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1 Indicators of Educational disparity Prepared for Technical Working Group workshop on Education Statistics 11-22 February 2002, Nairobi, Kenya By Tegegn Nuresu Wako Consultant National Educational Statistics Information Systems(NESIS) January 2002, Harare 2 Educational Indicators Measures of Disparity Introduction: is a basis for social development. Cambridge International Dictionary of English defines the word equality “Equality refers to the right of different groups of people to have a similar social position and receive the same treatment, regardless of their apparent differences. Another related word to the above is ””. This Equality(synonyms) again is defined as “…is a system of justice which allows a fair Parity judgement for a case where the laws that already exist are not Fairness Equal opportunity satisfactory”. Impartiality Egalitarianism Equality, therefore, refers to parity between men and women, boys Antonym: Inequality and girls, urban and rural and between regions districts, villages and Equity(synonyms) between ethnic and language groups etc. Inequality arises from Even-handedness mismanagement: lack of fair share of resources among different Fairness groups. In our case lack of even-handedness in the distribution of Impartiality educational resources(manpower, materials and facilities, books Justice Fair play etc). Inequality can also arise from social and cultural barriers: “I Justness want my daughter to learn how to cook in the house from her Antonym: Injustice mother rather than send her to school” type attitude. Inequality can also arise from lack of ability to pay school fees and didactic materials. Inequality can also arise from certain other conditions like urban and rural situation. Children in urban areas are in a better position compared to rural areas in that they can easily make friends who can assist them in their studies. They are also nearer to certain facilities such as libraries, newspapers etc. The objective of this paper is to enable statisticians and planners gain necessary skills to measure the extent of inequalities that exist among different groups. Every rational manager strives to bring about parity between different groups by sharing resources fairly between different groups. Unfortunately, in spite of this most desired phenomenon, we see differences in practice between different groups. Indicators of disparity tries to measure these inequalities between different groups with the objective of providing equal opportunity to all who are disadvantaged. This paper tries to outline the methodology of calculating indicators that measure the level of inequality. By so doing the measures of the level of equality are also addressed. What do we mean by disparity? The word disparity refers to lack of equality or parity among different groups. Obviously there must be some comparable things for us to be able to talk about parity or Disparity(synonyms) Difference disparity, equality or inequality. Parity between what? Or Inequality disparity between what?. Therefore, disparity is defined as the Discrepancy difference between two or more things. We want to talk about, Disproportion for example, the difference between urban and rural, between Gap Inconsistency gender, between regions and districts and villages etc in relation Lack of to educational opportunities available to these different groups. correspondence In other words we want to know, as planners and decision Antonym: Parity support systems, which region is disadvantaged when compared to the other regions and which one is more advantaged compared to other regions. Is the educational provision equal between urban and rural and between sexes etc. We assume planners, statisticians, decision makers and all educationists have a desire to provide equal opportunity for education. We provide such analysis to assist them formulate appropriate 3 action to achieve parity between different groups mentioned above and many more. When, on the contrary, such differences are brought about by mismanagement, deliberate act of favouring one at the expense of the other, again we use such analysis to expose their action and help them rectify the wrong actions. We do this in favour of equality between these different groups. Therefore, we need to take the responsibility of, not only providing the management with good analysis, but also base our analysis on correct data from the field that can be appreciated and applied in making correct decisions. We can also extend such discussion to trend analysis and ask: has system been improved over time? Has the situation of girls, for example, changed for the better or worse during the last few years? Which regions are doing better compared