (17 X 17)

Root Eigenvalue 

1 4,727540

2 2,373090

3 1,103630

4 1,002740

5 0,861917

6 0,837644

7 0,773758

8 0,695669

9 0,650908

10 0,620371

11 0,606981

12 0,532800

13 0,521134

14 0,466443

15 0,450356

16 0,411529

17 0,363493

Trace 17,000000 

Objective 3: Overcoming the effects of differential skewness

of test items in scale construction

As the procedure that was followed is fully described in the

method section, only the essential results are given here.

The items of the LCI were intercorrelated, and the eigenvalues

of the intercorrelation matrix were calculated. Nineteen of the

eigenvalues were greater than unity, accordingly 19 factors

were extracted and rotated to simple structure by means of a

Varimax rotation. 

Two of the factors had one loading each and were discarded. Next,

17 subscores were formed by adding all the items with substantial

loadings on a factor, together. The 17 subscores were then

intercorrelated. The matrix of intercorrelations is given in Table 15.

From Table 15 it is clear that the correlations of the subscores

with one another vary from moderate to low and from positive

to negative, suggesting more than one factor.

Next, the eigenvalues of the intercorrelation matrix were

calculated. The obtained eigenvalues are given in Table 16.

Four of the eigenvalues were greater than unity, suggesting four

factors (Kaiser, 1961).

Accordingly four factors were extracted and rotated to simple

structure by means of a Direct Oblimin rotation. The rotated

factor matrix is given in Table 17.

From an inspection of Table 17 it is clear that the first three

factors are well determined with four or more high loadings.

However, the fourth factor had only one high loading. A three-

factor-solution was therefore tried. The obtained factor matrix is

given in Table 18.

From Table 18 it is clear that all three factors are well

determined with four or more high loadings. From the

intercorrelations of the factors it is clear that External 

Control and Internal Control are essentially uncor related.

External Control is moderately negatively correlated with

Autonomy, and Internal Control is moderately positively

correlated with Autonomy.

SCHEPERS36

TABLE 15

MATRIX OF INTERCORRELATIONS OF THE SUBTESTS OF THE LOCUS OF CONTROL INVENTORY (1995)

Variable Subtest 1 Subtest 2 Subtest 3 Subtest 4 Subtest 5 Subtest 6 Subtest 7 Subtest 8 

Subtest 1 1,0000

Subtest 2 -0,0660 1,0000

Subtest 3 0,3673 -0,1870 1,0000

Subtest 4 -0,1112 0,3609 -0,2250 1,0000

Subtest 5 -0,1193 0,4261 -0,1823 0,3997 1,0000

Subtest 6 0,5352 -0,1262 0,4588 -0,1844 -0,1430 1,0000

Subtest 7 0,4443 -0,2841 0,2299 -0,3492 -0,3382 0,2890 1,0000

Subtest 8 0,2911 -0,1325 0,3450 -0,1474 -0,1492 0,3189 0,2669 1,0000

Subtest 9 0,5430 -0,1719 0,3074 -0,1914 -0,2084 0,3998 0,4747 0,3398

Subtest 10 0,3192 0,0411 0,3547 -0,0128 0,0477 0,3824 0,0900 0,1494

Subtest 11 0,3171 -0,0056 0,4365 -0,0927 0,0012 0,3209 0,1342 0,1703

Subtest 12 0,3335 -0,0827 0,4752 -0,0827 -0,0803 0,3639 0,1527 0,3046

Subtest 13 -0,1734 0,1992 -0,0888 0,2190 0,2050 -0,1807 -0,2647 0,0033

Subtest 14 -0,2024 0,3260 -0,1508 0,3008 0,3533 -0,2132 -0,3866 -0,1948

Subtest 15 -0,1889 0,1709 -0,1242 0,2012 0,3143 -0,1844 -0,2833 -0,2028

Subtest 16 0,0403 0,1888 0,1442 0,1621 0,3073 0,0453 -0,1050 0,0330

Subtest 17 0,5388 -0,1952 0,4480 -0,2378 -0,1795 0,4685 -0,4335 0,3277

Subtest 9 Subtest 10 Subtest 11 Subtest 12 Subtest 13 Subtest 14 Subtest 15 Subtest 16 Subtest 17 

