Cut-off of rank-size distribution in R

Question

How do you calculate the cut-off of a rank-size distribution, i.e. the x which splits the distribution so that the top x% categories contain 1-x% of observations? For example, if the cut-off is 20%, the top 20% categories contain 80% of the observations.

In the sample below, the categories are cbsa_code, frequency is Freq, and category rank by frequency is rank (some ranks can be equal and decimal because they were calculated with ties.method = "average"):

structure(list(cbsa_code = c("35620", "41860", "31080", "41940", 
"14460", "47900", "16980", "42660", "33100", "19100", "37980", 
"12420", "12060", "41740", "26420", "38900", "38060", "19740", 
"33460", "29820", "19820", "36740", "45300", "14500", "14860", 
"12580", "17140", "40900", "38300", "41620", "28140", "34980", 
"41180", "20500", "39580", "16740", "17460", "40140", "18140", 
"39340", "41700", "27260", "33340", "26900", "31540", "10580", 
"11460", "32820", "35380", "39300", "35300", "42100", "12540", 
"36420", "40060", "45940", "46140", "25540", "31140", "40380", 
"42200", "46520", "15380", "24340", "35840", "47260", "46060", 
"17820", "49340", "31700", "36540", "37100", "10420", "15980", 
"16700", "19780", "30780", "42220", "10740", "22660", "23540", 
"28940", "39900", "13820", "16580", "16860", "14260", "16940", 
"24860", "29620", "34940", "45780", "16820", "43620", "11700", 
"23420", "25420", "34820", "41500", "12940", "22220", "26620", 
"33860", "41540", "44060", "48900", "11260", "14740", "20100", 
"21660", "27060", "10900", "14580", "15540", "17900", "21340", 
"24660", "29540", "37340", "38860", "44140", "45220", "19380", 
"19660", "26980", "29940", "30460", "36260", "42020", "42340", 
"46340", "11540", "12020", "13380", "17020", "17200", "18700", 
"24580", "24780", "25500", "25900", "26820", "27540", "27940", 
"32780", "34900", "35980", "37860", "42540", "43900", "44300", 
"45060", "46700", "48620", "11100", "12220", "12300", "13460", 
"18180", "19340", "20780", "22140", "27140", "27220", "28420", 
"28700", "28740", "30700", "33260", "33780", "34580", "36860", 
"39460", "41420", "44700", "47380", "48660", "49180", "12700", 
"13100", "13780", "13980", "17660", "18580", "20020", "20260", 
"21140", "21500", "23060", "23860", "24060", "25180", "25200", 
"25860", "26380", "26500", "26740", "27980", "29340", "29460", 
"31060", "36100", "36780", "37460", "38940", "40420", "42140", 
"43780", "44180", "44500", "45860", "46660", "48700", "49620", 
"10780", "11900", "12260", "13220", "13900", "14100", "14540", 
"15940", "16180", "16300", "17340", "17780", "17860", "17980", 
"18660", "20140", "20220", "20420", "20700", "21260", "22020", 
"22500", "22520", "22900", "23180", "23580", "23820", "24300", 
"24420", "24540", "24900", "24940", "25060", "25940", "28020", 
"28060", "28580", "29300", "29660", "30020", "30140", "30420", 
"30860", "30980", "31180", "31340", "31900", "32140", "32580", 
"33140", "33220", "33540", "36140", "36500", "36700", "36980", 
"37900", "38540", "39140", "39420", "39540", "39660", "39740", 
"40660", "41140", "42700", "43060", "43100", "43380", "43580", 
"44100", "44260", "44920", "45460", "45900", "46020", "46540", 
"47460", "49300", "49420", "10100", "10140", "10460", "10620", 
"10700", "11500", "11740", "11820", "12620", "12740", "12900", 
"12980", "13140", "13180", "13740", "14010", "14140", "14380", 
"14700", "15220", "15260", "15500", "15620", "15740", "16100", 
"16220", "16540", "17220", "17300", "17580", "17740", "18020", 
"18220", "18260", "19060", "19140", "19180", "19300", "19580", 
"20380", "20660", "20740", "20820", "21220", "21540", "21580", 
"21700", "21780", "21840", "21900", "22180", "22420", "23300", 
"23460", "23660", "23900", "24020", "24220", "24260", "24620", 
"24640", "25620", "26780", "26860", "27420", "27460", "27900", 
"28100", "28380", "28660", "28780", "29060", "29100", "29200", 
"29380", "29700", "29980", "30300", "30340", "31020", "31300", 
"31380", "31940", "32220", "32260", "32380", "33700", "33740", 
"34100", "34340", "34460", "34620", "34700", "34780", "34860", 
"35460", "35740", "36460", "36940", "37120", "38240", "38380", 
"38820", "39020", "39060", "39820", "39940", "39980", "40220", 
