Quartiles, Quintiles, Deciles, and Percentiles

[Zuhumnan Dapel]

Dear All,
Want to confirm the following
1. Quintile - Divides the distribution into fifths,

sumdist x [aw=wght], n(5)

; 2. Decile-Divides the distribution into tenths,

sumdist x [aw=wght], n(10);

3. Percentile- Divides the distribution into hundredths

sumdist x [aw=wght], n(100);

. 4. Quartiles- Divides the distribution into quarters,

sumdist x [aw=wght], n(20)

Decile and Quantile are they thesame? Thanks.

[Zuhumnan Dapel]

Any help please?

[William Lisowski]

Points 1, 2 and 3 are correct as written. The description in point 4 is correct but the code should be

sumdist x [aw=wght], n(4)

"Quantile" encompasses all the others, and refers to the division of a distribution into any number of equal groups. The others terms are special cases of quantiles. For example, there is no commonly-used term for dividing a distribution into 42 equal groups, so if we do so using

sumdist x [aw=wght], n(42)

we then refer to the 42 quantiles where we might otherwise refer to the 4 quartiles or 10 deciles or 20 ventiles.

From the dictionary on my Mac:

quantile |ˈkwänˌtīl| noun Statistics
each of any set of values of a variate that divide a frequency distribution into equal groups, each containing the same fraction of the total population.
• any of the groups so produced, e.g., a quartile or percentile.

[Zuhumnan Dapel]

This is clear to me. Thank you very much

[Zuhumnan Dapel]

Dear All, a couple of issues for help

. sumdist income [aw=popwt2] if year==1980, n(10) qgp(gp)
Distributional summary statistics, 10 quantile groups

---------------------------------------------------------------------------
Quantile  |
group     |    Quantile  % of median     Share, %      L(p), %        GL(p)
----------+----------------------------------------------------------------
        1 |     3350.59        38.03         1.68         1.68       242.19
        2 |     4666.15        52.97         2.81         4.49       648.68
        3 |     5992.97        68.03         3.73         8.22      1187.47
        4 |     7096.29        80.55         4.52        12.75      1840.80
        5 |     8809.27       100.00         5.47        18.21      2630.46
        6 |    10689.07       121.34         6.70        24.91      3597.80
        7 |    13486.91       153.10         8.25        33.17      4790.01
        8 |    18280.20       207.51        10.68        43.85      6332.56
        9 |    29028.91       329.53        15.71        59.55      8601.15
       10 |                                 40.45       100.00     14442.95
---------------------------------------------------------------------------
Share = quantile group share of total pcepm; 
L(p)=cumulative group share; GL(p)=L(p)*mean(pcepm)

Is 3350.59 the average income of the first decile? If yes, what is the average income of the 10th decile as it is blank from the table of results above?
Thanks,
Dapel

[Joao Santos Silva]

Dear Dapel,

I am not familiar with this command, but since you are not getting any help...

My interpretation is that 3350.59 is the first decile, that is, a value such that 10% of the income observations are smaller than it. The 9th decile is 29028.91, so 10% of the observations are above it. There is no value in the last row because you only need 9 values to split the data in 10 parts (like you just need the median to split the sample in two). Quantiles do not involve averages at all.

All the best,

Joao

[Nick Cox]

Joao is right. In fact, this is all documented directly in the help for sumdist (SSC, was STB, as you are asked to explain):

sumdist estimates distributional summary statistics commonly used by income distribution
analysts, complementing those available via pctile, xtile, and summarize, detail.
Calculations are based on all non-missing values of varname. Use if if you wish to
exclude values less than or equal to zero.

For variable x and distribution function F(x), the statistics are:

(1) quantiles k = 1,2,...,K-1, for K = # quantile groups;

(2) the quantiles expressed as a percentage of median(x);

(3) the quantile group shares of x in total x (expressed as a %);

(4) the cumulative quantile group shares of total x (with cumulation in ascending order of
x), i.e. the Lorenz ordinates L(p_k) at each p_k = F(x_k) for quantile points x_k
(expressed as a %);

(5) the generalised Lorenz ordinates at each p_k = F(x_k), i.e. GL(p_k) = mean(x)*L(p_k).

[Zuhumnan Dapel]

Thank you all very much.

Is this Ok for the averages?

