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I want to calculate the median age of a population. Consider the following example data (downloaded from Statistics Canada's CANSIM database, Table 051-0001):

Table 051-0001 Estimates of population, by age group and sex for July 1, Canada, provinces and territories, annual (persons unless otherwise noted)                     
Survey or program details:                      
Estimates of Population by Age and Sex for Canada, Provinces and Territories - 3604                     
Geography      Age      2010      2011      2012      2013      2014
Canada           0    377861    376321    376593    381689    384566
Canada           1    380538    379990    379191    379400    384085
Canada           2    380634    383179    383355    382164    382320
Canada           3    371478    383741    386305    386406    384721
Canada           4    362882    375833    387091    389512    389258
Canada           5    356590    366757    379364    390614    392709
Canada           6    360976    361038    370316    382704    393919
Canada           7    355169    363440    364337    373378    385791
Canada           8    357366    358621    366509    367203    375972
Canada           9    361252    360577    361705    369300    369853
Canada          10    374141    365198    363513    364429    371892
Canada          11    376153    376458    368165    366089    366754
Canada          12    386812    379838    379378    370725    368303
Canada          13    400765    391245    382591    381913    373366
Canada          14    421855    405425    394646    385845    385158
Canada          15    434175    426802    409598    398813    390141
Canada          16    438635    440145    432615    415325    404911
Canada          17    446950    446524    446819    438832    422155
Canada          18    457629    455872    456107    456347    449780
Canada          19    472134    469609    469648    469727    471713
Canada          20    479938    479650    480075    478935    479825
Canada          21    465581    484077    487441    487222    487303
Canada          22    451825    470052    492089    494807    494773
Canada          23    457019    458775    478405    500371    502743
Canada          24    468109    461800    466140    485206    507465
Canada          25    472877    471522    468571    472175    491796
Canada          26    471323    475052    478176    474765    478485
Canada          27    470566    474287    481927    484782    481007
Canada          28    468454    475287    481774    488559    491073
Canada          29    474732    473693    482673    488843    495415
Canada          30    468396    479377    481415    489943    495625
Canada          31    457343    472955    486914    488681    496949
Canada          32    450990    462922    480345    493717    495002
Canada          33    452786    455951    469502    486633    499657
Canada          34    454337    456750    462188    475678    492631
Canada          35    456131    457914    462422    467980    481122
Canada          36    444761    458413    463276    467283    472694
Canada          37    447653    448563    463120    467676    471389
Canada          38    454974    450150    452853    467353    471303
Canada          39    479412    458047    454151    456406    470625
Canada          40    476941    480129    461608    457435    459218
Canada          41    473754    478010    483497    464567    459997
Canada          42    470960    475541    480964    486012    466836
Canada          43    476994    473014    478112    483096    487941
Canada          44    505091    479224    475161    480000    484701
Canada          45    543975    506180    480754    476662    481251
Canada          46    560528    541424    507546    481692    477527
Canada          47    565680    557834    542469    508247    482163
Canada          48    550871    562501    558604    542971    508357
Canada          49    559845    551970    563107    558829    542849
Canada          50    549137    558958    551916    563085    558602
Canada          51    535778    548847    558693    551419    562609
Canada          52    530407    536639    548423    557898    550499
Canada          53    518528    529913    535794    547485    556782
Canada          54    501147    516903    528855    534558    546135
Canada          55    496890    501158    515688    527602    532979
Canada          56    476058    493948    499663    514369    525968
Canada          57    452959    473623    492514    498260    512603
Canada          58    434140    451080    471946    491181    496564
Canada          59    426554    433281    449217    470450    489202
Canada          60    416427    424456    431432    447732    468404
Canada          61    406686    414501    422373    429874    445569
Canada          62    406030    405624    412256    420656    427664
Canada          63    409513    404282    403263    410629    418399
Canada          64    343036    401580    401573    401279    408094
Canada          65    319975    344130    398603    399294    398380
Canada          66    310757    317981    341249    395875    396003
Canada          67    297341    306903    314982    338733    392348
Canada          68    274022    292908    303625    312358    335276
Canada          69    262547    271018    289468    300739    308870
Canada          70    243273    257990    267625    286330    297083
Canada          71    233870    240638    254195    264252    282308
Canada          72    222439    230156    236842    250713    260198
Canada          73    209795    218638    226001    233119    246393
Canada          74    205522    206400    214332    222005    228699
Canada          75    196095    200626    201812    209949    217227
Canada          76    185698    190737    195686    197222    204987
Canada          77    183420    180672    185358    190437    191834
Canada          78    177259    176879    174946    179710    184600
Canada          79    170901    170424    170616    168901    173534
Canada          80    159809    163436    163608    163924    162272
Canada          81    146075    152274    156213    156439    156796
Canada          82    137515    138753    144686    148468    148791
Canada          83    125378    129386    131081    136719    140405
Canada          84    116263    117291    121359    122961    128353
Canada          85    105468    107226    109197    112966    114566
Canada          86     94205     96193     99099    100911    104508
Canada          87     82396     85051     88042     90690     92440
Canada          88     73145     73870     77032     79728     82232
Canada          89     62417     64399     66201     69033     71573
Canada         90+    200558    216331    233539    249385    265581
Canada   MedianAge      39.8        40      40.1      40.2      40.4
Source:                     
Statistics Canada. Table 051-0001 - Estimates of population, by age group and sex for July 1, Canada, provinces and territories, annual (persons unless otherwise noted)                        
(accessed: February 24, 2015)                       