"40260", "40340", "40860", "40980", "41060", "41820", "42820", 
"42860", "42940", "43140", "43260", "43340", "43420", "43740", 
"44220", "44420", "44460", "44660", "45180", "45340", "45380", 
"45620", "45820", "46220", "46300", "46740", "46980", "47180", 
"47220", "47240", "47300", "47620", "47700", "47940", "48020", 
"48060", "48140", "48260", "48580", "48780", "49080", "49220", 
"49660", "10180", "10220", "10300", "10500", "10540", "10660", 
"10820", "10860", "10940", "10980", "11020", "11060", "11140", 
"11180", "11220", "11380", "11420", "11580", "11620", "11660", 
"11680", "11780", "11860", "11940", "11980", "12100", "12140", 
"12180", "12380", "12460", "12660", "12680", "12780", "12820", 
"12860", "13020", "13060", "13260", "13300", "13340", "13420", 
"13500", "13540", "13620", "13660", "13700", "13720", "13940", 
"14020", "14180", "14220", "14340", "14420", "14620", "14660", 
"14720", "14780", "14820", "15020", "15060", "15100", "15180", 
"15340", "15420", "15460", "15580", "15660", "15680", "15700", 
"15780", "15820", "15860", "15900", "16020", "16060", "16260", 
"16340", "16380", "16460", "16500", "16620", "16660", "17060", 
"17260", "17380", "17420", "17500", "17540", "17700", "18060", 
"18100", "18300", "18380", "18420", "18460", "18500", "18620", 
"18740", "18780", "18820", "18860", "18880", "18900", "18980", 
"19000", "19220", "19260", "19420", "19460", "19500", "19540", 
"19620", "19700", "19760", "19860", "19940", "19980", "20060", 
"20180", "20300", "20340", "20460", "20540", "20580", "20900", 
"20940", "20980", "21020", "21060", "21120", "21180", "21300", 
"21380", "21420", "21460", "21740", "21820", "21980", "22060", 
"22100", "22260", "22280", "22300", "22340", "22380", "22540", 
"22580", "22620", "22700", "22780", "22800", "22820", "22860", 
"23140", "23240", "23340", "23380", "23500", "23620", "23700", 
"23780", "23940", "23980", "24100", "24140", "24380", "24460", 
"24500", "24700", "24740", "24820", "24980", "25100", "25220", 
"25260", "25300", "25460", "25580", "25700", "25720", "25740", 
"25760", "25780", "25820", "25840", "25880", "25980", "26020", 
"26090", "26140", "26220", "26300", "26340", "26460", "26540", 
"26580", "26660", "26700", "26940", "26960", "27020", "27100", 
"27160", "27180", "27300", "27340", "27380", "27500", "27600", 
"27620", "27700", "27740", "27780", "27860", "27920", "28180", 
"28260", "28300", "28340", "28500", "28540", "28620", "28820", 
"28860", "28900", "29020", "29180", "29260", "29420", "29500", 
"29740", "29780", "29860", "29900", "30060", "30220", "30260", 
"30280", "30380", "30580", "30620", "30660", "30820", "30880", 
"30900", "30940", "31220", "31260", "31420", "31460", "31500", 
"31580", "31620", "31660", "31680", "31740", "31820", "31860", 
"31930", "31980", "32000", "32020", "32100", "32180", "32280", 
"32300", "32340", "32460", "32500", "32540", "32620", "32660", 
"32700", "32740", "32860", "32900", "32940", "32980", "33020", 
"33060", "33180", "33300", "33420", "33500", "33580", "33620", 
"33660", "33940", "33980", "34020", "34060", "34140", "34180", 
"34220", "34260", "34300", "34380", "34420", "34500", "34540", 
"34660", "34740", "35020", "35060", "35100", "35140", "35220", 
"35260", "35420", "35440", "35500", "35580", "35660", "35700", 
"35820", "35860", "35900", "35940", "36020", "36220", "36300", 
"36340", "36380", "36580", "36620", "36660", "36820", "36830", 
"36840", "36900", "37020", "37060", "37080", "37140", "37220", 
"37260", "37300", "37420", "37500", "37540", "37580", "37620", 
"37660", "37740", "37780", "37940", "38100", "38180", "38220", 
"38260", "38340", "38420", "38460", "38500", "38580", "38620", 
"38700", "38740", "38780", "38840", "38920", "39220", "39260", 
"39380", "39500", "39700", "39780", "39860", "40080", "40100", 
"40180", "40300", "40460", "40540", "40580", "40620", "40700", 
"40740", "40780", "40820", "40940", "41100", "41220", "41400", 
"41460", "41660", "41760", "41780", "42300", "42380", "42420", 
"42460", "42620", "42680", "42740", "42780", "42900", "42980", 
"43020", "43180", "43220", "43300", "43320", "43460", "43500", 
"43660", "43700", "43760", "43940", "43980", "44020", "44340", 
"44540", "44580", "44620", "44740", "44780", "44860", "44900", 