. mean income [aw=popwt2] if year==1980, over(quantgp80)

Mean estimation                     Number of obs    =   10280

            1: quantgp80 = 1
            2: quantgp80 = 2
            3: quantgp80 = 3
            4: quantgp80 = 4
            5: quantgp80 = 5
            6: quantgp80 = 6
            7: quantgp80 = 7
            8: quantgp80 = 8
            9: quantgp80 = 9
           10: quantgp80 = 10

--------------------------------------------------------------
        Over |       Mean   Std. Err.     [95% Conf. Interval]
-------------+------------------------------------------------
pce          |
           1 |   11320.03   192.9415      10941.83    11698.24
           2 |   19012.88   80.62274      18854.84    19170.92
           3 |   25202.12   69.64595       25065.6    25338.64
           4 |   30402.86   64.84568      30275.75    30529.97
           5 |   37114.57   85.54648      36946.88    37282.25
           6 |   44980.68   98.77955      44787.05     45174.3
           7 |   55895.25   128.6133      55643.15    56147.36
           8 |   72392.55   197.6173      72005.18    72779.92
           9 |   105989.8   341.6985      105320.1    106659.6
          10 |   273538.6   5060.291      263619.4    283457.8
--------------------------------------------------------------

[Nick Cox]

One very general point and one specific one:

1. We don't have your dataset, so it is optimistic to expect that we can check anything specific to that.

2. The upper limit of the lowest decile (bin) was about 3350. That's only consistent with a bin mean below it. It's inconsistent with a mean of 11320, so far as I can see. I don't see that using analytic weights could be responsible. Same applies to other bins.

To make progress, invent a very, very simple dataset for yourself with say 20 observations where you can check independently what is being done. Or choose a publicly accessible dataset that others can work with to check anything.

[Zuhumnan Dapel]

Thank you very much. With respect to point 1 in #9, here is the difference between the results in posts #5 and #8:

1. I created deciles using

    sumdist income [aw=popwt2] if year==1980, n(10) qgp(quantgp80)

   . The variable representing the deciles is labeled "quantgp80"
2. In #8, I obtained the average income over quantgp80, i.e the deciles. Is this Okay if one is comparing the average incomes of the poorest 10% 11320.03 with the richest's 10%, 273538.6? using

    	
    mean income [aw=popwt2] if year==1980, over(quantgp80)

Dapel

[Zuhumnan Dapel]

Dear All,
any further help on this?

Thanks,
Dapel

[Zuhumnan Dapel]

Back to the issue

[Zuhumnan Dapel]

Hello

[David Benson]

I'm not sure what your exact question is, but I made up some toy data to test things out:

The only unfortunate thing is that I made it with n==100, so each bin will have exactly 10 obs.

* Example generated by -dataex-. To install: ssc install dataex
clear
input byte id int year long income
  8 2010  10146
 26 2010  11174
 67 2010  12490
 95 2010  14750
 17 2010  15392
 96 2010  15481
 68 2010  16039
 39 2010  16896
  3 2010  17656
 31 2010  17974
 78 2010  18385
 92 2010  18417
 64 2010  21173
 21 2010  22585
 93 2010  25789
 41 2010  29347
 52 2010  32706
 71 2010  33138
  9 2010  33310
 55 2010  33631
 35 2010  34246
 74 2010  34562
 47 2010  36497
 12 2010  39458
 42 2010  44494
 28 2010  44914
 50 2010  45368
 69 2010  46077
 24 2010  46679
 32 2010  47606
 83 2010  48673
 88 2010  50659
  5 2010  53302
 86 2010  53347
 73 2010  54214
 54 2010  54732
 85 2010  55895
 38 2010  56922
 29 2010  57893
 72 2010  60131
 16 2010  63376
 65 2010  64058
 90 2010  65238
 46 2010  66863
 76 2010  67018
100 2010  68745
 15 2010  69031
 63 2010  69923
  6 2010  71757
 53 2010  72277
 49 2010  74093
 79 2010  74296
 87 2010  75773
 40 2010  76920
 22 2010  77845
 33 2010  78005
 97 2010  80743
 27 2010  80949
 19 2010  83517
  4 2010  83694
 77 2010  83710
 82 2010  84529
 11 2010  84599
 44 2010  87208
 18 2010  87913
 48 2010  88552
 91 2010  88939
  7 2010  89292
 56 2010  89751
 81 2010  89834
 80 2010  90717
 70 2010  92030
 57 2010  94179
 14 2010  95399
 34 2010  95427
 99 2010  99290
 25 2010  99516
  1 2010 101805
 13 2010 102272
 66 2010 103682
 37 2010 104408
 98 2010 104637
 60 2010 105810
 20 2010 108473
 36 2010 110640
 62 2010 111550
 61 2010 111954
 30 2010 112570
 58 2010 113010
 23 2010 113743
 43 2010 113936
 94 2010 114439
 89 2010 115663
 84 2010 116281
  2 2010 116677
 45 2010 116889
 59 2010 121481
 10 2010 121729
 75 2010 122364
 51 2010 122740
end