In order to establish the median for (say) 2010, we calculate the cumulative totals:

Geography      Age      2010  Cumulative
Canada           0    377861      377861
Canada           1    380538      758399
Canada           2    380634     1139033
Canada           3    371478     1510511
Canada           4    362882     1873393
Canada           5    356590     2229983
Canada           6    360976     2590959
Canada           7    355169     2946128
Canada           8    357366     3303494
Canada           9    361252     3664746
Canada          10    374141     4038887
Canada          11    376153     4415040
Canada          12    386812     4801852
Canada          13    400765     5202617
Canada          14    421855     5624472
Canada          15    434175     6058647
Canada          16    438635     6497282
Canada          17    446950     6944232
Canada          18    457629     7401861
Canada          19    472134     7873995
Canada          20    479938     8353933
Canada          21    465581     8819514
Canada          22    451825     9271339
Canada          23    457019     9728358
Canada          24    468109    10196467
Canada          25    472877    10669344
Canada          26    471323    11140667
Canada          27    470566    11611233
Canada          28    468454    12079687
Canada          29    474732    12554419
Canada          30    468396    13022815
Canada          31    457343    13480158
Canada          32    450990    13931148
Canada          33    452786    14383934
Canada          34    454337    14838271
Canada          35    456131    15294402
Canada          36    444761    15739163
Canada          37    447653    16186816
Canada          38    454974    16641790
Canada          39    479412    17121202
Canada          40    476941    17598143
Canada          41    473754    18071897
Canada          42    470960    18542857
Canada          43    476994    19019851
Canada          44    505091    19524942
Canada          45    543975    20068917
Canada          46    560528    20629445
Canada          47    565680    21195125
Canada          48    550871    21745996
Canada          49    559845    22305841
Canada          50    549137    22854978
Canada          51    535778    23390756
Canada          52    530407    23921163
Canada          53    518528    24439691
Canada          54    501147    24940838
Canada          55    496890    25437728
Canada          56    476058    25913786
Canada          57    452959    26366745
Canada          58    434140    26800885
Canada          59    426554    27227439
Canada          60    416427    27643866
Canada          61    406686    28050552
Canada          62    406030    28456582
Canada          63    409513    28866095
Canada          64    343036    29209131
Canada          65    319975    29529106
Canada          66    310757    29839863
Canada          67    297341    30137204
Canada          68    274022    30411226
Canada          69    262547    30673773
Canada          70    243273    30917046
Canada          71    233870    31150916
Canada          72    222439    31373355
Canada          73    209795    31583150
Canada          74    205522    31788672
Canada          75    196095    31984767
Canada          76    185698    32170465
Canada          77    183420    32353885
Canada          78    177259    32531144
Canada          79    170901    32702045
Canada          80    159809    32861854
Canada          81    146075    33007929
Canada          82    137515    33145444
Canada          83    125378    33270822
Canada          84    116263    33387085
Canada          85    105468    33492553
Canada          86     94205    33586758
Canada          87     82396    33669154
Canada          88     73145    33742299
Canada          89     62417    33804716
Canada         90+    200558    34005274   <--- Total population

