Statistics

Disclamer from the Python source code

Notes on methods for computing quantiles

There is no one perfect way to compute quantiles. Here we offer two methods that serve common needs. Most other packages surveyed offered at least one or both of these two, making them "standard" in the sense of "widely-adopted and reproducible". They are also easy to explain, easy to compute manually, and have straight-forward interpretations that aren't surprising.

The default method is known as "R6", "PERCENTILE.EXC", or "expected value of rank order statistics". The alternative method is known as "R7", "PERCENTILE.INC", or "mode of rank order statistics".

For sample data where there is a positive probability for values beyond the range of the data, the R6 exclusive method is a reasonable choice. Consider a random sample of nine values from a population with a uniform distribution from 0.0 to 1.0. The distribution of the third ranked sample point is described by betavariate(alpha=3, beta=7) which has mode=0.250, median=0.286, and mean=0.300. Only the latter (which corresponds with R6) gives the desired cut point with 30% of the population falling below that value, making it comparable to a result from an inv_cdf() function. The R6 exclusive method is also idempotent.

For describing population data where the end points are known to be included in the data, the R7 inclusive method is a reasonable choice. Instead of the mean, it uses the mode of the beta distribution for the interior points. Per Hyndman & Fan, "One nice property is that the vertices of Q7(p) divide the range into n - 1 intervals, and exactly 100p% of the intervals lie to the left of Q7(p) and 100(1 - p)% of the intervals lie to the right of Q7(p)."

If needed, other methods could be added. However, for now, the position is that fewer options make for easier choices and that external packages can be used for anything more advanced.