with other regions etc. Deducing from such analysis, we have correctly assumed that as rational planners, we want equality between regions, between urban and rural and between gender etc. In this short paper we intend to look at the equality or inequality between three different groups. Our discussion is based on data published in annual education abstracts and can be used for reference. These groups are categorised as follows: Regions Urban Vs rural Sexes Objectives: The objective of this paper is to explain the methods of calculating disparity indicators, step by step, so that participants understand the underlining principles and apply it when compiling their reports to be submitted to planners, decision makers and other users. It is hoped that some of these methodologies will be used when organising an indicators report among others. It is also hoped that this paper provides a start for larger investigation to more comprehensive and practical analytical approach to the study of subject matter. Measures of inequality: The following measures of disparity are discussed briefly with illustrative examples. The examples are taken annually published data on education statistics. 1. Comparing figures 2. Graphical method 3. Representation index/selectivity index 4. Gender parity index 5. Lorenz Curves 6. Gini coefficient Method of comparing numbers: By comparing two or more figures, or columns of figures. We can see which one is greater or lower For example figures for girls can be compared with that of boys to see which one is greater or lower. This is the easiest way to look at disparities. It is a simple way that every one interested can work out and get a feeling of the level of disparity between different groups. However, it is crude, it may not tell us much. At times it is not possible to scan through all the figures at a time. Another alternative is to look at the absolute difference. The gap between, for example, boys and girls, urban and rural, region A and region B, or district X and district Y etc can be assessed by subtracting one figure from the other. This difference, in quantity, between urban and rural, between boys and girls, and between regions etc can give a better clue than 4 simple comparison of figures. However, it is again another coarse method to use but relatively easier to apply. Consider the table on the right. We can make two statements about the table: Gross enrolemnt ratio in Ethiopia 1. There is a marked difference between gross Yr Boys Girls GG enrolment ratios of boys and girls in Ethiopia. 1994 31.7 20.4 11.3 2. The difference increases with time thus widening 1995 36.6 22.7 13.9 the gap between the sexes. 1996 43.0 26.0 17.0 Note Yr=Year, GG=gender gap. 1997 52.0 31.0 21.0 We can use other methods discussed below to get a 1998 55.9 35.3 20.6 better picture. Other methods include percentages and 1999 60.9 40.7 20.2 ratios which cannot be neglected in terms of providing 2000 67.3 47.0 20.3 Source: Education statistics abstract, 2000. preliminary information about the level of inequality. The following example is taken from Education statistics annual abstract of Ethiopia of the year 2000/01. The graph below is used to discuss the level of disparity between regions and between sexes in Ethiopian context. Graphical method: use simple graphs to show the variation between regions, urban or rural and between sexes. Many users seem to be comfortable with graphs that show these different variations clearly. Today it is easy to use spreadsheet programs to draw such simple graphs. For example we can use enrolment ratio data and draw bar graphs to show the variation of enrolment ratios by region, district or even villages and schools. The heights of the bars can show the level of the ratios with respect to the regions. We can play around with numbers and draw graphs, and graphs can be used as analytical tools. See graph under RI below. Let us look at the following graph, which depicts the latest gross enrolment ratio for Ethiopia. The objective here is to compare ratios of boys to that of girls. We have plots of points for boys’ ratio against girls’ 120 Gambella Zim-Gross Ben_Gumuz ratio on scatter graph. The line Y=X bisects, diagonally from the origin, the 100 graph into two equal parts. Boys’ Country X Dire Dawa Zim Net SNNPR ratios are on the vertical axis and girls’ 80 Oromia Tigray Eth-Gross ratio on the horizontal axis. This is just 60 Amhara Eth-Net________ by chance. One can have it the other way round. You may wish to swap the 40 two, which has no bearing on the Country Y interpretation of the graph but change 20 Somale __________Af ar of position. However, the plots will BOYS fall on the opposite side of the 0 0 20 40 60 80 100 120 diagonal line directly symmetrical to their current position. Note also that GIRLS the scale on both axis ranges from 0 to 120. Source: Annual educational abstract 00/0 1 On the graph, the line Y=X and the line Y=2X are drawn along the co-ordinates. This is not without a purpose. We want to see where exactly the plots will be in relation to these lines. The following points are in order: 1. Any point on diagonal line, ie. Y=X, tells us the parity between boys and girls no matter where on the line. 2. On the other hand, any point on the line Y=2X, tells us that the ratio for boys is double that of the ratio for girls. 