1,0000

0,2278 1,0000

0,2094 0,3938 1,0000

0,2518 0,2506 0,2515 1,0000

-0,1470 -0,0074 -0,0225 -0,0553 1,0000

-0,2475 -0,0021 -0,0424 -0,0922 0,2427 1,0000

-0,2405 -0,0590 -0,0711 -0,1527 0,1579 0,2135 1,0000

-0,0269 0,1719 0,1824 0,1467 0,1067 0,1589 0,1321 1,0000

0,5171 0,3308 0,3401 0,3231 -0,2012 -0,2200 -0,2156 0,0121 1,0000



TABLE 18

ROTATED FACTOR MATRIX (DIRECT OBLIMIN)

VARIABLES K FACTOR I FACTOR II FACTOR III h2j

Subtest 1: Items 2, 3, 5, 13 0,146 0,160 0,754 0,634

14, 15, 22, 24, 28, 29, 

62, 64, 67 en 70

Subtest 2: Items 12, 34, 6 0,630 -0,083 0,082 0,366

35, 36, 41 en 79

Subtest 3: Items 10, 42, 6 -0,266 0,828 -0,105 0,678

49, 61, 63 en 75

Subtest 4: Items 20, 43, 6 0,579 -0,119 0,016 0,343

52, 53, 56 en 78

Subtest 5: Items 9, 51, 7 0,717 -0,017 0,024 0,502

57, 58, 65, 77 en 80

Subtest 6: Items 6, 7, 7 -0,052 0,411 0,374 0,462

16, 25, 37, 59 en 69

Subtest 7: Items 1, 17, 6 0,304 0,070 0,557 0,506

39, 44, 71 en 72

Subtest 8: Items 8, 40 3 -0,135 0,283 0,211 0,219

en 54

Subtest 9: Items 30, 46, 4 -0,037 0,065 0,665 0,505

73 en 74

Subtest 10: Items 26, 4 0,169 0,464 0,173 0,318

27, 31 en 32

Subtest 11: Items 18 2 0,056 0,529 0,083 0,324

en 19

Subtest 12: Items 48, 4 -0,064 0,514 0,074 0,312

55, 60 en 76

Subtest 13: Items 21, 3 0,262 0,057 -0,193 0,139

23 en 33

Subtest 14: Items 38 2 0,451 0,027 -0,198 0,309

en 45

Subtest 15: Items 4 2 0,291 -0,018 -0,203 0,176

en 11

Subtest 16: Items 47 2 0,346 0,307 -0,048 0,213

en 50

Subtest 17: Items 13, 3 -0,087 0,306 0,507 0,530

66 en 68

Number of items per 28 26 26 80

factor 

INTERCORRELATIONS OF FACTORS

VARIABLES FACTOR I FACTOR II FACTOR III

FACTOR I 1,000

FACTOR II -0,018 1,000

FACTOR III -0,393 0,438 1,000

Note: N = 1662

Factor I = External Control

Factor II = Internal Control

Factor III = Autonomy

Next, separate scales were formed, corresponding to each of the

factors, and subjected to item analysis.

The item statistics in respect of Scale I (External Control) are

given in Table 19.

From Table 19 it is clear that the item-total correlations range

from 0,341 to 0,611, with a mean of 0,459 and a standard

deviation of 0,080. The scale is therefore internally highly

consistent. Items 23, 33 and 50 were rejected, because their

indices of reliability were too low. The reliability of the scale

according to Cronbach’s coefficient alpha is 0,841.

The item statistics in respect of Scale II (Internal Control) are

given in Table 20.

According to Table 20 the item-total correlations range from

0,301 to 0,585, with a mean of 0,455 and a standard deviation of

0,069. The scale is therefore internally highly consistent. No

items were rejected. The reliability of the scale according to

Cronbach’s coefficient alpha is 0,832.

The item statistics in respect of Scale III (Autonomy) are given in

Table 21.

