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When a distribution is cut into percentiles, is there any rule whether it should be included the lower or the upper bound inside the percentile?

Example: 1,2,3,4,5,6,7

The median is obviously 4. If I wanted to cut the distribution in two parts should I include 4 in the lower or the higher percentile?

I'm sorting a long list of a daily observations of the same variable into daily deciles. I am going to include the boundaries in the lower decile for necessity, because the first decile is often equal to 0, which is also the minimum value that the variable can have.

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Division of an ordered list of values into bins, classes, or intervals with ideally equal frequencies will be frustrated in practice whenever

  1. The number of data points is not a multiple (meaning, integer multiple) of the number of bins needed. As in the example, if you want two bins and the number of values is odd, the median can be included in the lower or the upper bin. Or, suppose you have 15 values; then any division into quartile bins is at best some variation on bins with frequencies 3, 4, 4, 4.

  2. There are ties, as the only consistent rule is that identical values are assigned to the same bin.

Otherwise

(1) you decide on the rules

(2) you should make them explicit

(3) why are you doing this any way? If you had data on say people's heights, dividing them arbitrarily into tall and short groups isn't going to help much statistically, regardless of whether height is a response or a predictor.

Historically, quartiles, deciles, percentiles and so forth were points within a distribution, either individual values or summaries or estimates interpolated somehow between them. Only by extension are they the bins, classes or intervals so delimited. So, some people move back and forth between talk of lower (first) quartile, median and upper (third) quartile as three summary points -- and talk of the four bins they define. Usually the equivocation doesn't bite, but history is clear on the terminology.

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  • $\begingroup$ I'm interested in categorizing stock prices according to firms' characteristics $\endgroup$
    – Mr Frog
    Commented May 28, 2020 at 15:51
  • $\begingroup$ I feared as much. Why categories use information better than the variables they are based on beats me. $\endgroup$
    – Nick Cox
    Commented May 28, 2020 at 15:56
  • $\begingroup$ On Statalist almost all the interest in this question seems to come from people wanting to bin firms or their stocks into groups, sometimes grandly called portfolios. People are often puzzled when subdivision fails to yield bins with equal frequencies. Some discussion is visible in stata-journal.com/article.html?article=pr0054 (despite the title of the paper being quite different) and in stata-journal.com/article.html?article=dm0095 (although that may be behind a pay wall as far as you are concerned). $\endgroup$
    – Nick Cox
    Commented May 28, 2020 at 16:06

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