5 3. Any point that falls on the bottom left corner of the graph tells us the low level of enrolment ratios (or whatever subject we want to show) The regions Afar and Somale fall under this in our example. 4. On the other hand, any point that falls on the top right corner of the graph, tells us that the ratios are higher(Addis Abeba). 5. Finally, any point that falls below the equality line indicates that the boys are disadvantaged in that they have lower enrolment ratio than girls. Having said that, lets go back to the Ethiopian example above. There are several points that can be raised. I will select only some of them. Some more points are included in the exercises. The two regions, Afar and Somale are the most unfortunate areas. They are the ones most closest to the origin of the graph. This means, they have lowest ratios both for boys and girls. These are the most disadvantaged areas which need special attention in terms of coverage1. On the opposite corner, top right, lies Addis Ababa. This is the most advantaged region. Moreover, disparity between the ratio of boys and girls is nearly zero. In fact, the enrolment ratio of girls is slightly higher than that of boys. The reason is clear. Addis Ababa is the capital city. The Harari and Gambella regions have relatively higher ratios. However, the difference between the sexes is higher, especially for Gambella. Harari is a town where the majority of the residents are Muslim population. Gambella, on the other hand, is one of the low lands in the Sudan border with several refugee camps. This explains little about the actual situation. A further study is recommended to find out the reasons behind low enrolment ratio for girls. Dire Dawa is another town near Harari region. It has lower enrolment ratio with a lower gap between boys and girls. One wonders why? What is the reason behind lower enrolment ratio in Dire Dawa as compared with Harari region and much wider gap between the sex ratios in Harari compared to Dire Dawa. The other category of regions, as shown on the graph, are Tigray and Amhara. The two are similar in that they both have lower gap between the sexes. However Amhara region has a much lower enrolment ratios compared to Tigray region. Note that both of them are nearer to the equality line. This means we are less concerned about gender gap in the two regions compared to other regions. The last category of regions, I would like to consider are Oromiya and SNNPRG2. The latter region has a relatively higher enrolment ratio than the former. However, they both have higher gap between the sex ratios with Oromiya slightly in a better position. The above graph is also used to compare two countries. These are Ethiopia and Zimbabwe. The Zim-Gross label indicates the gross enrolment ratio for Zimbabwe and Zim-Net label indicates the net enrolment ratio for Zimbabwe. Similarly Eth-net indicates the net enrolment ratio for Ethiopia. The plots exhibit that the two countries are considerably far apart in terms of coverage. Considering the gross enrolment ratio, Zimbabwe has 110(115 1 Such attention should also include investigation of quality of data used for calculating the ratios(school age population and enrolment data) 2 Southern Nations, Nationalities and Peoples Regional Government 6 for boys and 105 for girls) and Ethiopia has 57(67 for boys and 47 for girls). The gap between two countries is 57%. Obviously Zimbabwe is in a better position than Ethiopia in terms of coverage in primary education as judged by gross enrolment ratios and net enrolment rations. Again Zimbabwe has narrower gender gap(5.0) compared to Ethiopia(20.3) using gross enrolment ratios as a measure. This can be seen from the graph. As we said earlier, the nearer is the point to the line of equality, the lesser is the gender gap. The ratio for Ethiopia is further away from the line of equality compared to that of Zimbabwe using both gross and net enrolment ratios. Please note that the labels for regions afar and Somale and Eth-net and Amhara regions overlap on the graph. Using the net enrolment ratio for comparing the two countries, we arrive at 47% difference between the two countries. It can be