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<table width="100%" summary="page for quantile {stats}"><tr><td>quantile {stats}</td><td style="text-align: right;">R Documentation</td></tr></table>

<h2>Sample Quantiles</h2>

<h3>Description</h3>

The generic function <code>quantile</code> 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.

<h3>Usage</h3>

<pre>
quantile(x, ...)

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

<h3>Arguments</h3>

<table summary="R argblock">
<tr valign="top"><td><code>x</code></td>
<td>
numeric vector whose sample quantiles are wanted, or an
object of a class for which a method has been defined (see also
&lsquo;details&rsquo;). <code><a href="../../base/html/NA.html">NA</a></code> and <code>NaN</code> values are not
allowed in numeric vectors unless <code>na.rm</code> is <code>TRUE</code>.
</td></tr>
<tr valign="top"><td><code>probs</code></td>
<td>
numeric vector of probabilities with values in
[0,1]. (Values up to 2e-14 outside that
range are accepted and moved to the nearby endpoint.)
</td></tr>
<tr valign="top"><td><code>na.rm</code></td>
<td>
logical; if true, any <code><a href="../../base/html/NA.html">NA</a></code> and <code>NaN</code>'s
are removed from <code>x</code> before the quantiles are computed.
</td></tr>
<tr valign="top"><td><code>names</code></td>
<td>
logical; if true, the result has a <code><a href="../../base/html/names.html">names</a></code>
attribute. Set to <code>FALSE</code> for speedup with many <code>probs</code>.
</td></tr>
<tr valign="top"><td><code>type</code></td>
<td>
an integer between 1 and 9 selecting one of the
nine quantile algorithms detailed below to be used.
</td></tr>
<tr valign="top"><td><code>...</code></td>
<td>
further arguments passed to or from other methods.
</td></tr>
</table>

<h3>Details</h3>

A vector of length <code>length(probs)</code> is returned;
if <code>names = TRUE</code>, it has a <code><a href="../../base/html/names.html">names</a></code> attribute.

<code><a href="../../base/html/NA.html">NA</a></code> and <code><a href="../../base/html/is.finite.html">NaN</a></code> values in <code>probs</code> are
propagated to the result.

The default method works with classed objects sufficiently like
numeric vectors that <code>sort</code> and (not needed by types 1 and 3)
addition of elements and multiplication by a number work correctly.
Note that as this is in a namespace, the copy of <code>sort</code> in
base will be used, not some S4 generic of that name. Also note
that that is no check on the &lsquo;correctly&rsquo;, and so
e.g. <code>quantile</code> can be applied to complex vectors which (apart
from ties) will be ordered on their real parts.

There is a method for the date-time classes (see
<code>"<a href="../../base/html/DateTimeClasses.html">POSIXt</a>"</code>). Types 1 and 3 can be used for class
<code>"<a href="../../base/html/Dates.html">Date</a>"</code> and for ordered factors.

<h3>Types</h3>

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

All sample quantiles are defined as weighted averages of
consecutive order statistics. Sample quantiles of type i
are defined by:

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

where 1 &le; i &le; 9,
(j-m)/n &le; p &lt; (j-m+1)/n,
x[j] is the jth order statistic, n is the
sample size, the value of &gamma; is a function of
j = floor(np + m) and g = np + m - j,
and m is a constant determined by the sample quantile type.

Discontinuous sample quantile types 1, 2, and 3

For types 1, 2 and 3, Q[i](p) is a discontinuous
function of p, with m = 0 when i = 1 and i =
 2, and m = -1/2 when i = 3.

<dl>
<dt>Type 1</dt><dd>Inverse of empirical distribution function.
&gamma; = 0 if g = 0, and 1 otherwise.
</dd>
<dt>Type 2</dt><dd>Similar to type 1 but with averaging at discontinuities.
&gamma; = 0.5 if g = 0, and 1 otherwise.
</dd>
<dt>Type 3</dt><dd>SAS definition: nearest even order statistic.
&gamma; = 0 if g = 0 and j is even,
and 1 otherwise.
</dd>
</dl>

Continuous sample quantile types 4 through 9

For types 4 through 9, Q[i](p) is a continuous function
of p, with gamma = g and m given below. The
sample quantiles can be obtained equivalently by linear interpolation
between the points (p[k],x[k]) where x[k]
is the kth order statistic. Specific expressions for
p[k] are given below.

<dl>
<dt>Type 4</dt><dd>m = 0. p[k] = k / n.
That is, linear interpolation of the empirical cdf.

</dd>
<dt>Type 5</dt><dd>m = 1/2.
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.

</dd>
<dt>Type 6</dt><dd>m = p. p[k] = k / (n + 1).
Thus p[k] = E[F(x[k])].
This is used by Minitab and by SPSS.

</dd>
<dt>Type 7</dt><dd>m = 1-p.
p[k] = (k - 1) / (n - 1).
In this case, p[k] = mode[F(x[k])].
This is used by S.

</dd>
<dt>Type 8</dt><dd>m = (p+1)/3.
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 <code>x</code>.

</dd>
<dt>Type 9</dt><dd>m = p/4 + 3/8.
p[k] = (k - 3/8) / (n + 1/4).
The resulting quantile estimates are approximately unbiased for
the expected order statistics if <code>x</code> is normally distributed.

</dd>
</dl>

Further details are provided in Hyndman and Fan (1996) who recommended type 8.
The default method is type 7, as used by S and by R &lt; 2.0.0.

<h3>Author(s)</h3>

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

<h3>References</h3>

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
The New S Language.
Wadsworth &amp; Brooks/Cole.

Hyndman, R. J. and Fan, Y. (1996) Sample quantiles in statistical
packages, American Statistician 50, 361&ndash;365.
doi: <a href="https://doi.org/10.2307/2684934">10.2307/2684934</a>.

<code><a href="ecdf.html">ecdf</a></code> for empirical distributions of which
<code>quantile</code> is an inverse;
<code><a href="../../grDevices/html/boxplot.stats.html">boxplot.stats</a></code> and <code><a href="fivenum.html">fivenum</a></code> for computing
other versions of quartiles, etc.

<h3>Examples</h3>

<pre>
quantile(x &lt;- rnorm(1001)) # Extremes &amp; Quartiles by default
quantile(x, probs = c(0.1, 0.5, 1, 2, 5, 10, 50, NA)/100)

### Compare different types
quantAll &lt;- function(x, prob, ...)
 t(vapply(1:9, function(typ) quantile(x, prob=prob, type = typ, ...), quantile(x, prob, type=1)))
p &lt;- c(0.1, 0.5, 1, 2, 5, 10, 50)/100
signif(quantAll(x, p), 4)
## for complex numbers:
z &lt;- complex(re=x, im = -10*x)
signif(quantAll(z, p), 4)
</pre>

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