The estimate types and interpolation schemes used include: Notes: R‑1 through R‑3 are piecewise constant, with discontinuities. R‑4 and following are piecewise linear, without discontinuities, but differ in how is computed. R‑3 and R‑4 are not symmetric in that they do not give when . Excel's PERCENTILE.EXC and Python's default "exclusive" method are equivalent to R‑6. Excel's PERCENTILE and PERCENTILE.INC and Python's optional "inclusive" method are equivalent to R‑7. This is R's default method. Packages differ in how they estimate quantiles beyond the lowest and highest values in the sample, i.e. and . Choices include returning an error value, computing linear extrapolation, or assuming a constant value. Of the techniques, Hyndman and Fan recommend R-8, but most statistical software packages have chosen R-6 or R-7 as the default. The standard error of a quantile estimate can in general be estimated via the bootstrap. The Maritz–Jarrett method can also be used. Approximate quantiles from a stream Computing approximate quantiles from data arriving from a stream can be done efficiently using compressed data structures. The most popular methods are t-digest and KLL. These methods read a stream of values in a continuous fashion and can, at any time, be queried about the approximate value of a specified quantile. Both algorithms are based on a similar idea: compressing the stream of values by summarizing identical or similar values with a weight. If the stream is made of a repetition of 100 times v1 and 100 times v2, there is no reason to keep a sorted list of 200 elements, it is enough to keep two elements and two counts to be able to recover the quantiles. With more values, these algorithms maintain a trade-off between the number of unique values stored and the precision of the resulting quantiles. Some values may be discarded from the stream and contribute to the weight of a nearby value without changing the quantile results too much. The t-digest maintains a data structure of bounded size using an approach motivated by k-means clustering to group similar values. The KLL algorithm uses a more sophisticated "compactor" method that leads to better control of the error bounds at the cost of requiring an unbounded size if errors must be bounded relative to . Both methods belong to the family of data sketches that are subsets of Streaming Algorithms with useful properties: t-digest or KLL sketches can be combined. Computing the