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I am not particularly familiar with statistics and am looking at methods for analysing numbers that have been broken into different "brackets" or "groups" for different entities.

Consider three companies.

  • A announces that they have 10 factories generating between 10,000 and 20,000 widgets per year and 15 factories generating between 20,000 to 40,000

  • B announces that they have 7 factories generating between 15,000 and 25,000 widgets and 5 factories generating between 25,000 and 35,000

  • C announces they have 5 factories generating between 10,000 and 23,000 widgets and 14 between 23,000 and 40,000

To be clear all widget production is included in my simplified example above.

What statistical methods can be used (and do any exist) that would allow you to aggregate this data together and make generalised predictions (all companies have on average x factories that generate 15,000 widgets per year and y factories that generate 40,000)?

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    $\begingroup$ It's a good question (+1), but I would urge caution: unless those companies are using some standard intervals for reporting their statistics, there may be substantial bias introduced by their choices of breakpoints. You're probably familiar with the myriad sports statistics flung about the media of the form "the team has won five of their last seven games--they're on a streak!" that exist because, of course, the team lost the eighth game back. Aggregation can be similarly jiggered to make data look better (or worse) than they really are. Statistical procedures won't cure that problem. $\endgroup$ – whuber May 2 '13 at 13:51
  • $\begingroup$ @whuber totally agree with your point on bias. I edited the question to be clear that all data is included, so setting the break points in this example probably wouldn't be based on bias. $\endgroup$ – Robert Anton Reese May 2 '13 at 14:30
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    $\begingroup$ You might be interested in Sheppard's corrections. For a recent exposition, see the first couple of pages of arxiv.org/pdf/1004.4989.pdf. $\endgroup$ – whuber May 2 '13 at 14:38
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You have, at a minimum, points on the cumulative distribution function for each, i.e. the cumulative frequency or the cumulative probability of factories producing less than # widgets. So you can plot those. If any firm is very different, it will stand out. Otherwise, you don't have enough information.

Personally I prefer to invert this and think of quantile functions, but it's the same information.

Otherwise there is no white magic to restore detail omitted in data production.

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summary: If you can get the number of widgets for each factory then you can just use the counts to do what you want, if you are stuck with the binned data you are probably out of luck. Details below.

The bins (10,000 to 20,000 and 20,000 to 40,000) are different widths which is going to make any sort of comparison or prediction very hard since you have will have to make some assumption about how the data are distributed within each bin. It seems like if you know how many factories are in each bin then you (or someone) must know how many widgets each factory produces otherwise how are you getting the numbers to put in each bin. If you have that data you could just use simple counts to answer the sort of questions you wanted, especially since then all widget production is covered, so you don't even have to worry about sample error.

Assuming that information is hidden from you as I said you would have make some assumptions about how factories are distributed within bins, if you assume they are evenly distributed then you could estimate how many factories would be greater or less than any specified number of widgets by just multiplying the number of factories by the proportion of the bin above (or below) the specified number of widgets. But this assumption will probably not work very well since your bins are wide relative to the total maximum number of widgets, and the break point is near the centre (to see why this is problem imagine a normal distribution, imagine splitting it into two bins with the break point in the middle, the two bins will be the same height, i.e. they will hide the fact that most factories widget production is near average).

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