From Table 21 it is clear that the item-total correlations 

range from 0,370 to 0,575, with a mean of 0,488 and a

standard deviation of 0,074. Therefore, the scale is 

internally highly consistent. No items were rejected. The

reliability of the scale according to Cronbach’s coefficient

alpha is 0,866.

DIFFERENTIAL SKEWNESS OF TEST TERMS 37

TABLE 17

ROTATED FACTOR MARIX (DIRECT OBLIMIN)

VARIABLES K FACTOR 1 FACTOR 2 FACTOR 3 FACTOR 4 h2j

Subtest 1: Items 2, 3, 5, 14, 15, 22, 24, 28, 29, 62, 64, 67 and 70 13 0,738 0,131 +0,192 0,016 0,634

Subtest 2: Items 12, 34, 35, 36, 41 and 79 6 0,081 0,627 -0,039 -0,050 0,365

Subtest 3: Items 10, 42, 49, 61, 63 and 75 6 -0,090 -0,275 +0,694 0,244 0,656

Subtest 4: Items 20, 43, 52, 53, 56 and 78 6 0,027 0,607 -0,123 0,024 0,360

Subtest 5: Items 9, 51, 57, 58, 65, 77 and 80 7 0,022 0,698 +0,050 -0,080 0,499

Subtest 6: Items 6, 7, 16, 25, 37, 59 and 69 7 0,362 -0,080 +0,403 0,067 0,464

Subtest 7: Items 1, 17, 39, 44, 71 and 72 6 0,546 0,296 -0,070 0,020 0,504

Subtest 8: Items 8, 40 and 54 3 0,228 -0,040 +0,040 0,516 0,405

Subtest 9: Items 30, 46, 73 and 74 4 0,669 -0,001 +0,004 0,155 0,521

Subtest 10: Items 26, 27, 31 and 32 4 0,155 0,108 +0,537 -0,060 0,357

Subtest 11: Items 18 and 19 2 0,059 -0,010 +0,595 -0,050 0,368

Subtest 12: Items 48, 55, 60 and 76 4 0,073 -0,040 +0,374 0,308 0,346

Subtest 13: Items 21, 23 and 33 3 -0,185 0,315 -0,045 0,201 0,191

Subtest 14: Items 38 and 45 2 -0,194 0,442 +0,050 -0,040 0,308

Subtest 15: Items 4 and 11 2 -0,203 0,262 +0,050 -0,130 0,183

Subtest 16: Items 47 and 50 2 -0,050 0,331 +0,295 0,054 0,211

Subtest 17: Items 13, 66 and 68 3 0,494 -0,109 +0,310 0,045 0,532

Number of items per factor 26 28 23 3

Note: Factor 3 has been reflected



SCHEPERS38

TABLE 19

ITEM STATISTICS IN RESPECT OF SCALE I OF THE LCI: EXTERNAL CONTROL

DESCRIPTION OF ITEM N MEAN OF STANDARD ITEM-TEST INDEX OF  

ITEM DEVIATION CORRELATION RELIABILITY 

OF ITEM OF ITEM

(Xg) (sg) (rgx) (rgxsg ) 

Q4 Convinced that a person without money will get nowhere, no matter how 1662 3,620 1,835 0,341 0,625

hard he/she works

Q9 People obtain good positions simply because they know the right people 1662 5,039 1,291 0,392 0,506

Q11 Convinced that if once he/she has failed in something it is virtually 1662 2,471 1,392 0,379 0,528

impossible to achieve in it again

Q12 Convinced that he/she is subject to the whims of fate 1662 2,788 1,518 0,573 0,869

Q20 Failed because other people interfered in his/her business 1662 3,397 1,367 0,409 0,560

Q21 Dependent on the advice and cues of others in order to produce 1662 3,426 1,325 0,398 0,527

quality work

Q23 Will readily support a group decision even if he/she disagrees with it 1662 2,978 1,444 **** ****

Q33 Will readily accept responsibility for errors in work situation even if 1662 2,801 1,460 **** ****

he/she is innocent

Q34 Lady Luck plays an important role in his/her life 1662 2,644 1,518 0,471 0,715

