**Absolute Values**
                Pakistan        Cities          Rest of Urban   Rural   
                W-01    W-02    W-01    W-02    W-01    W-02    W-01    W-02
Market Share    58616   52123   33081   34375   4452    4125    21083   13623
Brand A         15933   14372   9213    9270    1253    1189    5467    3913
Brand B         14132   11926   6379    6489    1114    1074    6639    4363
Brand C         9206    8747    4710    5698    1059    895     3437    2154
Brand D         9249    8534    6810    6897    539      474    1900    1163
Brand E         10096   8543    5969    6021    488      493    3639    2030

                Pakistan        Cities      Rest of Urban       Rural   
                W-01    W-02    W-01    W-02    W-01    W-02    W-01    W-02
Brand's Share   100.0%  100.0%  100.0%  100.0%  100.0%  100.0%  100.0%  100.0%
Brand A         27.2%   27.6%   27.9%   27.0%   28.1%   28.8%   25.9%   28.7%
Brand B         24.1%   22.9%   19.3%   18.9%   25.0%   26.0%   31.5%   32.0%
Brand C         15.7%   16.8%   14.2%   16.6%   23.8%   21.7%   16.3%   15.8%
Brand D         15.8%   16.4%   20.6%   20.1%   12.1%   11.5%   9.0%    8.5%
Brand E         17.2%   16.4%   18.0%   17.5%   11.0%   11.9%   17.3%   14.9%

Above mention is Pakistan data of 5 brands. The absolute values are given in Table 1 and percentages are given in Table 2. There are four cuts of this data i.e., Cities, Rest of Urban and Rural and by adding these 3 cuts we got an overall data of Pakistan for 5 brands.

If we compare Brand D percentage of Pakistan it is increasing from 15.8% to 16.4%, however at Cities, Rest of Urban and Rural level the percentages are decreasing. My question is if the Brand D is performing well at overall (Pakistan) level, how is it possible to be under-performing at subsequent/ lower levels (Cities, Rest of Urban and Rural)? How can we justify this difference as in a layman point of view, difference in every cut should reflect the overall brand performance?

  • 2
    $\begingroup$ This question is clearly on-topic - it's asking to resolve a statistical paradox which becomes more obvious now I've edited so the tables show more clearly! The Brand D market share has increased overall from Year 1 to Year 2, yet it has decreased in each of the three market segment (Cities, Rest of Urban, Rural). The OP is asking how this is possible. The first table, which has the raw sales figures, gives the answer to the question: the market segments have changed in relative size. In particular "Cities" where, Brand D is strong, has grown; "Rural" where D is weak has shrunk. $\endgroup$
    – Silverfish
    Apr 23, 2015 at 9:13
  • 6
    $\begingroup$ The OP may want to look up Simpson's paradox - it's possible that this question should be closed as a duplicate of a related question. $\endgroup$
    – Silverfish
    Apr 23, 2015 at 9:17
  • 1
    $\begingroup$ see also similar example: stats.stackexchange.com/questions/125683/… $\endgroup$
    – Tim
    Apr 23, 2015 at 10:38
  • 1
    $\begingroup$ Even closer example might be Basic Simpson's Paradox, which was an exam-style question on hospital operation success rates. I'd lean against closing this as a duplicate though: that question had a rather confusing multiple choice set-up, and as both Peter Flom and Jonathan note in their answers, had some complicating "real world" issues. More superficially, the "Year 1 vs Year 2" comparison in this question introduces the interesting aspect of "upwards vs downwards trend", which I can't recall seeing in the best-known Simpson's paradox examples. $\endgroup$
    – Silverfish
    Apr 23, 2015 at 12:40

1 Answer 1


The Brand D market share has increased overall from Year 1 to Year 2, yet it has decreased in each of the three market segment (Cities, Rest of Urban, Rural). How is this possible?

Note that Brand D's decline in each segment is pretty small (20.6% to 20.1% in Cities, 12.1% to 11.5% in Rest of Urban, 9.0% to 8.5% in Rural) so its performance is actually fairly static. It's even possible they may be due to random fluctuations rather than a systematic downwards trend; at any rate, the falls were not dramatic.

