where is the distribution's standard deviation. In particular, the median is never more than one standard deviation from the mean. The above formula can be used to bound the value in terms of quantiles. When , the value that is standard deviations above the mean has a lower bound For example, the value that is standard deviation above the mean is always greater than or equal to , the median, and the value that is standard deviations above the mean is always greater than or equal to , the fourth quintile. When , there is instead an upper bound For example, the value for will never exceed , the first decile. Estimating quantiles from a sample One problem which frequently arises is estimating a quantile of a (very large or infinite) population based on a finite sample of size . The asymptotic distribution of the -th sample quantile is well-known: it is asymptotically normal around the -th population quantile with variance equal to where is the value of the distribution density at the -th population quantile. However, this distribution relies on knowledge of the population distribution; which is equivalent to knowledge of the population quantiles, which we are trying to estimate! Modern statistical packages thus rely on a different technique — or selection of techniques — to estimate the quantiles. Hyndman and Fan compiled a taxonomy of nine algorithms used by various software packages. All methods compute , the estimate for the -quantile (the -th -quantile, where ) from a sample of size by computing a real valued index . When is an integer, the -th smallest of the values, , is the quantile estimate. Otherwise a rounding or interpolation scheme is used to compute the quantile estimate from , , and . (For notation, see floor and ceiling functions). The first three are piecewise constant, changing abruptly at each data point, while the last six use linear interpolation between data points, and differ only in how the index used to choose the point along the piecewise linear interpolation curve, is chosen. Mathematica, Matlab, R and GNU Octave programming languages support all nine sample quantile methods. SAS includes five sample quantile methods, SciPy and Maple both include eight, EViews includes the six piecewise linear functions, Stata includes two, Python includes two, and Microsoft Excel includes two. Mathematica and SciPy support arbitrary parameters for methods which allow for other, non-standard, methods.