"44940", "44980", "45000", "45020", "45140", "45500", "45520", 
"45540", "45580", "45660", "45700", "45740", "45980", "46100", 
"46180", "46380", "46460", "46500", "46620", "46780", "46820", 
"46860", "46900", "47020", "47080", "47340", "47420", "47540", 
"47580", "47660", "47780", "47820", "47920", "47980", "48100", 
"48180", "48220", "48300", "48460", "48540", "48820", "48940", 
"48980", "49020", "49100", "49260", "49380", "49460", "49700", 
"49740", "49780", "49820"), Freq = c(1812L, 1558L, 1052L, 622L, 
514L, 455L, 395L, 393L, 311L, 266L, 261L, 259L, 249L, 213L, 204L, 
156L, 151L, 141L, 95L, 92L, 91L, 91L, 84L, 76L, 71L, 70L, 68L, 
66L, 64L, 64L, 61L, 59L, 52L, 50L, 46L, 45L, 44L, 44L, 40L, 38L, 
38L, 36L, 35L, 34L, 32L, 31L, 30L, 30L, 29L, 28L, 27L, 26L, 25L, 
25L, 25L, 24L, 23L, 21L, 21L, 21L, 21L, 21L, 20L, 20L, 20L, 20L, 
19L, 17L, 17L, 16L, 16L, 16L, 15L, 15L, 15L, 15L, 15L, 15L, 14L, 
14L, 14L, 14L, 14L, 13L, 13L, 13L, 12L, 12L, 12L, 12L, 12L, 12L, 
11L, 11L, 10L, 10L, 10L, 10L, 10L, 9L, 9L, 9L, 9L, 9L, 9L, 9L, 
8L, 8L, 8L, 8L, 8L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 
6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 
5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA), rank = c(1, 2, 3, 
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 
21.5, 21.5, 23, 24, 25, 26, 27, 28, 29.5, 29.5, 31, 32, 33, 34, 
35, 36, 37.5, 37.5, 39, 40.5, 40.5, 42, 43, 44, 45, 46, 47.5, 
47.5, 49, 50, 51, 52, 54, 54, 54, 56, 57, 60, 60, 60, 60, 60, 
64.5, 64.5, 64.5, 64.5, 67, 68.5, 68.5, 71, 71, 71, 75.5, 75.5, 
75.5, 75.5, 75.5, 75.5, 81, 81, 81, 81, 81, 85, 85, 85, 89.5, 
89.5, 89.5, 89.5, 89.5, 89.5, 93.5, 93.5, 97, 97, 97, 97, 97, 
103, 103, 103, 103, 103, 103, 103, 109, 109, 109, 109, 109, 117, 
117, 117, 117, 117, 117, 117, 117, 117, 117, 117, 127, 127, 127, 
127, 127, 127, 127, 127, 127, 143, 143, 143, 143, 143, 143, 143, 
143, 143, 143, 143, 143, 143, 143, 143, 143, 143, 143, 143, 143, 
143, 143, 143, 166.5, 166.5, 166.5, 166.5, 166.5, 166.5, 166.5, 
166.5, 166.5, 166.5, 166.5, 166.5, 166.5, 166.5, 166.5, 166.5, 
166.5, 166.5, 166.5, 166.5, 166.5, 166.5, 166.5, 166.5, 196.5, 
196.5, 196.5, 196.5, 196.5, 196.5, 196.5, 196.5, 196.5, 196.5, 
196.5, 196.5, 196.5, 196.5, 196.5, 196.5, 196.5, 196.5, 196.5, 
196.5, 196.5, 196.5, 196.5, 196.5, 196.5, 196.5, 196.5, 196.5, 
196.5, 196.5, 196.5, 196.5, 196.5, 196.5, 196.5, 196.5, 254.5, 
254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 
254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 
254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 
254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 
254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 
254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 
254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 
254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 
254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 254.5, 370.5, 370.5, 
370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 
370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 
370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 
370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 
370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 
370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 
370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 
370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 
370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 
370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 
370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 
370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 
370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 
370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 
370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 
370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 
370.5, 370.5, 370.5, 370.5, 370.5, 370.5, 447, 448, 449, 450, 
451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 
464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 
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EDIT: The provided ranks should be disregarded, as ties.methods = "average" is the wrong method to use in this case.