The total population contains $34\,005\,274$ individuals, with the "middle (or median) person" being person number $34\,005\,274 \div 2 = 17\,002\,637 = M$. A lookup shows that this person is said to be between 38 and 39 years old since $$ \text{Cumulative}(38) = \overbrace{16\,641\,790}^{C(38)} < \overbrace{17\,002\,637}^{M} < \overbrace{17\,121\,202}^{C(39)} = \text{Cumulative}(39). $$ To estimate the proportion between 38 and 39, we assume that the population is evenly distributed between the age cohorts based on the cohort size. As such, this "median person" has age $$ 38 + \frac{\overbrace{17\,002\,637}^M - \overbrace{16\,641\,790}^{C(38)}} {\underbrace{17\,121\,202}_{C(39)} - \underbrace{16\,641\,790}_{C(38)}} = 38 + \frac{360\,847}{479\,412} \approx 38.8 = \bar{M}. $$ In the above calculation, the "median cohort portion" is calculated to estimate the age above the cohort division.

To me this approach seems intuitive and sound. However, Statistics Canada reports the median age as $$ 39.8 = \bar{M} + 1. $$ Why is that?

I assume this has to do with the fact that the population cohort aged 20 (say) is in their 21st year of life? Even if this is the case, a median age of 39.8 points to someone that is older than 39*, yet the "median person" is somewhere between 38 and 39 years old; i.e., younger than 39. Unless, of course, that proposition is incorrect.

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  • $\begingroup$ Your reasoning seems spot on to me. I did not check your numbers, and it is always possible that there's some sort of arithmetic error. I'm also a bit intrigued by the fact that they list 0-year olds, which isn't wrong, but which seems like a good opportunity for an off-by-1 error $\endgroup$ – davidlowryduda Feb 25 '15 at 0:00
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The trick here is understanding the grouping categories of age here. The age group "39" is people aged 39 to less than 40 (i.e. 39 - 39.99999999). So if you look at the cumulative total for the 38 year old group, you will see the value is 16,641,790 as you've correctly noted. The cumulative total for the 39 year old group, which is really the "39 to < 40 year old group) is 17,121,202. There are a total of 479,412 individuals who are older than 39 and less than 40. We know from your table that there are 16,641,790 individuals who are less than 39 years old. So starting at person 16,641,791 and going through person 17,121,202 we find our 39 to less than 40 year olds. There are 479,412 people in this group. By your faulty assumption of equal distribution of ages, if we take $479,412\over10$, we can construct the following table, roughly, by adding this amount to the end of the cumulative figure of the 38th age group in an iterative fashion:

Age     Begin       End 
39.0    16,641,791  16,689,731
39.1    16,689,732  16,737,672
39.2    16,737,673  16,785,614
39.3    16,785,615  16,833,555
39.4    16,833,556  16,881,496
39.5    16,881,497  16,929,437
39.6    16,929,438  16,977,378
39.7    16,977,379  17,025,320
39.8    17,025,321  17,073,261
39.9    17,073,262  17,121,202

According to these calculations, the median would fall somewhere around 39.7. Of course, birthday months and ages are not uniformly distributed, so this accounts for the 0.2 discrepancy we see between this "on the back of a napkin" calculation and the official statistic.

I hope this helps.

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  • 1
    $\begingroup$ So, fundamentally, it boils down to my incorrect interpretation of the age associated with the given cohorts. The correct choice would have been 39 to start with, rather than my 38. That is, "A lookup shows that this person is said to be at least 39 years old..." $\endgroup$ – Werner Feb 25 '15 at 8:19
  • $\begingroup$ Yes, that's essentially it -- it's all about category definitions here. $\endgroup$ – StatsStudent Feb 25 '15 at 8:36

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