I think the best methodologycal context within which to position your problem is decision making theory, particularly multiple criteria decision making (MCDM). You will probably find examples in this corps of theory of abusive usage of statistics and mathematics without an understanding of what is in stake. Basically ranking is deciding, and one can not expect a mathematical tool to magically do something only a human being can do. Math helps but can not replace the decision maker role. Check for example the work of Carlos Bana e Costa, maybe he has some examples.  

To help tackle your problem consider the following excellent (theoretically sound and practical) manual on MCDM:

http://eprints.lse.ac.uk/12761/1/Multi-criteria_Analysis.pdf

What you want to do is to find an agregate model of several rankings or criteria. A simple average would be a wrong simplification because the different ranks surely have diferent weigths in a final ranking. Another reason (besides them being ordinal like you said) is that you have different variables or ranks, measuring different value dimensions, while a statistical average is something reporting to a single variable. You have a multiple dimension problem - MCDM.

Maybe you can use the indicators or metrics (quantitative) used to build the rankings instead of the rankings themselves (ordinal).

A simple additive model for your problem is expressing in the same scale (0-100 ) all your indicators or metrics (criteria) behind each rank i. You can use value functions to do this (check the manual). Then you can weight each criteria by its importance or weight i (in %) and sum over all criteria i, to obtain a final ranking:

Σ (criteria i * weight i)

where total weigths    Σ weight i = 100%

Then perform sensitivity analysis.

The reason your problem is a decision problem is because somebody (a decision maker) has to assign weights to each rank.

There are simple methodologies like "swing weigths method" to do that. I think this is exemplified in the manual above.

If you want to be able to defend the consistency of your global ranking you can use the software MacBeth that checks the mathematical consistency of your judgments (for example if you prefer a to b and b to c and then you say you prefer c to a this is inconsistent)

If you prefer a qualitative approach to the criteria weights trade off you can take a look at this modest paper I wrote suggesting a way to do it:

http://www.fm-kp.si/zalozba/ISSN/1854-4231/6_127-146.pdf

Basically I suggest taking as models novel criteria in the value dimension of which every pair of conflicting criteria becomes automatically ranked (one becomes more important than the other). For example, according to the value dimension of quality (in a delay of gratification or investment perspective) maybe the Rank of "Scores" becomes more important than the rank of "Graduation Rates". I am not sure if this is a good example, but the approach is relying more on the modelling perspective.

I am also attaching some references, since I did my Masters and worked in Universities Funding incorporating indicators/criteria into one single value many years ago:

 3] Barr, N. , 2003. Financing Higher Education in the UK: The 2003 white paper. House of Commons Education and Skills Committee, The Future of Higher Education, Fifth Report of Session 2002-03, Volume II, Oral and written evidence, HC 425-II, (TSO), pp. Ev 292-309.

[5] Philips, L., Bana e Costa, C. B., 2005. Transparent prioritisation, budgeting and resource allocation with multi-criteria decision analysis and decision conferencing (working paper). Department of Operational Research, London School of Economics and Political Science, pp.2.

[6] Sanyal, B. C., 2005. Financing Higher Education: International Perspectives, 2nd International Barcelona Conference on Higher Education, Barcelona, Spain.

[7] Johnstone, D. B., 2004. The economics and politics of cost sharing in higher education: comparative perspectives, Economics of Education Review, Nº 23, pp. 403-410.

[9] Varghese, N. V., 2001. The limits to diversification of sources of funding in higher education. UNESCO: International Institute for Educational Planning, Paris.

[11] Johnstone, D. B., 2005. The role of student loans in financing higher education: student access and institutional viability, 2nd International Barcelona Conference on Higher Education, Barcelona, Spain.

[13] Escotet, M. A., 2005. Gobierno, rendicion de cuentas e financiamiento de la universidad, 2nd International Barcelona Conference on Higher Education, Barcelona, Spain.

[14] Maldonado-Maldonado, A., 2005. International cooperation and an overview on higher education financing, 2nd International Barcelona Conference on Higher Education, Barcelona, Spain.

[15] Kent, R, 1995. Two Positions in the International Debate About Higher Education: The World Bank and UNESCO. XIX Latin American Studies Association Congress, Washington DC

[16] European Research Associates (EU-RA), 2004. Projects in ‘Economics of Education’. Study on ‘The Financing of Higher Education in Europe’. European Commission - Directorate General Education and Culture, pp. xx-yy.

[19] López, S., 2004. Vínculos entre financiación y organizacion universitaria: análisis comparado de la universidad española, Archivos Analíticos de Políticas Educativas, Volume 12, Nº 63.

