Parameters α and β to the continuousBy function. Exact meaning of parameters is described in [Hyndman1996] in section "Piecewise linear functions"

A class for types with a default value.

O(n·log n). Estimate the kth q-quantile of a sample x, using the continuous sample method with the given parameters.

The following properties should hold, otherwise an error will be thrown.

• input sample must be nonempty
• the input does not contain NaN
• 0 ≤ k ≤ q

O(k·n·log n). Estimate set of the kth q-quantile of a sample x, using the continuous sample method with the given parameters. This is faster than calling quantile repeatedly since sample should be sorted only once

The following properties should hold, otherwise an error will be thrown.

• input sample must be nonempty
• the input does not contain NaN
• for every k in set of quantiles 0 ≤ k ≤ q

O(k·n·log n). Same as quantiles but uses Vector container instead of Foldable one.

California Department of Public Works definition, α=0, β=1. Gives a linear interpolation of the empirical CDF. This corresponds to method 4 in R and Mathematica.

Hazen's definition, α=0.5, β=0.5. This is claimed to be popular among hydrologists. This corresponds to method 5 in R and Mathematica.

Definition used by the SPSS statistics application, with α=0, β=0 (also known as Weibull's definition). This corresponds to method 6 in R and Mathematica.

Definition used by the S statistics application, with α=1, β=1. The interpolation points divide the sample range into n-1 intervals. This corresponds to method 7 in R and Mathematica and is default in R.

Median unbiased definition, α=1/3, β=1/3. The resulting quantile estimates are approximately median unbiased regardless of the distribution of x. This corresponds to method 8 in R and Mathematica.

Normal unbiased definition, α=3/8, β=3/8. An approximately unbiased estimate if the empirical distribution approximates the normal distribution. This corresponds to method 9 in R and Mathematica.

O(n·log n). Estimate the kth q-quantile of a sample, using the weighted average method. Up to rounding errors it's same as quantile s.

The following properties should hold otherwise an error will be thrown.

• the length of the input is greater than 0
• the input does not contain NaN
• k ≥ 0 and k ≤ q

O(n·log n) Estimate median of sample

O(n·log n). Estimate the median absolute deviation (MAD) of a sample x using continuousBy. It's robust estimate of variability in sample and defined as:

\[ MAD = \operatorname{median}(| X_i - \operatorname{median}(X) |) \]

O(n·log n). Estimate the range between q-quantiles 1 and q-1 of a sample x, using the continuous sample method with the given parameters.

For instance, the interquartile range (IQR) can be estimated as follows:

midspread medianUnbiased 4 (U.fromList [1,1,2,2,3])
==> 1.333333