read from the above graph that the gender gap is narrower in Zimbabwe than in Ethiopia. Note that Zimbabwe has 7 years of primary education and Ethiopia 8 years of primary education. How this does not affect our analysis as long as the corresponding school age population is used to calculate the ratios. Finally, let me say a word about the two points on the graph (country X and country Y). One is located above the equality line and the other below the equality line. These are hypothetical countries meant only for illustration. Country Y is below the equality line showing that boys are the more disadvantaged groups compared to the girls. Country X is above the equality line showing that the boys are more advantaged. Moreover, the hypothetical country X is above the line Y=2X, indicating the ratio for boys is more than double that of the girls. I am sure there are more points to be raised about the above analysis, but this is enough for our discussion. The main question remaining is: what follows next? What should we do about each of the above situations? Should we take part of the problem and address that or all of the above can be addressed? This depends on budget availability. If we have enough budget we may address all problems. However, in practice, there is budget shortage in which case we are forced to prioritise the problems. In a way the priorities are already set in the above analysis. In my opinion the issue of the two regions Afar and Somale should be taken up first. They are exceptionally low in terms of coverage in primary education. Considering gender disparity as the main issue, SNNPRG and Oromiya are our first priority regions next to Benshangul_Gumuz and Gambella. The latter two, in my opinion, needs a special investigation. The two regions boarder neighbouring Sudan. Gambella is a refugee area while Benshangul_Gumuz has fewer refugees. However, they both are among those regions classified as most disadvantaged regions. Their ratios seem inflated. The ratios are inflated whenever the enrolment data is inflated or conversely, whenever population data is under estimated. Inflation could also be due to mechanical error in data collection, processing and analysis. For example, when breaking population figures, five-year age group, into single age groups errors may have been introduced. Nevertheless, the two regions need further study as to why the enrolment ratios are inflated. Note that the underlining table in the above analysis is simpler to present. However, the graph glows the issues up for the reader to see easily. There is more information conveyed through the graph than the table. Note also that, we can use such graphical analysis for 7 many different situations: urban and rural, enrolment ratios over time, comparing boys and girls and countries etc. Exercises The reader is encouraged to do the following exercises to consolidate his knowledge about graphical analysis. 1. Draw similar graph like the above using your own country data and interpret the result. Point out extreme cases and discuss. 2. List the first four countries in order of priority needs with highest gender disparity and discuss about each one of them. 3. Draw two similar graphs, one for boys and the other for girls showing the result over time period. Take the example from your own country statistics annual bulletin. The next indicator we consider is the representation index. 3. Representation index(RI): Representation index is defined as the proportion of characteristics divided by the proportion of criterion3. RI is the proportion of characteristics divided by the proportion of criterion expressed in percentages. A characteristics, Johnstone explains, is a variable whose equality is being investigated while criterion is a variable acting as (criterion) standard. The following graph shows the representation index for Ethiopia, 1986 e.c./93-94/. The proportion of enrolment in primary grades is divided by the proportion of corresponding school age population for primary/age 7-12/. The data is ranked by RI index. The results are shown on the following graph. We may pick up many points about the graph but the 300.0 Representation Index following two points are selected for illustration. 250.0 200.0 Addis Ababa region is RI the most over 150.0 represented region of 100.0 all other regions. Addis Abeba is the 50.0 capital of the country. 