[21] Canton, E., Venniker, R., Jongbloed, B., Koelman, J, Meer, P. ,Vossensteyn, H, 2002. Higher Education Reform: Getting the Incentives Right. Netherlands Bureau for Policy analysis and the Center for Higher Education Policy Studies.

Disponível em:

http://www.utwente.nl/cheps/publications/downloadable_publications/

[24] Vossensteyn, J. J., 2005. Perceptions of student price-responsiveness.

Disponível em:

http://www.utwente.nl/cheps/publications/downloadable_publications/

[26] Cave, M., Hanney, S., Henkel, M., Kogan, M., 1997. The Use of Performance Indicators in Higher Education – The Challenge of the Quality Movement. Higher Education Policy Series 3, Jessica Kingsley Publishers.

[27] 2006. Higher Education in The World 2006. The Financing of Universities. Guni Series on The Social Commitment of Universities 1. Palgrave Macmillan.

[28] Jongbloed, B., Vossensteyn, H., 2001. Keeping up performances: an international survey of performance-based funding in higher education, Journal of Higher Education Policy and Management, Vol. 23, Nr. 2, pp. 127-144.

[29] Lang, D., 2004. The political economy of performance funding, Taking Public Universities Seriously Conference, University of Toronto.

[30] Jongbloed, B., Koelman, J., 2000. Vouchers for higher education? Center for Higher Education Policy Studies.

Disponível em:

http://www.utwente.nl/cheps/publications/downloadable_publications/

[31] Jongbloed, B., 2004. Funding Higher Education: options, trade-offs and dilemmas, Fullbright Brainstorms 2004 – New Trends in Higher Education.

Disponível em:

http://www.utwente.nl/cheps/publications/downloadable_publications/

[32] Kaiser, F., Vossensteyn, H., Koelman, J., 2001. Public Funding of Higher Education. A Comparative Study of Funding Mechanisms in Ten Countries. Center for Higher Education Policy Studies, pp. xx-yy.

Disponível em:

http://www.utwente.nl/cheps/publications/downloadable_publications/

[33] Perotti, R., 2002. The Italian University System: rules vs. incentives, 1st conference on Monitoring Italy, ISAE, Rome

[34] Armenteros, J., 2003. La Financiación de los sistemas universitarios autonómicos: objetivos e instrumentos. Encuentro “La financiación de las universidades madrileñas en una sociedad avanzada”, Universidad Politécnica de Madrid, Madrid.

[36] Salermo, C., 2003. What we know about the efficiency of higher education

institutions: the best evidence. Center for Higher Education Policy Studies.

Disponível em:

http://www.utwente.nl/cheps/publications/downloadable_publications/

[38] Bana e Costa, C., Thore, S., 2002. The many dimensions of an R&D project: multi-criteria analysis, in Technology Commercialization: DEA and Related Analytical Methods for Evaluating the Use and Implementation of Technical Innovation. Kluwer Academic Publishers, pp. 23-44.

[42] Burke, J., and Associates, 2002. Funding Public Colleges and Universities for Performance. Popularity, Problems and Prospects. The Rockefeller Institute Press, pp. xx-yy.

[43] Shin, J., Milton, S., 2004. The effects of performance budgeting and funding programms on graduation rate in public four-year colleges and universities, Education Policy and Analysis Archives, volume 12, nº 22.

[44] OECD, 2006. Funding systems and their effects on higher education systems, international study draft report and power-point presentations in the meeting of 14th September in Paris.

[46] Goodwin, P., Wright, G., 1991. Decision Analysis for Management Judgment. John Wiley & Sons, pp. xx-yy.

[47] Hamrick, F., Schuh, J., Shelley, M., 2004. Predicting higher education graduation rates from institutional characteristics and resource allocation, Education Policy and Analysis Archives, volume 12, nº 19.

[50] HEFCE, 2006. How HEFCE (Higher Education Funding Council for England) allocates its funds.

Disponível em:

http://www.hefce.ac.uk

Good luck.

I think the best methodologycal context within which to position your problem is decision making theory, particularly multiple criteria decision making (MCDM). You will probably find examples in this corps of theory of abusive usage of statistics and mathematics without an understanding of what is in stake. Basically ranking is deciding, and one can not expect a mathematical tool to magically do something only a human being can do. Math helps but can not replace the decision maker role. Check for example the work of Carlos Bana e Costa, maybe he has some examples.  

To help tackle your problem consider the following excellent (theoretically sound and practical) manual on MCDM:

http://eprints.lse.ac.uk/12761/1/Multi-criteria_Analysis.pdf

What you want to do is to find an agregate model of several rankings or criteria. A simple average would be a wrong simplification because the different ranks surely have diferent weigths in a final ranking. Another reason (besides them being ordinal like you said) is that you have different variables or ranks, measuring different value dimensions, while a statistical average is something reporting to a single variable. You have a multiple dimension problem - MCDM.

