THE USE OF ENTROPY INDEX FOR GENDER INEQUALITY

In the Millennium Declaration, the proportion (or ratio) of females to males in a given sector is used as the indicator for gender inequality. A critical limitation in the proportion (or ratio) statistic is that it does not include randomness in gender disparity over a period of time. It is well-known that randomness, which is a key ingredient in studying the behaviour of a system, is measured using entropy. This paper presents the use of entropy as an alternative measure for gender inequality in a sector of an economy.


INTRODUCTION
Discrimination against women in all fields of human endeavour has resulted to a crushing problem of gender inequality.As stated in the Millennium Declaration, gender inequality is indicated in the ratios of females to males enrolled at all levels of education (primary, secondary, and tertiary levels); the ratio of literate females to males 15-24 years old; the share of females in wage employment in the non-agricultural sector; and the proportion of seats held by women in national parliament (Ogbonna, 2006).Earlier studies on inequalities in Nigerian education show that males have greater access to education than females and the gap in access between males and females tends to widen as they transit from primary to tertiary level of education (Nwagwu, 1998).Nwagwu (1998) identified enrolment ratio as a relevant index to judge the extent of female access to education in Nigeria, and also reported that data on enrolment ratios are often scanty.
The motivation for this study is based on the priority of the Millennium Declaration for gender equality as the target of achieving gender equality by 2015 is fast approaching, and on the assertion by Oroge (1999) that many legislations have been enacted which provide equal opportunities for women and men and also protect women from being discriminated against, but in practice, these may not necessarily be so.In this study therefore, we develop an entropy index as an alternative measure for gender inequality in a sector of an economy, and applied it to enrolment in primary and secondary schools in Nigeria using data in Nwagwu (1998).

CONCEPTUAL FRAMEWORK
It is a well-known result that for a probability distribution associated with the discrete random variable X , the entropy is given by where k is a constant.This is a measure of the degree of randomness in the system (Vassiliou, 1984).For k =1, equation ( 1) is referred to as Shannon entropy (see Ciuperca and Girardin, 2005).
Shannon entropy has the property of symmetry i.e. the measure is unchanged if the outcomes i x are re-ordered e.g.,...) , ( ,...) , ( The properties of Shannon entropy are stated in the publication of Departmento de Matematica (2001).Shannon entropy as used in operations research and in hypothesis selection has been critically examined by White (1975).With Shannon formula the entropy of a system can be calculated from the entropy of its sub-systems if we know how the sub-systems interact with each other.For instance, for positive integers i b where Shannon entropy is given by Girardin and Limnios (2003) for a finite measure * µ absolutely continuous with respect to a σ -finite measure µ with Radon-Nikodym derivative f as Applications of entropy abound in literature.Wilson (1970) had earlier discussed the use of entropy in system modelling.Entropy has been applied to oil spill problem (Freund and Saxena, 1984;and Thomas, 1979), cost functions (Dinkel and Kochenberger, 1979;and Paris and Howitt, 1998), hazard rate function estimation (Ebrahimi, 2000), international trade (Pulliainen, 1970), combat degradation (Rodrigues, 1989), queuing systems (Guiasu, 1986), manpower systems (McClean and Abodunde, 1978;McClean, 1986;and Omosigho and Osagiede, 1999), external debt profile (Ekhosuehi and Osagiede, 2007a;2007b;Osagiede and Ekhosuehi, 2007), industrial and employment concentration (Horowitz, 1970), unemployment scenario (Ekhosuehi and Osagiede, 2007c) and income inequality (Mussard et al, 2003).

THE MODEL
In order to derive a measure which accounts for the degree of randomness for gender inequality in a given sector, say sector i , over the time interval , where i T is the maximum period of time for which data are available for sector i , we need to compare the changes in population of males and females for period t , } 1 : , in sector i .The change in population in any sector is often due to wastages and new recruits employed to fill vacancies.Let ) (t n m i be the number of males in sector i , at period t , and let ) (t n f i be the number of females in sector i , at period t .For , and The number of times for which the change in number of females is not less than the change in number of males in From equation (1), we obtain To constrain the entropy measure to a finite range of [0,1], we apply Jensen inequality (Ruszczynski and Shapiro, 2003).Thus, ( ) ( ) H is the entropy index for gender inequality in sector i .
and represents the situation of no randomness and a very strong dominance of males in sector i .
and the total number of females equals the total number of males in the period under consideration; it represents perfect randomness and gender equality sector i .For intermediate values, the entropy index summarizes in one figure the degree of randomness and gender inequality in the sector.If there are s sectors in the economy, we define the average entropy index as

APPLICATION
In this section, we illustrate the use of the entropy index to measure gender inequality in primary and secondary schools in Nigeria.There has been an impressive increase in enrolments in primary and secondary schools in Nigeria over the years.In spite of the tremendous growth at these levels, the female gender may not have had equal opportunity to education as their male counterparts.Since data on enrolments for males and females are usually given (or sometimes computed from information provided) the entropy index developed in section 3 can be applied to school enrolment.More so, the entropy index comprises pooled proportion of females in the sector and degree of randomness of the change in population of both genders over the years.We now illustrate the use of the entropy index using enrolment data as presented in Tables I and II.The enrolments for each gender are computed from the tables using: . Similarly, other results are obtained.i θ is computed using the maximum likelihood value:

H H
, the entropy indices imply that gender inequality (against females) in the periods under review are not severe, and that greater educational opportunities abound at the secondary level.
Using the enrolment stock in Tables I and II, Nwagwu (1998) analysed gender inequality in both primary and secondary schools using enrolment ratio statistics as presented in Tables III and IV.
Table III The results (enrolment ratios) in Tables III and IV show some ambiguity.For instance, how do an individual interpret enrolment ratio for males of 105% and that for females of 81% in the 1983/84 session in Table III?From the acronym NA in Tables III and IV, it can be inferred that data on enrolment ratios are scanty; as earlier reported by Nwagwu (1998).This is also a limitation to the use of enrolment ratios for gender inequality in the Nigerian education sector.Furthermore, inferences from enrolment ratios as mentioned in Nwagwu (1998) do not state the degree of randomness of gender disparity.For this reason, the entropy index developed in section 3 is a better measure than enrolment ratio statistics.

CONCLUSION
In this study, the fundamental statistical entropy measure has been modified and used as an alternative measure for gender inequality.The entropy index developed in this study is mathematically tractable and it encapsulates pooled proportion of females in the sector and degree of randomness of the change in population of both genders over the years.It application to primary and secondary enrolments in Nigeria shows that gender inequality is not severe at these levels and that greater educational opportunities are found at the secondary level than at the primary level.More efforts are however required to eliminate the inequality gap at all levels of education by the year 2015, and preferably at all sectors of the economy in general.

..
Using the formula in equation (9), we obtain the entropy index for primary and secondary schools respectively for each time interval as As earlier mentioned, the entropy index for intermediate values summarizes in one figure the degree of randomness and gender inequality in the sector.Since

V. U. Ekhosuehi, Department of Maths, University of Benin, Nigeria, A. A. Osagiede, Department
of Maths, University of Benin, Nigeria,

Table I :
Enrolments in primary schools