Q35 Strongly believes in fatalism 1662 2,867 1,525 0,472 0,720

Q36 Life influenced to a great extent by coincidences 1662 3,195 1,455 0,529 0,770

Q38 Other people are responsible for his/her wellbeing 1662 3,581 1,437 0,360 0,518

Q41 Convinced that failures in life could be attributed to fate 1662 2,549 1,441 0,570 0,821

Q43 Present achievements adversely affected by negative experiences in past 1662 3,127 1,626 0,468 0,761

Q45 Other people are in charge of his/her life and determine outcome of issues 1662 2,505 1,359 0,536 0,729

Q47 Convinced that a person cannot achieve without the right opportunities 1662 4,619 1,445 0,366 0,529

Q50 Convinced that success depends mainly upon equal opportunities in life 1662 4,757 1,376 **** ****

Q51 Advancement in life is determined by your superiors 1662 3,652 1,375 0,496 0,683

Q52 Parents/guardians negatively influenced his/her achievement at school 1662 2,142 1,467 0,391 0,574

Q53 Present achievement negatively influenced by people who are not 1662 2,172 1,304 0,485 0,632

favourably disposed towards him/her

Q56 His/her progress in the past thwarted by people that were hostile towards 1662 3,035 1,498 0,462 0,692

him/her

Q57 Convinced that only people who are at the right place at the right time 1662 3,282 1,465 0,569 0,833

get promoted

Q58 Only people who belong to the right political party have a chance in life 1662 2,342 1,564 0,484 0,757

Q65 Convinced that clique formation is the most important determinant of 1662 3,710 1,676 0,398 0,668

social acceptance

Q77 Convinced that promotion in the new South Africa will depend largely 1662 4,306 1,784 0,423 0,755

on skin-colour

Q78 Convinced that it is impossible to rise above your own environment 1662 2,936 1,690 0,344 0,582

Q79 Convinced that his/her fate is determined by coincidental events over 1662 2,690 1,463 0,611 0,894

which he/she has no control

Q80 Convinced that his/her advancement in life will be determined by certain 1662 3,634 1,541 0,540 0,831

influential people

MEANS AND STANDARD DEVIATIONS OF ITEM STATISTICS

(ONLY IN RESPECT OF ITEMS INCLUDED IN TEST SCORE)

Xg sg rgx rgxsg

Mean 3,189 1,494 0,459 0,683

SD 0,736 0,141 0,080 0,120

Cronbach alpha = 0,841

Mean of test = 79,730

Standard deviation = 17,079

Number of items = 25



DIFFERENTIAL SKEWNESS OF TEST TERMS 39

TABLE 20

ITEM STATISTICS IN RESPECT OF SCALE II OF THE LCI: INTERNAL CONTROL

DESCRIPTION OF ITEM N MEAN OF STANDARD ITEM-TEST INDEX OF  

ITEM DEVIATION CORRELATION RELIABILITY 

OF ITEM OF ITEM

(Xg) (sg) (rgx) (rgxsg ) 

A6 Convinced that personal insight is a pre-requisite for good interpersonal 1662 5,686 1,096 0,458 0,502

relationships

A7 The structure and routine of a person’s work should be determined by 1662 5,664 1,045 0,465 0,486

himself/herself

A8 Readily accepts responsibility for mistakes that appear in his/her work 1662 5,607 1,149 0,386 0,444

A10 Convinced that success is mainly related to a person’s ability and dedication 1662 6,126 0,930 0,475 0,442

A16 Decides on matters himself/herself, rather than waiting for others to take 1662 5,180 1,619 0,343 0,556

decisions on his/her behalf

A18 Recognition encourages him/her to perform even better 1662 6,063 1,018 0,461 0,470

A19 Success encourages him/her to work harder and achieve greater heights 1662 6,329 0,854 0,525 0,448

A25 Prefers to follow his/her own mind, rather than following someone else’s 1662 5,274 1,303 0,389 0,507

instructions

A26 Insists on recognition of his/her own individual achievements 1662 4,655 1,571 0,301 0,473