Maybe you can use the indicators or metrics (quantitative) used to build the rankings instead of the rankings themselves (ordinal).

A simple additive model for your problem is expressing in the same scale (0-100 ) all your indicators or metrics (criteria) behind each rank i. You can use value functions to do this (check the manual). Then you can weight each criteria by its importance or weight i (in %) and sum over all criteria i, to obtain a final ranking:

Σ (criteria i * weight i)

where total weigths    Σ weight i = 100%

Then perform sensitivity analysis.

The reason your problem is a decision problem is because somebody (a decision maker) has to assign weights to each rank.

There are simple methodologies like "swing weigths method" to do that. I think this is exemplified in the manual above.

If you want to be able to defend the consistency of your global ranking you can use the software MacBeth that checks the mathematical consistency of your judgments (for example if you prefer a to b and b to c and then you say you prefer c to a this is inconsistent)

If you prefer a qualitative approach to the criteria weights trade off you can take a look at this modest paper I wrote suggesting a way to do it:

http://www.fm-kp.si/zalozba/ISSN/1854-4231/6_127-146.pdf

Basically I suggest taking as models novel criteria in the value dimension of which every pair of conflicting criteria becomes automatically ranked (one becomes more important than the other). For example, according to the value dimension of quality (in a delay of gratification or investment perspective) maybe the Rank of "Scores" becomes more important than the rank of "Graduation Rates". I am not sure if this is a good example, but the approach is relying more on the modelling perspective.

I am also attaching some references, since I did my Masters and worked in Universities Funding incorporating indicators/criteria into one single value many years ago:

 3] Barr, N. , 2003. Financing Higher Education in the UK: The 2003 white paper. House of Commons Education and Skills Committee, The Future of Higher Education, Fifth Report of Session 2002-03, Volume II, Oral and written evidence, HC 425-II, (TSO), pp. Ev 292-309.

[5] Philips, L., Bana e Costa, C. B., 2005. Transparent prioritisation, budgeting and resource allocation with multi-criteria decision analysis and decision conferencing (working paper). Department of Operational Research, London School of Economics and Political Science, pp.2.

[6] Sanyal, B. C., 2005. Financing Higher Education: International Perspectives, 2nd International Barcelona Conference on Higher Education, Barcelona, Spain.

[7] Johnstone, D. B., 2004. The economics and politics of cost sharing in higher education: comparative perspectives, Economics of Education Review, Nº 23, pp. 403-410.

[9] Varghese, N. V., 2001. The limits to diversification of sources of funding in higher education. UNESCO: International Institute for Educational Planning, Paris.

[11] Johnstone, D. B., 2005. The role of student loans in financing higher education: student access and institutional viability, 2nd International Barcelona Conference on Higher Education, Barcelona, Spain.

[13] Escotet, M. A., 2005. Gobierno, rendicion de cuentas e financiamiento de la universidad, 2nd International Barcelona Conference on Higher Education, Barcelona, Spain.

[14] Maldonado-Maldonado, A., 2005. International cooperation and an overview on higher education financing, 2nd International Barcelona Conference on Higher Education, Barcelona, Spain.

[15] Kent, R, 1995. Two Positions in the International Debate About Higher Education: The World Bank and UNESCO. XIX Latin American Studies Association Congress, Washington DC

[16] European Research Associates (EU-RA), 2004. Projects in ‘Economics of Education’. Study on ‘The Financing of Higher Education in Europe’. European Commission - Directorate General Education and Culture, pp. xx-yy.

[19] López, S., 2004. Vínculos entre financiación y organizacion universitaria: análisis comparado de la universidad española, Archivos Analíticos de Políticas Educativas, Volume 12, Nº 63.

[21] Canton, E., Venniker, R., Jongbloed, B., Koelman, J, Meer, P. ,Vossensteyn, H, 2002. Higher Education Reform: Getting the Incentives Right. Netherlands Bureau for Policy analysis and the Center for Higher Education Policy Studies.

Disponível em:

http://www.utwente.nl/cheps/publications/downloadable_publications/

[24] Vossensteyn, J. J., 2005. Perceptions of student price-responsiveness.

Disponível em:

http://www.utwente.nl/cheps/publications/downloadable_publications/

[26] Cave, M., Hanney, S., Henkel, M., Kogan, M., 1997. The Use of Performance Indicators in Higher Education – The Challenge of the Quality Movement. Higher Education Policy Series 3, Jessica Kingsley Publishers.