0.0 In fact many city areas NP ra ri ba a uz ia lla a ay ra have a better iy op w ha be um ba SN gr m Ha Da hi m am Ti ro A G Et A O re is G advantage compared n. Di dd Be Region A to rural areas. The other point is about the two most under represented regions. Amhara and Oromiya. The two regions are the largest, in terms of both population and area, regions of the country. The next disadvantaged region is SNNPRG. This region has a slightly higher RI index than the above two regions. However the three regions are the largest and most populous regions, (the sum of the school age population of the three regions is over 85% of the total primary school age population of the country). Representation index is a simple indicator we can use to analyse a situation. Regions can be ranked according to RI to identify those advantaged and disadvantaged in terms of 3 Characteristics is the variable we want to measure(e.g. enrolment) while the criterion is the variable against which we compare the characteristic to measure(e.g school age population). 8 coverage, budget allocation, distribution of qualified teachers, and other necessary resources etc. Exercises: Percent of 18- Percent of 2 1. Calculate Representation index using 24 years year college your own data on enrolment in Sex Race population enrolment primary grades and the corresponding Males white 38.9 35.9 school age population of your country. Asian 0.8 1.5 Graph and interpret the result. Black 6.2 4.4 2. The following table contains data for Hispaic 3.9 2.9 college students by race and sex. American Indian 0.3 0.5 Calculate the representation index and discuss the points. Which race is under Females white 38.4 43.8 represented? Discuss different Asian 0.8 1.4 scenarios by sex. Note that college Black 6.7 6.1 students are characteristics and 18-24 Hispaic 3.7 3.1 year population as criterion. American Indian 0.3 0.6 Total 100.0 100.0 3. Use education budget and school age population by region of your own country and calculate representation index. Which Source: www.nsf.com region is under represented? Or over represented? Discuss. In each case sort your data by RI and graph the result. 4. Sex parity index (SPI): This index is commonly called Gender Parity Index. The word gender has a broader meaning nowadays. Therefore, I felt calling this indicator as ‘sex parity index (SPI)’ instead of gender parity index. This may solve the problem of interpretative capacity of this indicator. SPI is defined the ratio between the female and the male rates. For example, female net intake ratio divided by male net intake ratio. The value of SPI is mostly between 0 and 1. The value goes above one whenever the female rate is higher than the male rate. However, when there is a perfect equality between the two(male and female), the SPI is equal to 1. When there is absolute inequality between the two, the value of SPI is equal to 0. The following table shows the evolution of net intake rate in Zimbabwe for the last three years. The SPI is given in the last column. There are two points to be raised about the table below: 1. The increase in net intake rate between 1998 and 1999 is much smaller than the NIR evolution by sex(Zimbabwe) increase between 1999 and 2000 in Boys Girls Total SG SPI Zimbabwe. This is true both for boys and 1998 42.3 41.7 42.0 0.60 0.99 girls. An increase of 8.5% in one year 1999 43.5 43.0 43.3 0.50 0.99 seems high to me. We may have to go 2000 50.8 51.9 51.8 -1.10 1.02 back and check the figures we used to SG=Sex gap; SPI=Sex Parity Index calculate the rates for correctness or asses Source: Annual statistics abstract, 1999 the efforts made to result in such high increase. It may also be necessary to check if the calculations are correct. 2. Enrolment rate for girls is greater than enrolment rate for boys for the year 2000. Looking at the past trends for girls’ intake rate, the girls’ rate has not exceeded that of boys’. This may not pose a problem at first glance, but one could suspect the result. However, we have little information to say more about this situation. We need more time series data to get more information on how the changes occurred 9 over the years for both boys and girls. If such data is available additional information and wider knowledge could be gained. On the other hand we may believe the efforts made during the previous years have shown great increase on the number of new entrants and hence, this result came about. This is fine as long as we reach an understanding to substantiate the result. However, in the absence of such common understanding we may resort to further study which could mean more knowledge about the level of intake rate. This indicator can also be used to investigate the urban and rural differences. The following table shows literacy rates in Zimbabwe for the year 1999. The rural /urban gap is 10.1 for NIR evolution by sex(Zimbabwe) male and 18.4 for female. Obviously the female gap is Urban Rural RUG RUPI higher compared to male gap. This shows that the rural Male 96.4 86.3 10.1 0.90 females are more disadvantaged compared to urban Female 94.2 75.8 18.4 0.80 males. On the other hand the gap