A27 Takes responsibility for his/her own intellectual development 1662 5,623 1,088 0,483 0,526

A31 Important for him/her to receive feedback on tasks which he/she has 1662 5,821 1,122 0,471 0,528

performed

A32 Convinced that reward for achievement is earned 1662 5,252 1,234 0,454 0,560

A37 Convinced that achievement of personal objectives depends on oneself 1662 5,954 1,043 0,539 0,562

A40 Readily accepts responsibility for his/her own poor performance 1662 5,306 1,329 0,407 0,541

A42 Convinced that the respect one receives is directly related to one’s 1662 6,100 1,059 0,538 0,570

behaviour

A48 Agrees that failure in life can be attributed to a lack of dedication 1662 5,354 1,303 0,392 0,511

A49 Convinced that success depends mainly on hard work 1662 6,220 0,899 0,515 0,463

A54 Takes personal responsibility for things that go wrong in his/her life 1662 5,427 1,177 0,427 0,503

A55 Convinced that the outcome of matters is determined by his/her own inputs 1662 5,589 1,000 0,585 0,585

*A59 Believes that his/her own input bears no relation to the outcome of matters 1662 5,744 1,351 0,390 0,527

A60 Convinced that achievement depends upon his/her utilising his/her 1662 6,133 1,120 0,446 0,499

God-given talents to the full

A61 Convinced that the achievements he/she has obtained were deserved 1662 5,968 1,131 0,490 0,554

A63 Convinced that promotions are earned through hard work and perseverance 1662 6,194 0,900 0,483 0,434

A69 Convinced that he/she is sufficiently qualified for the work he/she is doing 1662 5,742 1,107 0,493 0,546

A75 Convinced that the achievements he/she has obtained are the results of 1662 5,969 1,087 0,552 0,600

hard work and dedication

A76 Convinced that failures in life are due to a lack of perseverance 1662 5,023 1,519 0,356 0,541

MEANS AND STANDARD DEVIATIONS OF ITEM STATISTICS

(ONLY IN RESPECT OF ITEMS INCLUDED IN TEST SCORE)

Xg sg rgx rgxsg

Mean 5,692 1,156 0,455 0,515

SD 0,418 0,201 0,069 0,047    

Cronbach alpha = 0,832

Mean of test = 148,001

Standard deviation = 13,359

Number of items = 26



SCHEPERS40

TABLE 21

ITEM STATISTICS IN RESPECT OF SCALE III OF THE LCI: AUTONOMY

DESCRIPTION OF ITEM N MEAN OF STANDARD ITEM-TEST INDEX OF  

ITEM DEVIATION CORRELATION RELIABILITY 

OF ITEM OF ITEM

(Xg) (sg) (rgx) (rgxsg ) 

*Q1 Doubts his/her own capabilities when his/her work is being criticised 1662 4,796 1,372 0,468 0,642

Q2 Geared towards ensuring that his/her case triumphs during a conflict 1662 5,494 1,277 0,429 0,547

situation

Q3 Would readily take risks 1662 4,500 1,363 0,370 0,505

Q5 Can readily convince someone else of his/her viewpoint 1662 5,256 1,017 0,559 0,569

Q13 Convinced that he/she will succeed when undertaking important tasks 1662 5,797 0,951 0,566 0,539

Q14 Makes things happen through his/her own input, rather than waiting for 1662 5,201 1,165 0,575 0,670

things to happen

*Q15 Waits for other people to take charge, rather than taking charge 1662 4,691 1,503 0,543 0,816

Q17 Failure spurs him/her on to improve his/her performance 1662 5,315 1,288 0,392 0,505

Q22 Likes taking decisions himself/herself 1662 5,804 1,102 0,574 0,633

Q24 Would readily air his/her views when they differ from someone else’s 1662 5,440 1,249 0,549 0,686

