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I'm taking a first course in statistics. I have been given a dataset with the number of companies falling into different categories of revenues (in thousands of euros):

<1               64108
1-39             73121
40-99            67532
100-399          68613
400-1999         33760
2000-9999        10293
10000-39999       2493
40000-199999       779
200000-infinity    235

Now I was asked to count the lower quartile and how much the revenue should be if a company belonged to the two highest deciles.

Is it reasonable to assume that the revenue of those 235 companies is 200,000 thousand euros? And does the 39 mean 39,000 euros so if one of those 73,121 companies has revenue 39,600, it actually belongs to the range 40-99? By assuming the data is uniformly distributed, I can modify the data such that I know how many companies belongs to every class but is this a standard way to do such problems?

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It is not clear exactly how much data you have on the companies. You just know in which "window" they fall, but not the actual revenues? For your first and second question, I think the answer is yes: 235 companies have revenues of 200k and above, if a company had revenues of 39600, the revenues are probably rounded to the closest 1000, so in this case it would probably be rounded to 40000 and fall in the 40-99 range. – Antoine Vernet Sep 27 '12 at 12:47
That is correct. Only "windows" are known. There was said "magnitude of revenue". – Jaakko Seppälä Sep 27 '12 at 12:49
In this case, I think you are right in the sense that you need some sort of assumption on how the revenues of companies are distributed in each classes. One assumption could be that the companies revenues are normally distributed around the center of each range. This would then give you a global distribution of revenues. From there it is straightforward to calculate quartile and decile. I am curious to know if there is a way of doing it without making an assumption on the distribution of the revenues (which obviously would be more satisfying.) – Antoine Vernet Sep 27 '12 at 12:59
"235 companies have revenues of 200k and above" But I would like to know am I allowed to assume that all of them has revenue exactly 200000k. If not, it makes the computation of deciles harder. – Jaakko Seppälä Sep 27 '12 at 13:00
I think you don't really have a choice because you need an upper bound for revenues. I am sure that if you wait a bit, someone more expert than me will probably chip in a more satisfying solution. – Antoine Vernet Sep 27 '12 at 13:12

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You certainly should not assume that all companies that have income of 200,000 or above have income of 200,000 exactly. That makes no sense. But what should you assume? I am not sure what answer would be expected in a first course in statistics.

For the first question (find the lowest quartile) you can probably assume that the data are uniformly distributed in each category - that makes more sense than normal distribution. So, you would figure the total number of categories, then figure which category had the 20th percentile, then interpolate.

For the second question, though, it's much trickier. The desired answer might be "I don't have enough information" or something like that. Or possibly you ought to say "at least XXX". But, if you want to get fancy, distributions like this often follow some sort of power law. But that may be beyond what is being asked.

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