between male and Diff 2.2 10.5 female in rural areas is much higher compared to urban RUG=Rural/Urban Gap; RUPI=Rural/urban Parity Index areas. Both male and females are disadvantaged in rural Source: Annual statistics abstract, 1999 areas. This indicates that the Ministry of Education officials in Zimbabwe should give priority to rural females. They are the ones most disadvantaged. However, this does not mean neglecting the men in the rural areas. They also need due attention but with greater priority given to rural females. On the whole, the literacy level in Zimbabwe is very high compared to Ethiopia. Exercises: 1. The following example is NER by Economic Sector and Data Source: 1999 (Enrolment) taken from Zimbabwe Settlement areas Male Female Both Annual Education Communal Land 95.0 90.9 93.5 Statistics Abstract of Resettlement Area 104.2 121.0 111.9 the year 1999. Calculate Commercial Farming Area 77.1 75.9 76.5 the Sex parity Index Urban Area 80.8 79.7 80.3 and comment on the result. What steps should be taken in order to improve the situation? Discuss. 2. Compile data on enrolment ratio for the last five years by gender from your own country example. Calculate sex parity index and comment interpret the result. Draw the graph of SPI in both cases. 5. Lorenz curve(LC): Lorenz curve is developed by Max O. Lorenz to describe the X-Axis Y-Axis extent of income inequality in a society. Characteristic (Vertical Criterion (horizontal axis) Economists use LCs to measure income axis) inequality among households. In this case, the Proportion of enrolment in cumulative percentage of the households and the Amount of education offered each grade cumulative percentage of the income are used to Proportion of live births Proportion of live deaths draw the Lorenz Curve graph. The cumulative Percent of 18-24 years Percent of 2 year college percentage of the household is drawn on the population enrolment horizontal axis and the cumulative percentage of Proportioon of school age Proportion of enrolment the income on the vertical axis. The population characteristic, the variable to be measured is drawn on the vertical line and the criterion variable on the horizontal line. The figure on the right illustrates this point. 10 Johnstone used distribution of enrolment across grades. He put the cumulative proportion of the enrolment on the x-axis and the grade on y-axis. Then the cumulative proportion of both variables were calculated and used to draw the LC graph. The following summarise the necessary steps needed to draw the Lorenz curves: 1. Identify the criterion and the characteristic variables. 2. Sort the variables by the characteristics. 3. Calculate the proportion of each of the two variables. 4. Calculate the cumulative proportions of the two variables. 5. Graph the curve using the x-axis for the cumulative proportion of the characteristic and the y-axis for cumulative proportion of the criterion. The following curve is drawn using data from annual abstract, Ethiopia 1993 e.c. /00-01/. The example tries to show the measure of inequality of enrolment Lorenz Curves 8 distribution across grades. The further away the Lorenz curve is 7 Line of equality from the line of equality, the higher 6 the level of inequality. Conversely, the closer the Lorenz curve is to 5 Grade the line of equality, the higher the 4 level of equality. The best use of 3 the Lorenz curve is made when two or more curves are drawn on 2 one graph then comparison. This 1 way comparison can be made e.g. Lorenz Curve over time as in the example below. 10 20 30 40 50 60 70 80 90 100 As it is now, we can only say there Cumulative proportion of enrolment is no equality in the distribution of enrolment by grade in Ethiopian Source: Annual Abstract, 1992 e.c./1999-00/ primary schools. I suggest you draw the graph manually to understand better the underlining principles. Y Distribution of enrolment by gade in Ethiopian primaryschhols: Lorenz Curves The best use of LC is made when 8 comparing scenarios like: male and 7 female, urban and rural, between regions, over time etc. Then we 6 Enrolment by grade 1993 e.c./00-01/ know which LC is nearer to the line 5 of equality and judge a better or a worse situation. We can use the 4 same graph to plot the points for 3 different years or urban/rural situations. The example on the left 2 is again taken from Ethiopia: 1 Enrolment distribution by grade. Enrolment by grade 1990 e.c./97-98/ Two years data are compared to see 10 20 30 40 50 60 70 80 90 100 the changes. The intention here is to Source: Education Statistics annual abstract, 1993 e.c/1990 e.c. X compare enrolment distribution by grade for the two years indicated 11 and see if the distribution of enrolment by grade has improved or not. The LCs show the distribution (lower) for the year 1990/97 and the upper for the year 2000/014. The distribution for the latter year, obviously is closer to the line of equality. Hence, the enrolment distribution by grade, for Ethiopia, has improved slightly over the last four years. An interesting exercise will be to draw Lorenz curves for boys and girls and between urban and rural by five year gap over several years and see the outcome. The reader is encouraged to do the following exercises to master the subject. Exercises: Grade Enrolment 1. The following data, distribution of enrolment by grade, is taken 1 416139 from Zimbabwe, annual statistical abstract. Draw the Lorenz 2 358133 Curve and comment on the result. 