quantile                package:stats                R Documentation

_S_a_m_p_l_e _Q_u_a_n_t_i_l_e_s

_D_e_s_c_r_i_p_t_i_o_n:

     The generic function 'quantile' produces sample quantiles
     corresponding to the given probabilities. The smallest observation
     corresponds to a probability of 0 and the largest to a probability
     of 1.

_U_s_a_g_e:

     quantile(x, ...)

     ## Default S3 method:
     quantile(x, probs = seq(0, 1, 0.25), na.rm = FALSE,
              names = TRUE, type = 7, ...)

_A_r_g_u_m_e_n_t_s:

       x: numeric vectors whose sample quantiles are wanted.  Missing
          values are ignored.

   probs: numeric vector of probabilities with values in [0,1].

   na.rm: logical; if true, any 'NA' and 'NaN''s are removed from 'x'
          before the quantiles are computed.

   names: logical; if true, the result has a 'names' attribute.  Set to
          'FALSE' for speedup with many 'probs'.

    type: an integer between 1 and 9 selecting one of the nine quantile
          algorithms detailed below to be used.

     ...: further arguments passed to or from other methods.

_D_e_t_a_i_l_s:

     A vector of length 'length(probs)' is returned; if 'names = TRUE',
     it has a 'names' attribute.

     'NA' and 'NaN' values in 'probs' are propagated to the result.

_T_y_p_e_s:

     'quantile' returns estimates of underlying distribution quantiles
     based on one or two order statistics from the supplied elements in
     'x' at probabilities in 'probs'.  One of the nine quantile
     algorithms discussed in Hyndman and Fan (1996), selected by
     'type', is employed.

     Sample quantiles of type i are defined by

              Q[i](p) = (1 - gamma) x[j] + gamma x[j+1],

     where 1 <= i <= 9, (j-m)/n <= p < (j-m+1)/ n, x[j] is the jth
     order statistic, n is the sample size, and m is a constant
     determined by the sample quantile type. Here gamma depends on the
     fractional part of g = np+m-j.

     For the continuous sample quantile types (4 through 9), the sample
     quantiles can be obtained by linear interpolation between the kth
     order statistic and p(k):

             p(k) = (k - alpha) / (n - alpha - beta + 1),

     where alpha and beta are constants determined by the type. 
     Further, m = alpha + p(1 - alpha - beta), and gamma = g.

     *Discontinuous sample quantile types 1, 2, and 3*

     _T_y_p_e _1 Inverse of empirical distribution function.

     _T_y_p_e _2 Similar to type 1 but with averaging at discontinuities.

     _T_y_p_e _3 SAS definition: nearest even order statistic.

     *Continuous sample quantile types 4 through 9*

     _T_y_p_e _4 p(k) = k / n. That is, linear interpolation of the
          empirical cdf.


     _T_y_p_e _5 p(k) = (k - 0.5) / n. That is a piecewise linear function
          where the knots are the values midway through the steps of
          the empirical cdf. This is popular amongst hydrologists.


     _T_y_p_e _6 p(k) = k / (n + 1). Thus p(k) = E[F(x[k])]. This is used by
          Minitab and by SPSS.


     _T_y_p_e _7 p(k) = (k - 1) / (n - 1). In this case, p(k) =
          mode[F(x[k])]. This is used by S.


     _T_y_p_e _8 p(k) = (k - 1/3) / (n + 1/3). Then p(k) =~ median[F(x[k])].
          The resulting quantile estimates are approximately
          median-unbiased regardless of the distribution of 'x'.


     _T_y_p_e _9 p(k) = (k - 3/8) / (n + 1/4). The resulting quantile
          estimates are approximately unbiased if 'x' is normally
          distributed.

     Hyndman and Fan (1996) recommend type 8. The default method is
     type 7, as used by S and by R < 2.0.0.

_A_u_t_h_o_r(_s):

     of the version used in R >= 2.0.0, Ivan Frohne and Rob J Hyndman.

_R_e_f_e_r_e_n_c_e_s:

     Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S
     Language_. Wadsworth & Brooks/Cole.

     Hyndman, R. J. and Fan, Y. (1996) Sample quantiles in statistical
     packages, _American Statistician_, *50*, 361-365.

_S_e_e _A_l_s_o:

     'ecdf' for empirical distributions of which 'quantile' is the
     "inverse"; 'boxplot.stats' and 'fivenum' for computing "versions"
     of quartiles, etc.

_E_x_a_m_p_l_e_s:

     quantile(x <- rnorm(1001))# Extremes & Quartiles by default
     quantile(x,  probs=c(.1,.5,1,2,5,10,50, NA)/100)

     ### Compare different types
     p <- c(0.1,0.5,1,2,5,10,50)/100
     res <- matrix(as.numeric(NA), 9, 7)
     for(type in 1:9) res[type, ] <- y <- quantile(x,  p, type=type)
     dimnames(res) <- list(1:9, names(y))
     round(res, 3)