Q28 Likes occupying a leadership position 1662 5,463 1,366 0,533 0,728

Q29 Would stick to his/her viewpoint when someone for whom he/she has 1662 5,288 1,211 0,494 0,598

great respect disagrees with him/her

Q30 Likes solving complex problems 1662 5,206 1,325 0,518 0,686

*Q39 Feels that he/she has no control over his/her own circumstances 1662 4,779 1,400 0,407 0,570

Q44 Often achieves set objectives, irrespective of the conditions 1662 5,187 1,023 0,533 0,545

Q46 Convinced that he/she can solve most problems, irrespective of the 1662 5,400 1,113 0,504 0,561

conditions

Q62 Can predict the outcome of an examination he/she has just written 1662 5,623 1,205 0,335 0,403

Q64 Finds it easy to satisf y choosy people 1662 4,345 1,296 0,381 0,494

Q66 Convinced that he/she possesses the ability to produce work of the highest 1662 6,090 0,902 0,517 0,466

quality

Q67 Would strongly defend his/her actions if the appropriateness thereof were to 1662 5,764 1,055 0,517 0,545

be questioned by others

Q68 Convinced that he/she is sufficiently qualified for the work that he/she is 1662 5,739 1,007 0,546 0,550

doing

Q70 Prefers challenging work to routine work 1662 5,649 1,217 0,518 0,630

*Q71 Subsequently doubts the correctness of the decisions he/she has taken 1662 4,430 1,292 0,433 0,559

*Q72 Dependent on the support and goodwill of others in the execution of tasks 1662 4,294 1,337 0,365 0,488

*Q73 Would readily quit if he/she is battling with a complex problem 1662 5,199 1,282 0,483 0,619

Q74 Often takes the initiative in finding solutions for troublesome problems 1662 5,210 1,129 0,575 0,649

MEANS AND STANDARD DEVIATIONS OF ITEM STATISTICS

(ONLY IN RESPECT OF ITEMS INCLUDED IN TEST SCORE)

Xg sg rgx rgxsg

Mean  5,229 1,210 0,488 0,585

SD   0,489 0,153 0,074 0,090 

Cronbach alpha = 0,866

Mean of test = 135,958

Standard deviation = 15,189

Number of items = 26



DISCUSSION

From the findings of the study it is clear that the degree of

skewness (marginal splits) of binary items places an upper limit

on the correlations bet ween the items. Furthermore,

intercorrelation matrices based on such items, arranged

according to their degree of skewness, have the typical structure

of a simplex, quasi-simplex or pseudosimplex. Factoring such

matrices result in factors of skewness, regardless of the contents

of the items.

As far as the first objective of the study is concerned, it was

found that similar gradients exist in respect of the correlations in

an intercorrelation matrix based on continuous variables,

arranged in terms of skewness.

To achieve the second objective, the items of the LCI 

were grouped into parcels of four items each. The parcels

were intercorrelated and subjected to factor analysis. Three

factors were obtained. These factors were also largely 

factors of skewness. Content also played a role in so far as the

contents of the items were associated with their degree of

skewness. The items loading on Factors I and III (Autonomy

and Internal Control) are essentially negatively skewed,

whereas those loading on Factor II (External Control) are

positively skewed.

As far as the third and major objective of the study is concerned,

the procedure that was followed, yielded three factors that were

well determined. The corresponding scales that were produced

are internally highly consistent, with reliabilities that range

from 0,832 to 0,866.

Autonomy is positively correlated with Internal Control (r =

0,438; p < 0,001). There is thus approximately 19% common

variance between the two constructs. However, the reliable

variance of Autonomy is 87% and that of Internal Control 83%.

The specific variances of the two scales therefore vary from

64%to 68%.

Autonomy and External Control are negatively correlated 

(r = – 0,393; p < 0,001). There is thus approximately 15%

common variance between the two constructs. The reliable

variance of External Control is 84%. The specific variances of the

two constructs (scales) therefore vary from 69% to 72%.

Internal Control and External Control are essentially

uncorrelated (r = -0,018; p > 0,05). This is in keeping with Social

Learning Theory and Attribution Theory: The causes of human

behaviour can be divided into two broad classes, namely those

that pertain to one’s natural disposition and those that pertain

to the external world (Roediger III et al., 1991). It is therefore not

surprising that items from these two domains are essentially

independent of one another.

