Median Age - Demographic definition

[Reeza]

I need to find an efficient way to calculate the median age from summary tables. The method is explained in the link below, and is pretty straightforward, but I'm looking for efficiency. I'm good at brute force, moreso than efficiency ... or at least brain dead at the moment.

The correct answer for the data below is 39.8. I was also wondering if there was a way to do this using a Stat proc, but an obvious method didn't occur to me.

Methodology in quesion and first answer.

http://stats.stackexchange.com/questions/139132/how-to-calculate-the-median-age-of-a-population

A quick description would be to find the two values that surround the median, find the corresponding ages that align with that number. Since it's on a scale the actual number is somewhere between the two numbers so weight it based on the difference betwen the two numbers. Since it's a continuous scale, add 1 to scale up. 

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 

[PGStats]

Depending on how much precision you need, brute force can be fairly efficient:

data test / view=test;
set pop;
dumPop = round(pop/1000);
do i = 0 to dumPop-1;
    dumAge = age + i / dumPop;
    output;
    end;
keep dumAge;
run;

proc univariate data=test;
var dumAge;
run;

[PGStats]

But more seriously, you can do the math:

data medPop / view=medPop;
set pop end=done;
if done then do;
    medPop  = cumulative / 2;
    output;
    end;
keep medPop;
run;

data medAge;
set medPop;
do until(cumulative > medPop);
    set pop;
    end;
medAge = (age*(cumulative-medPop) + (age+1)*(medPop - cumulative + pop)) / pop;
output;
stop;
keep medAge;
run;

[Ksharp]

Correct answer has already given the point.

data have;
input Geography  $    Age    _2010  Cumulative;
cards;
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 
;
run;
data temp;
 set have;
 do i=0.1 to 1 by 0.1;
  new_age=age+i;
  output;
 end;
run;
proc means data=temp median;
var new_age;
freq _2010;
run;

[Doc_Duke]

Reeza,

One note on the Median.  Statistically, it is not well defined when there are ties.  From the formal definition, any value between 39 and 40 is "correct".  The reference that you cite assumes that the distribution of the values between 39 and 40 is uniform.  That may be a reasonable assumption here, but less so elsewhere. 

[Reeza]

Thanks @Doc_Duke for the information.

I think the uniformity assumption for birthdates is probably fair. 

[Ksharp]

I don't think so . More like US has more September Baby.

Check Rick's blog :

http://blogs.sas.com/content/iml/2011/09/09/the-most-likely-birthday-in-the-us.html

http://blogs.sas.com/content/iml/2011/09/16/the-effect-of-holidays-on-us-births.html

http://blogs.sas.com/content/iml/2011/09/13/the-birthday-controversy-are-more-people-born-in-april-o...

[ballardw]

And some of the issues around quantiles are why Procs Means, Summary and Tabulate have the QMETHOD and QNTLDEF options to clarify the calculation methods and results a bit.

[Reeza]

@Ksharp - thanks for the reference and point!
Sometimes its easy to accept the way things are and not question them 😉