3 357069 2. Use the above data under representation index and draw Lorenz 4 345538 curve and interpret the result. 5 333321 3. Use your own country data on enrolment and school age 6 327901 population by region and draw Lorenz Curve. Interpret the result. 7 311139 4. Take two time points in time(say five years), and obtain data for enrolment and school age population by region from your own country. Draw the Lorenz Curves using the data under exercise 3 above. Has the situation improved over time or not? Discuss. The next indicator takes the notion of the LC further and gives a mathematical expression of the level of inequality. 6. Gini coefficient(GC): This coefficient gives mathematical expression of the level of concentration. By going through the steps for while drawing the Lorenz curve, we have already gone half way for calculating the Gini coefficient. We add a few steps to obtain the coefficient that quantifies the level of inequality using the method of Gini coefficient. Scientists still use this coefficient to measure the level of wealth distribution between nations, income between households, health among community etc. Educationists use the coefficient to measure the equality between boys and girls, urban and rural etc. Gini coefficient is an expression of the ratio of the area between the line of equality and the Lorenz curve(the shaded area of the graph on the right). When we have a perfectly equal distribution, the value of Gini coefficient is 0. On the contrary when we have a perfectly unequal distribution, the value of the Gini coefficient is 1. In the former case the Lorez Curve is further away from the line of equality and in the latter case it is closer to the line of equality. following formula may be used to calculate the Gini coefficient. We can use the LC graphs and show the level of variation visually. However if we want to quantify the result, or the curves don’t clearly show the level of variation, we may need 4 You need a colour printer to be able to see. Otherwise you need rely on the labels. 12 to calculate the Gini coefficient and compare the result. The first formula is taken from Johnstone’s book “Indicators of Education Systems”. Johnstone used it to draw Lorenz Curves and calculation of Gini coefficient. The following is a formula for the calculation of Gini coefficient. Since this is a training exercise, the reader is encouraged to practice examples using both formulas. The system in country X has six years of primary education. Enrolment by grade is given in column 2. Proportion of enrolment by grade is given in column 3. Proportion of total education is given in column 4. The other two columns 5 and 6 contain cumulative proportion of enrolment by grade and total education n respectively. The cumulative proportions are used to G in i ( p i 1 q i p i q i 1 ) draw the LC. Column 7 takes the first portion of the i 1 formula and obtains the product pi-1qi. Column 8 takes Whe re pi = Cumulative pro po rtio n o f the characteristic the second part of the formula and obtains the product who se equality is being investigated. qi = Cumulative pro po rtio n o f the variable which is piqi-1. Finally column 8 is subtracted from column 7 in acting as a criterio n fo r the measurement. column 9. The Gini coefficient is shown as a sum of the last column 0.297. Obviously this figure is nearer to 0 than 1. This implies that the distribution of enrolment across grades in country x is nearly equality distributed. However there is still a room for improvement. We keep on improving the system until the value equals or at least approaches 0. Create the following table on a spreadsheet and study through. Enrolemnt by grade: Country X (1) (2) (3) (4) (5) (6) (7) (8) (9)=(7)-(9) Grade pi pp i pq i cpp i cpq i p i-1 q i p i q i-1 (p i-1qi-piqi-1) 0 0 0 0 0 0 0 0 1 895904 0.338 0.167 0.338 0.167 0.000 0.000 0.000 2 575857 0.217 0.167 0.555 0.334 0.113 0.093 0.020 3 437052 0.165 0.167 0.719 0.501 0.278 0.240 0.038 4 329360 0.124 0.167 0.843 0.668 0.480 0.422 0.058 5 241584 0.091 0.167 0.934 0.835 