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If I have a data set with many values repeated (example below) how would it be divided into quartiles? I assumed at first that the quartiles would be uneven, but every single definition I've read makes it sound like they must be equal.

Example:

{ 1, 1, 4, 4, 4, 4, 8, 8}
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You can arbitrrily split the ties into groups below and above the given quartile to maintain the right proportions. – Michael Chernick Aug 10 '12 at 16:35
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Martin, there's no question or ambiguity about the quartiles here, because they would comprise the first and second smallest, the third and fourth, and so on, giving ({1,1}, {4,4}, {4,4}, {8,8}), from lowest to highest. When the amount of data is not a multiple of four, however, then (somewhat) arbitrary choices have to be made. But what are the definitions you have read? Can you cite some of them? – whuber♦ Aug 10 '12 at 17:20
To provide more context, these numbers would represent performance indicators of employees. The problem I'm having is arbitrarily placing certain employees in a lower/higher quartile simply because many other employees have the same rating. The grouping you provided whuber, would make employees in the 3rd quartile look worse even though they performed just as well as the 2nd quartile. – Martin K. Aug 10 '12 at 19:15
"The problem I'm having is arbitrarily placing certain employees in a lower/higher quartile simply because many other employees have the same rating." This is not really a statistics question. What would you do if all eight employees had the same rating? – gam Aug 10 '12 at 22:43
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Martin, the problem isn't with the quartiles, it's with your intent to use them to rank employees. You have pointed out how arbitrary and unfair that is. The solution is not to lump employees into such bins--which loses crucial information--but to use their performance indicators directly. – whuber♦ Aug 10 '12 at 23:08

1 Answer

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Quartiles take the observed data and provide the three cut points that theoretically would divide the population into quartiles. There are a number of different approaches to estimating the appropriate value for a quartile cut-point. C.f. http://stat.ethz.ch/R-manual/R-patched/library/stats/html/quantile.html for some examples from one popular statistics program.

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