There are two more important features to notice: Brand D's share varies substantially between market segments with (in both years) a strength in the Cities segment, a weakness in Rest of Urban, and worst performance in Rural. Also, market segments show dramatic year-on-year change in market share, far more so than Brand D's. The overall market is declining from 58,616 in Year 1 to 52,123 in Year 2. Despite this the Cities segment has risen from 33,081 to 34,375 so its share has increased from 56.44% to 65.95%, a rise of 9.5 percentage points. Rest of Urban fell from 4,452 to 4,125; proportionately, its share rose slightly from 7.60% to 7.91%. The more important Rural sector fell dramatically from 21,083 to 13,623 so its share dropped from 37.20% to 26.14%.

These two variations, together with its relatively static performance within each segment, account for Brand D's overall improved share: its best segment, $\color{blue}{\text{Cities}}$, has risen markedly in market share while its worst segment, $\color{red}{\text{Rural}}$, has collapsed in share. Then Brand D's overall share is given as a weighted average of its share across each segment, weighted by the relative size of each segment, and in Year 2 those weightings are more favourable to Brand D:

$$\text{Year 1 share} = \color{blue}{0.5644} \times 20.6\% + 0.0760 \times 12.1\% + \color{red}{0.3720} \times 9.0\% = 15.8\%$$

$$\text{Year 2 share} = \color{blue}{0.6595} \times 20.1\% + 0.0791 \times 11.5\% + \color{red}{0.2614} \times 8.5\% = 16.4\%$$

We can visualise these proportions using a spineplot (if you are unfamiliar there's some discussion of them on this thread) in which the shaded area, showing the overall market share of Brand D, is formed from three rectangles whose individual areas correspond to the terms in the above sum. We can now see how slight the proportionate decline of Brand D in each segment is (the rectangles all decrease a little in height from Year 1 to Year 2), and how this is compensated for by the increased share of Brand D's best segment (the tallest rectangle becomes wider) and the decreased share of Brand D's worst segment (the shortest rectangle becomes narrower), so that the total area of the three rectangles increases.

Spineplot showing Simpson's paradox

This is an example of Simpson's Paradox, also known as the "amalgamation paradox": the proportion or probability of one event may be lower than another across all subgroups, but higher overall. A very similar example is given in the Wikipedia example, using medical data from a paper about kidney stones. Success rates for treatments A and B varied depending on the size of the kidney stone:

                Treatment A     Treatment B 
Small stones    93% (81/87)     87% (234/270)
Large stones    73% (192/263)   69% (55/80)
All stones      78% (273/350)   83% (289/350)

We see that whichever the size of kidney stone, Treatment B has a lower success rate than A. However, overall, Treatment B appears to have a higher success rate than A — analogous to how Brand D had a higher overall percentage share in Year 2 than in Year 1, despite having lower market share proportions in each segment. The reason for this apparent reversal upon amalgamation is that the stone sizes were more favourable to Treatment B than to Treatment A: success rates are higher for small stones, and Treatment B was mostly used on small stones whereas Treatment A was disproportionately used on large stones. This matches the way market segmentation was more favourable in Year 2 than in Year 1, as a greater proportion of sales took place in urban areas, while the poor performance in rural areas became less important.

R code for spine plots


segment <- factor(rep(1:3, times = c(33081, 4452, 21083)),
    labels=c("Cities", "Rest", "Rural"))

brand <- factor(c(rep(1:2, times = c(6810, 33081-6810)), #Cities
           rep(1:2, times = c(539, 4452-539)),           #Rest
           rep(1:2, times = c(1900, 21083-1900))),       #Rural
           labels=c("Brand D", "Other")
spineplot(brand~segment, main="Year 1", xlab="Market segment", ylab="Brand")

segment <- factor(rep(1:3, times = c(34375, 4125, 13623)),
    labels=c("Cities", "Rest", "Rural"))

brand <- factor(c(rep(1:2, times = c(6897, 34375-6897)), #Cities
           rep(1:2, times = c(474, 4125-474)),           #Rest
           rep(1:2, times = c(1163, 13623-1163))),       #Rural
           labels=c("Brand D", "Other")
spineplot(brand~segment, main="Year 2", xlab="Market segment", ylab="Brand")
  • $\begingroup$ Thank you for the answer and for the excellent explanation, this was indeed a paradox for us and your answer helped us to jump out of that paradox, thank you for sharing the R-code as well. Best Regards, Asad $\endgroup$
    – asad
    Apr 24, 2015 at 5:44

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