In the procedure followed in the present study, the items were

grouped according to the factors they loaded on. Each subscore

was therefore internally consistent. The fact that a Varimax

rotation was used, kept the factors relatively independent of

one another, and simplified the procedure. In intercorrelating

the items of the LCI both content and degree of skewness of

the items must have played a role. However, the degree of

skewness of the subtests are considerably smaller than those of

the single items.

Forming parcels on a priori theoretical grounds does not

guarantee internal consistency within parcels or eliminate 

the effects of differential skewness completely. For this, a 

new measure of association that is independent of skewness, 

is required.

The procedure described, has been used at the Rand Afrikaans

University since 1992, and has consistently produced scales of

high reliability and validity.

ACKNOWLEDGEMENTS

I hereby wish to thank all the members of the Statistical

Consultation Service of the Rand Afrikaans University for 

all the hours of computational work done for me. I value it 

very highly.

A special word of thanks to Annetjie Boshoff for the typing of

the manuscript. Nobody can type tables better than her.

REFERENCES

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Bohrnstedt, G.W. & Knoke, D. (1988). Statistics for social data

analysis (2nd ed.) Itasca, Ill.: F.E. Peacock Publishers, Inc..

Cronbach, L.J. (1951). Coefficient alpha and the internal

structure of tests. Psychometrika, 16, 297-334.

Ferguson, G.A. (1941). The factorial interpretation of test

difficulty. Psychometrika, 6, 323-329.

Gorsuch, R.L. (1974). Factor analysis. Philadelphia: W.B. Saunders

Company.

Guilford, J.P. (1950). Fundamental statistics in psychology and

education (2nd ed.) New York: McGraw-Hill.

Guttman, L.A. (1954). A new approach to factor analysis: the

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Guttman, L.A. (1955). A generalized simplex for factor analysis.

Psychometrika, 20, 173-192

Guttman, L.A. (1957). Empirical verification of the radex

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Educational and Psychological Measurement, 17, 391-407.

Horst, P. (1953). Correcting the Kuder-Richardson reliability 

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371-374.

Horst, P. (1965). Factor analysis of data matrices. New York: Holt,

Rinehart and Winston.

Jöreskog, K.G. (1970). Estimation and testing of simplex models.

British Journal of Mathematical and Statistical Psychology, 23,

121-145.

Jöreskog, K.G. & Sorbom, D. (1982). LISREL V: Analysis of linear

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Kaiser, H.F. (1961). A note on Guttman’s lower bound for the

number of common factors. British Journal of Statistical

Psychology, 14 (1), 1.

Kuder, G.F. & Richardson, M.W. (1937). The theory of the

estimation of test reliability. Psychometrika, 2, 151-160.

Magnusson, D. (1967). Test theory. Reading Mass.: Addison-

Wesley.

Mischel, W. (1979). On the interface of cognition and

personality: Beyond the person-situation debate. American

Psychologist, 34, 740-754.

Roediger III, H.L., Capaldi, E.D., Paris, S.G. & Poliv y, J. 

(1991). Psychology (3rd ed.) New York: Harper Collins

Publishers.

Rotter, J.B. (1966). Generalized expectancies for internal versus

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Schepers, J.M. (1962). A components analysis of a 

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Schepers, J.M. (1992). Toetskonstruksie: Teorie en praktyk.

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DIFFERENTIAL SKEWNESS OF TEST TERMS 41



APPENDIX 1

Cronbach’s coefficient alpha and Kuder-Richardson Formula 20

(KR20), for continuous data, are formally the same, and can be

written as follows:

, where                      (1)

K = number of test items

= sum of item variances

= test variance

For binary data KR20 can be written as follows:

, where                 (2)

K = number of test items

pg = proportion of subjects endorsing item g according to the key

The variance of the p-values can be written as

(3)

From this it follows that

Substitution for in (2) gives

,                  (4)

,                     (5)

Because (mean of test)

and 

(6)

Formula 6 applies only to binary data.