0.704 0.624 0.080 6 174340 0.066 0.167 1.000 1.002 0.936 0.835 0.101 Total 2654097 1.000 0.297 The second example uses another formula. This is Brown’s formula obtained from www.paho.org/English/SHA/be_v22n1-Gini.htm. The example below is also obtained from the above website. Both formulas should lead to the same result. It is about live deaths in some Latin American countries. The idea is to investigate the distribution of deaths across these countries in order to find out if infant k1 deaths are equally distributed or not. This example Gini 1 (Yi 1 Yi )( X i1 X i ) i0 is chosen to show how this method is applied in Whe re another discipline other that education on than one Y=Cumulated pro po rtio n o f the health variable had and also to show the different nature of the X=Cumulated pro po rtio n o f the po pulatio n variable Lorenz Curve. This is given as an exercise to the reader. 13 EXAMPLE: The following table summarises the steps involved in calculating the Gini coefficient using the health example. All necessary calculations are made to obtain the coefficient. All you need is to plug it in to the above Brown’s formula. The intention is to measure the distribution of infant deaths across Latin American countries listed above. I chose this example to illustrate the steps involved in calculating the Gini coefficient because, in a way, the steps are clearly stated in the manuscript and to show how the Gini coefficient is applied in areas other than education. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Cumulative Cumulative Live births Proportion Proportion Infant proportion proportion (1,000) live births (Infant Xi+1-Xi Yi+1+Yi deaths live births (Infant 1997 (X1) deaths) Country (X1) deaths) (A) (B) A*B Bolivia 250 14750 0.09 0.17 0.09 0.17 0.09 0.17 0.02 Peru 621 26703 0.24 0.31 0.33 0.48 0.24 0.65 0.16 Ecuador 308 12012 0.12 0.14 0.45 0.62 0.12 1.10 0.13 Colombia 889 21336 0.34 0.24 0.79 0.86 0.34 1.48 0.50 Venezuela 568 12496 0.22 0.14 1.0 1 0.22 1.86 0.41 Total 2636 87297 1 1.2 Note that the Lorenz curve drawn from the above data looks a little strange. It falls above the equality line unlike many other Lorenz curves I have drawn before which often lie below the equality line. This has to do with the nature of the data. The literature5 has it that "when the variable is beneficial to the population, the curve lies below the diagonal line. On the contrary, when the variable is prejudicial, as in the case of deaths, it is found above the line". One of the exercises in the afternoon is to draw the Lorenz curve of the above data. Now, let us summarise the steps involved. The steps involved in calculating the Gini coefficient is given below 1. Sort the units by the health variable (infant mortality rate) from the worst situation(highest rate) to the best situation (lowest rate). 2. Calculate the proportions of infant deaths. 3. Calculate the proportion of live births. 4. Calculate the cumulative proportion for both live births and infant deaths. 5. Calculate the Gini coefficient using the above formula You should get Gini=0.20. This is not a high value. The level of infant deaths is similar among the above countries. The interesting point will be to compare this value with the values obtained from North America, Europe, Africa, Asia etc whenever data is available. 5 www.paho.org 14 Exercises: You are encouraged to do the following exercises in order to understand the methodology. , you may wish to do the following exercises to get hands on practice on the exercise. The following exercise is taken from Zimbabwe. Table 1 gives the evolution of net enrolment ratio in Zimbabwe. 1. Calculate the Gini coefficient for the exercises under the Lorenz curve above and compare your result. Grade Enrolment 2. Use the following data from Country Y enrolment by 1 1751600 grade and calculate the Gini coefficient and draw the 2 951944 Lorenz curve. 3 721587 3. Draw a Lorenz Curve for the exercise on live deaths 4 515300 and comment on the result. 5 384756 6 282546 7 264518 8 218419 Total 3224065 References: 1. Johnstone, J.N., Indicators of education systems, Kogan Page UNESCO, London. 2. Women, minorities, and persons with disabilities in Science and mathematics. http://www.nsf.gov/ 3. UNESCO, Basic Education Indicators, Division of Statistics, UNESCO, Paris. 4. [March 2001], Measuring Health Inequalities: Gini Coefficient and Concentration index, Epidemiological Bulletin, vol. 22 No. 1, http://www.paho.org/ 5. Johnstone, J.N, Indicators of the performance of educational systems, IIEP occasional papers No. 41, UNESCO, Paris. 6. Thiessen H. , Measuring the real world: A text book of applied Statistical Methods, John Wiley, and sons, United Kingdom.
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