Answer 1

Based on the very nice solution by @JorysMeys, here is a leaner solution, which only requires the frequency distribution as an input and which returns the matching share of observations in addition to the cutoff (because the sum will seldom be 100%).

get_cutoff <- function(freq){
  # remove NA values from distribution (make sure NA doesn't mean zero before running the function)
  freq <- freq[!is.na(freq)]
  # order distribution by decreasing frequency
  freq <- freq[order(-freq)]
  # subtract 100% from cumulative frequency share plus rank share
  pdiff <- cumsum(freq/sum(freq)) + seq(1,length(freq))/length(freq) - 1
  # position (=rank) of smallest absolute difference (generally not 0 since ranks are discrete)
  pos <- which.min(abs(pdiff))
  # return cutoff of rank share and matching cumulative frequency share
  return(c(pos/length(freq), sum(freq[1:pos])/sum(freq)))
}

The reason why there is no need to calculate rank beforehand is that the cutoff should not depend on the ties.method argument of rank(). If you do calculate rank beforehand and apply the other solution, then you should use ties.method = "random". Other methods will give uninterpretable results. This solution calculates a simple rank with seq(1,length(freq)), which is strictly equivalent to ties.method = "random".

Answer 2

I reckon that the top x% categories are defined by the rank, so that the top x% categories is the categories with a rank lower than max(rank)*x/100. Then it's actually quite easy to do.

• You calculate cumulative proportions from the counts and convert the ranks to a percentage of the total categories as well,
• You add these to together and check which one is the closest to 1

In R code this could be done like this :

get_cutoff <- function(rank, freq){

  counts <- tapply(freq, rank, sum, na.rm = TRUE)
  ranks <- as.numeric(names(counts))

  pdiff <- cumsum(counts / sum(counts)) +  ranks/max(ranks) - 1
  pos <- which.min(abs(pdiff))
  return(ranks[pos]/max(ranks))
}

Storing your structure in a data frame called mydf gives the following:

get_cutoff(mydf$rank, mydf$Freq) [1] 0.08833152

To check for yourself this is correct, you can do:

> counts <- with(mydf, tapply(Freq, rank, sum, na.rm = TRUE))
> ranks <- as.numeric(names(counts))
> get_cutoff(mydf$rank, mydf$Freq) * max(ranks)
[1] 81
> which(ranks == 81)
[1] 57
> sum(counts[1:57])/sum(counts)
[1] 0.915586
> sum(counts[1:57])/sum(counts) + 81/max(ranks)
[1] 1.003918

Due to the discrete nature of ranks, only in specific cases this will be a 100% perfect solution. The algorithm above finds the fraction tied to the rank that gives you the result closest to the perfect solution.