For continuous data Kuder-Richardson Formula 20 and

Cronbach’s coefficient alpha can be written as follows:

, where                      (7)

K = number of test items

= sum of item variances

= test variance

Formula 7 can be transformed as follows:

,                      (8)

,                             (9)

,                              (10)

The total test variance can never exceed therefore the

greater the test variance, the higher the reliability (Schepers,

1992, p. 33).

2

1

K

g

g

K
=

σ∑

� �
��

�

�

�

�
�
�
�

�

	










�
1

1

2 2

1

2

K

K

g x
g

K

x

� �

�( )

2 2

1

2

( )

1
( 1)

K

g x

g

x

K

K

=

 
σ − σ 

 = − − σ 
  

∑

2

1
20 2

1 1
1

1

K

g

g

x

K
KR

K K K

=

 
σ 

 = − − + − σ 
  

∑

2
xσ

2
gσ∑

2

20 2
1

1

g

x

K
KR

K

 σ
= − 

− σ  

∑

2
2

20 2
1–

– 1

p

x

K
K KKR

K

 µ
µ − − σ 

 ∴ =
σ 

  

p
K

�
�

2

1

K

g

g

p
=

= µ∑

2
2

2

2

– (

1–
– 1

p

x

K
K K

K

 µ
µ σ + 

 =
 σ
  

2 2

1
20 2

– ( )

1–
– 1

K

g p

g

x

p K p
K

KR
K

=

 
σ + 

 =  σ 
  

∑

2

1

K

g

g

p
=

∑

2 2 2

1

( )
K

g p

g

p K p
=

= σ +∑

2

12

2
2

( – )

,

–

K

g

g
p

g

p p

K

p
p

K

=σ =

=

∑

∑

2

1
20 2

( – )

1–
– 1

K

g g

g

x

p p
K

KR
K

=

 
 
 =  σ 
  

∑

� g
2

2
gσ∑

2

1
20 2

1–
– 1

K

g

g

x

K
KR

K

=

 
σ 

 =  σ 
  

∑

SCHEPERS42



APPENDIX 2

PHI COEFFICIENT IN RESPECT OF BINARY TEST ITEMS

Item 1 

+ – Total 

– A B A+B

30 20 50

(40) (10) (50)

qk = 0,5 

+ C D C+D

10 40 50

(0) (50) (50)

pk = 0,5 

Total A+C B+D N

40 60 100

(40) (60) (100)

pg = 0,4 qg = 0,6 

Note that the maximum value of � is a direct function of the

marginal splits .

Example

�max
,

,

,

,

,

,
,�



�
�

�

�
�


�
�

�

�
� � �

0 4

0 6

0 5

0 5

0 20

0 30
0 81649658

( )g g k kp q and q p

2 2
2
max

max

g k

g g k k

g k

g k

g k

g k

p q

p q p q

p q

q p

p q

q p

φ =

φ =

   
=        

max

(1 )

g g k

g g k k

g k

g g k k

g k

g g k k

p p p

p q p q

p p

p q p q

p q

p q p q

−
φ =

−
=

=

max , , ,

gk g k

g g k k

gk g g k

p p p

p q p q

To obtain set p p where p p

−
φ =

φ = ≤

0,4 0,4 0,5

0,4 0,6 0,5 0,5

0,20

0,244948974

0,81649658

gk g k
phi

g g k k

p p p

p q p q

−
=

− ×
=

× × ×

=

= ⇒

0,

0 2000

2449,489743

0,81649658

Maximum value of

Set C then

φ
=

−
φ =

= − ⇒

−
φ =

+ + + +

−
=

= − ⇒

( )( )( )( )

200 1200

2449,489743

0,40824829

BC AD

A B C D A C B D

0,4 0,5 0,10 0,20

0,2449489740,4 0,6 0,5 05

0,40824829

gk g k
phi

g g k k

gk

p p p
r

p q p q

p

−
=

− × −
= =

× × ×

= −

DIFFERENTIAL SKEWNESS OF TEST TERMS 43

It
e
m

 2