* Summary statistics
format income %10.0gc
summ income, detail format

                           income
-------------------------------------------------------------
      Percentiles      Smallest
 1%       10,660         10,146
 5%     15,436.5         11,174
10%     18,179.5         12,490       Obs                 100
25%       44,704         14,750       Sum of Wgt.         100

50%       73,185                      Mean           69,739.1
                        Largest       Std. Dev.      33,496.6
75%     97,358.5        121,481
90%      113,840        121,729       Variance       1.12e+09
95%      116,783        122,364       Skewness      -.1833772
99%      122,552        122,740       Kurtosis       1.858154


sumdist income if year==2010, n(10) qgp(gp)  // if year==2010 isn't going to matter here, since they are all year==2010
* So this gives the upper bound of each decile (i.e. top earner in lowest decile earned $17,974 in my madeup data)
*  $17,974 / $73,185 = 24.56% of the median
Distributional summary statistics, 10 quantile groups

---------------------------------------------------------------------------
Quantile  |
group     |    Quantile  % of median     Share, %      L(p), %        GL(p)
----------+----------------------------------------------------------------
        1 |   17974.000       24.560        2.122        2.122     1479.980
        2 |   33631.000       45.953        3.850        5.972     4164.790
        3 |   47606.000       65.049        6.021       11.993     8363.800
        4 |   60131.000       82.163        7.826       19.819    13821.480
        5 |   72277.000       98.759        9.726       29.545    20604.340
        6 |   83694.000      114.359       11.268       40.813    28462.690
        7 |   89834.000      122.749       12.537       53.350    37205.960
        8 |  103682.000      141.671       13.971       67.321    46949.130
        9 |  113743.000      155.418       15.727       83.048    57917.080
       10 |                                16.952      100.000    69739.070
---------------------------------------------------------------------------
Share = quantile group share of total income;
L(p)=cumulative group share; GL(p)=L(p)*mean(income)


mean income if year==2010, over(gp)

Mean estimation                   Number of obs   =        100

--------------------------------------------------------------
        Over |       Mean   Std. Err.     [95% Conf. Interval]
-------------+------------------------------------------------
income       |
           1 |    14799.8   850.6192      13111.99    16487.61
           2 |    26848.1   2005.814      22868.13    30828.07
           3 |    41990.1    1660.62      38695.07    45285.13
           4 |    54576.8    1067.61      52458.43    56695.17
           5 |    67828.6   967.4502      65908.97    69748.23
           6 |    78583.5     1109.6      76381.81    80785.19
           7 |    87432.7   735.7576       85972.8     88892.6
           8 |    97431.7   1423.667      94606.84    100256.6
           9 |   109679.5   1131.362      107434.6    111924.4
          10 |   118219.9   1093.665      116049.8      120390
--------------------------------------------------------------


tabstat income, by(gp) stats(n mean median min max sum) format(%10.1gc)
// this confirms that bottom decile goes from $10,146 to $17,974, with mean==$14,799.8
// 147,998 / 6,973,907 = 0.02122 which matches the Share% from the sumdist table above
Summary for variables: income
     by categories of: gp (Quantile group)

      gp |         N      mean       p50       min       max       sum
---------+-----------------------------------------------------------------
       1 |        10    14,800    15,437    10,146    17,974     147,998
       2 |        10    26,848    27,568    18,385    33,631     268,481
       3 |        10    41,990    44,704    34,246    47,606     419,901
       4 |        10    54,577    54,473    48,673    60,131     545,768
       5 |        10    67,829    67,882    63,376    72,277     678,286
       6 |        10    78,584    77,925    74,093    83,694     785,835
       7 |        10    87,433    88,233    83,710    89,834     874,327
       8 |        10    97,432    97,359    90,717   103,682     974,317
       9 |        10   109,680   111,095   104,408   113,743   1,096,795
      10 |        10   118,220   116,783   113,936   122,740   1,182,199
---------+-----------------------------------------------------------------
   Total |       100    69,739    73,185    10,146   122,740   6,973,907
---------------------------------------------------------------------------

