World Rankings - Stanford 3rd after Caltech and Oxford.

Colleges and Universities A-Z Stanford University
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Stanford= # 3 World Rankings, #3 Prestige Rankings, #2 physical Sciences, #1 Engineering and Tech, #2 social sciences, # 4 life sciences, #1 arts and humanities, #7 clinical precliinical health.

Stanford and Berkeley highlighted with -
2016 results by times higher ed world rankings

World Rankings
1 California Institute of Technology
2 University of Oxford
3 Stanford University -
4 University of Cambridge
5 Massachusetts Institute of Technology
6 Harvard University
7 Princeton University
8 Imperial College London
9 ETH Zurich – Swiss Federal Institute of Technology Zurich
10 University of Chicago

Prestige Rankings
1 Harvard University
2 Massachusetts Institute of Technology
3. Stanford University -
4 University of Cambridge
5 University of Oxford

6 University of California, Berkeley -
7 Princeton University

8 Yale University
9 Columbia University

Physical Sciences

  1. California Institute of Technology
    2 Stanford University -
    3 Massachusetts Institute of Technology
    4 University of California, Berkeley -
    5 Princeton University
    =6 University of Cambridge
    =6 Harvard University
    8 University of Oxford
    9 Cornell University
    10 Carnegie Mellon University

Engineering and Tech

  1. Stanford University -
    2 California Institute of Technology
    3 Massachusetts Institute of Technology
    4 University of Cambridge
    5 University of California, Berkeley -
    6 University of Oxford
    7 Princeton University
    8 ETH Zurich – Swiss Federal Institute of Technology
    9 Imperial College London
    10 Carnegie Mellon University

Social Sciences

  1. Massachusetts Institute of Technology
    2 Stanford University -
    3 Princeton University
    4 University of Oxford
    5 Yale University
    6 Harvard University
    7 University of Chicago
    =8 University of Pennsylvania
    =8 University of Michigan
    10 University of California, Berkeley -

Life Sciences

  1. University of Oxford
    =2 University of Cambridge
    =2 Harvard University
    4 Stanford University -
    5 California Institute of Technology
    6 Imperial College London
    7 University of California, Berkeley -
    =8 Massachusetts Institute of Technology
    =8 Princeton University
    10 Yale University

Arts and Humanities
1 Stanford University -
2 Harvard University
3 Massachusetts Institute of Technology
4 University of Oxford
5 University College London
6 University of Cambridge
7 University of California, Berkeley -
8 University of Chicago
=9 Columbia University
=9 Princeton University
11 Yale University

clinical, pre-clinical and health

  1. University of Oxford
    2 Harvard University
    3 University of Cambridge
    4 University College London
    =5 University of California, Berkeley -
    =5 Imperial College London
    7 Stanford University -
    8 King’s College London
    9 Johns Hopkins University
    10 Columbia University
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No source given above, but I bet it’s an English ranking, because they tend to do some serious boosterism in favor of UK schools, and in the process, they tweak the methodology to produce some very silly results.

When it comes to international rankings, the Chinese seem to have the most objective methodology, and here are the results (Stanford comes out even better). Field specific rankings are also available.

http://www.shanghairanking.com/ARWU2015.html

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the source was given - 2016 results by times higher ed world rankings

link is here

https://www.timeshighereducation.com/world-university-rankings/2016/world-ranking#!/page/0/length/25/sort_by/rank_label/sort_order/asc/cols/rank_only