[27] 2006. Higher Education in The World 2006. The Financing of Universities. Guni Series on The Social Commitment of Universities 1. Palgrave Macmillan.

[28] Jongbloed, B., Vossensteyn, H., 2001. Keeping up performances: an international survey of performance-based funding in higher education, Journal of Higher Education Policy and Management, Vol. 23, Nr. 2, pp. 127-144.

[29] Lang, D., 2004. The political economy of performance funding, Taking Public Universities Seriously Conference, University of Toronto.

[30] Jongbloed, B., Koelman, J., 2000. Vouchers for higher education? Center for Higher Education Policy Studies.

Disponível em:

http://www.utwente.nl/cheps/publications/downloadable_publications/

[31] Jongbloed, B., 2004. Funding Higher Education: options, trade-offs and dilemmas, Fullbright Brainstorms 2004 – New Trends in Higher Education.

Disponível em:

http://www.utwente.nl/cheps/publications/downloadable_publications/

[32] Kaiser, F., Vossensteyn, H., Koelman, J., 2001. Public Funding of Higher Education. A Comparative Study of Funding Mechanisms in Ten Countries. Center for Higher Education Policy Studies, pp. xx-yy.

Disponível em:

http://www.utwente.nl/cheps/publications/downloadable_publications/

[33] Perotti, R., 2002. The Italian University System: rules vs. incentives, 1st conference on Monitoring Italy, ISAE, Rome

[34] Armenteros, J., 2003. La Financiación de los sistemas universitarios autonómicos: objetivos e instrumentos. Encuentro “La financiación de las universidades madrileñas en una sociedad avanzada”, Universidad Politécnica de Madrid, Madrid.

[36] Salermo, C., 2003. What we know about the efficiency of higher education

institutions: the best evidence. Center for Higher Education Policy Studies.

Disponível em:

http://www.utwente.nl/cheps/publications/downloadable_publications/

[38] Bana e Costa, C., Thore, S., 2002. The many dimensions of an R&D project: multi-criteria analysis, in Technology Commercialization: DEA and Related Analytical Methods for Evaluating the Use and Implementation of Technical Innovation. Kluwer Academic Publishers, pp. 23-44.

[42] Burke, J., and Associates, 2002. Funding Public Colleges and Universities for Performance. Popularity, Problems and Prospects. The Rockefeller Institute Press, pp. xx-yy.

[43] Shin, J., Milton, S., 2004. The effects of performance budgeting and funding programms on graduation rate in public four-year colleges and universities, Education Policy and Analysis Archives, volume 12, nº 22.

[44] OECD, 2006. Funding systems and their effects on higher education systems, international study draft report and power-point presentations in the meeting of 14th September in Paris.

[46] Goodwin, P., Wright, G., 1991. Decision Analysis for Management Judgment. John Wiley & Sons, pp. xx-yy.

[47] Hamrick, F., Schuh, J., Shelley, M., 2004. Predicting higher education graduation rates from institutional characteristics and resource allocation, Education Policy and Analysis Archives, volume 12, nº 19.

[50] HEFCE, 2006. How HEFCE (Higher Education Funding Council for England) allocates its funds.

Disponível em:

http://www.hefce.ac.uk

Good luck.

In actuality, there is no INHERENT reason why you can't use the mean of variables with different meanings or of ordinal scaled variables. For example, it is common in certain types of medical assessment (for example, stroke severity) for a doctor to provide a "score" that is the average of a number of different ordinal rankings assessing different measures of performance (e.g. speech, locomotion, coordination, etc.). Grade point averages are another example of this, in fact. It is a fairly common approach.

That said, you need to exercise great caution in doing so and how you decide to interpret the results. Whether or not it is appropriate in your case has to do with the exact analysis goals. The biggest problem with these combinations is that you lose information (to be more precise, you actual underestimate the variability) of your outcome. There is a lot of literature on the subject, and in fact it is a rather contentious issue all around. A couple of good starting points are:

Knapp, T.R. (1990) "Treating Ordinal Scales as Interval Scales: An Attempt To Resolve the Controversy." Nursing Research, 39(2): 121-3. Online: http://www.mat.ufrgs.br/~viali/estatistica/mat2282/material/textos/treating_ordinal_scales%5B1%5D.pdf

Marcus-Roberts, H.M. and Roberts, F.S. (1987). "Meaningless statistics." Journal of Educational Statistics, 12(4): 383-394. Online: http://www.jstor.org/stable/1165056?seq=1#page_scan_tab_contents

In addition, each of these provides a number of other, more specific citations for various issues. You may also find more recent pieces on the subject by using Google's "cited by" search option for these papers.

I think Fernando and Ryan has really done justice to your question