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 Use/Abuse Stat.Book Beginners Stats & R
"It has long been an axiom of mine that the little things are infinitely the most important" (Sherlock Holmes)

### Example, with R

Quantiles are values chosen to divide ordered values into predefined portions.

The median (1.1), their 50% quantile, divides these 5 ordered values into 2 equal groups:

If you rank the values in order, the median is their middle-most (= least deviant).

 group 1 (<1.1) group 2 (>1.1) -999999 0 1.1 2 2.002 2 values 2 values 50% of 5 values 50% of 5 values

Or you could find their 50% (the p = 0.5 th) quantile with ### Definition and Use

1. Quantiles are commonly assumed to divide sets of ordered numbers into equal-sized groups.
• Quartiles are expected to divide them into 4 equal groups.
• Deciles are supposed to divide them into 10 equal groups.
• Percentiles should divide them into 100 equal-sized groups.
For the 5 numbers listed above, this reasoning may seem of academic interest.
2. More practically perhaps, you can regard a set of n different values as n different quantiles.
3. The most commonly encountered quantiles are the maximum (the 100% quantile) and minimum (the 0% quantile).
• Since the maximum has none of the values above it, and the minimum has none of them below it, these are called 'extreme' or 'divergent' quantiles.
• Conversely, the range enclosed by the first and third quantiles - termed the interquartile range - can be said to typify the distribution. It is commonly used as a summary statistic of spread. Values lying outside that range may be regarded as unrepresentative or outlying.

### Simple formula

Given each item's rank (r) gives the number of items of less than or equal value, this is a usable approximation for large sets of values.

 the rank of the pth quantile is pn

When pn does not correspond to the rank of any value of y you have to interpolate, or choose the best value.

When n is small you may prefer R's default quantile formula (type ?quantile to R for more):

1+p(n-1)

### Tips and Notes

• Instead of percentages, quantiles are commonly expressed using proportions: Thus the first quartile is the 25% or p = 0.25 th quantile.
• Beware, the simple definitions run into difficulties when some of the numbers have equal values (tied), or where only certain numbers can be observed (discrete variables).
• When applied to sufficiently large sets of (un-tied) values, the relative rank is virtually indistinguishable from p, the proportion of values below that value.
When applied to small and/or heavily-tied sets of numbers, these ways of defining quantiles may differ quite noticeably!

### Useful references

Altman, D.G. & Bland, J.M. (1994) Quartiles, quintiles, centiles and other quantiles. BMJ 309, 996 (15 October). Full text A good introduction to the use of quantiles in medical statistics.
Wikipedia: Quantile. Full text A bit heavy on formulae and a bit light on explanation.
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