* Creating my own summary stats for each gp (decile, in this case)
sort income id
bysort gp (income): gen n = _n
​​​​​​​egen gp_avg = mean(income), by(gp)
egen gp_count = count(income), by(gp)
egen gp_min = min(income), by(gp)
egen gp_max = max(income), by(gp)

. list n income gp gp_avg gp_count gp_min gp_max if gp<=2, sepby(gp) noobs

  +--------------------------------------------------------------+
  |  n   income   gp     gp_avg   gp_count     gp_min     gp_max |
  |--------------------------------------------------------------|
  |  1   10,146    1   14,799.8         10   10,146.0   17,974.0 |
  |  2   11,174    1   14,799.8         10   10,146.0   17,974.0 |
  |  3   12,490    1   14,799.8         10   10,146.0   17,974.0 |
  |  4   14,750    1   14,799.8         10   10,146.0   17,974.0 |
  |  5   15,392    1   14,799.8         10   10,146.0   17,974.0 |
  |  6   15,481    1   14,799.8         10   10,146.0   17,974.0 |
  |  7   16,039    1   14,799.8         10   10,146.0   17,974.0 |
  |  8   16,896    1   14,799.8         10   10,146.0   17,974.0 |
  |  9   17,656    1   14,799.8         10   10,146.0   17,974.0 |
  | 10   17,974    1   14,799.8         10   10,146.0   17,974.0 |
  |--------------------------------------------------------------|
  |  1   18,385    2   26,848.1         10   18,385.0   33,631.0 |
  |  2   18,417    2   26,848.1         10   18,385.0   33,631.0 |
  |  3   21,173    2   26,848.1         10   18,385.0   33,631.0 |
  |  4   22,585    2   26,848.1         10   18,385.0   33,631.0 |
  |  5   25,789    2   26,848.1         10   18,385.0   33,631.0 |
  |  6   29,347    2   26,848.1         10   18,385.0   33,631.0 |
  |  7   32,706    2   26,848.1         10   18,385.0   33,631.0 |
  |  8   33,138    2   26,848.1         10   18,385.0   33,631.0 |
  |  9   33,310    2   26,848.1         10   18,385.0   33,631.0 |
  | 10   33,631    2   26,848.1         10   18,385.0   33,631.0 |
  +--------------------------------------------------------------+


. list n income gp gp_avg gp_count gp_min gp_max if inlist(gp, 9, 10), sepby(gp) noobs

  +------------------------------------------------------------------+
  |  n    income   gp      gp_avg   gp_count      gp_min      gp_max |
  |------------------------------------------------------------------|
  |  1   104,408    9   109,679.5         10   104,408.0   113,743.0 |
  |  2   104,637    9   109,679.5         10   104,408.0   113,743.0 |
  |  3   105,810    9   109,679.5         10   104,408.0   113,743.0 |
  |  4   108,473    9   109,679.5         10   104,408.0   113,743.0 |
  |  5   110,640    9   109,679.5         10   104,408.0   113,743.0 |
  |  6   111,550    9   109,679.5         10   104,408.0   113,743.0 |
  |  7   111,954    9   109,679.5         10   104,408.0   113,743.0 |
  |  8   112,570    9   109,679.5         10   104,408.0   113,743.0 |
  |  9   113,010    9   109,679.5         10   104,408.0   113,743.0 |
  | 10   113,743    9   109,679.5         10   104,408.0   113,743.0 |
  |------------------------------------------------------------------|
  |  1   113,936   10   118,219.9         10   113,936.0   122,740.0 |
  |  2   114,439   10   118,219.9         10   113,936.0   122,740.0 |
  |  3   115,663   10   118,219.9         10   113,936.0   122,740.0 |
  |  4   116,281   10   118,219.9         10   113,936.0   122,740.0 |
  |  5   116,677   10   118,219.9         10   113,936.0   122,740.0 |
  |  6   116,889   10   118,219.9         10   113,936.0   122,740.0 |
  |  7   121,481   10   118,219.9         10   113,936.0   122,740.0 |
  |  8   121,729   10   118,219.9         10   113,936.0   122,740.0 |
  |  9   122,364   10   118,219.9         10   113,936.0   122,740.0 |
  | 10   122,740   10   118,219.9         10   113,936.0   122,740.0 |
  +------------------------------------------------------------------+

Hope that helps!

[Zuhumnan Dapel]

Wow! This is really comprehensive. Will study it. Thanks a million!