quantile
Empirical Quantiles

### Description

Calculates a vector of the desired quantiles of the data.

### Usage

    quantile(x, ...)
    quantile.default(x, probs = 0:4/4, na.rm = FALSE,
        names = TRUE, type = 7, ...)

### Arguments

x | a numeric vector. Missing values (NAs) are only allowed if you set na.rm = TRUE.
probs | a vector of desired probability levels where the values are must be between or equal to 0 and 1. The default value, seq(0, 1, 0.25), produces a "five number summary" that includes: the minimum, lower quartile, median, upper quartile, and maximum of x.
na.rm | a logical value that specifies if missing values (NAs) are removed before the function performs the computation.
names | a logical value that specifies if a "names" attribute should be included in the result.
type | an integer value between 1 and 9 that is used to select one quantile algorithm. For more information about the definition of each value, see the Details section.
... | methods may have other arguments. The following arguments are available in the default method.

### Details

The type argument, depending on the value, returns a discontinuous or continuous sample quantile type. You can assign a value from the range one to nine and the sample quantiles of type i are defined by: Q[i](p) = (1 - y) x[j] + y x[j+1]
where:

- i <= i <= 9
- (j - m)/n <= p < (j - m + 1)/n
- x[j] is the jth order statistic
- n is the sample size
- the value of y is a function of j = floor(np+m) and g = np + m -j
- m is a constant determined by the sample quantile type

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

type | Calculation
1 | y = 0 if g = 0, and 1 otherwise.
2 | y = 0.5 if g = 0, and 1 otherwise.
3 | SAS definition: nearest even order statistic. y = 0 if g = 0 and j is even, and 1 otherwise.

Continuous
For types 4 to 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.

type | Calculation
4 | m = 0. p[k] = k / n.
5 | m = 1/2. p[k] = (k - 0.5) / n.
6 | m = p. p[k] = k / (n + 1). Thus p[k] = E[F(x[k])]. This is used by Minitab and by SPSS.
7 | m = 1-p. p[k] = (k - 1) / (n - 1). In this case, p[k] = mode[F(x[k])]. This is used by S.
8 | m = (p+1)/3. p[k] = (k - 1/3) / (n + 1/3). Then p[k] =~ median[F(x[k])].
9 | m = p/4 + 3/8. p[k] = (k - 3/8) / (n + 1/4).

Value
returns a vector of empirical quantiles that correspond to the probs levels in the sorted x data. The length of the return value is the same as the probs vector. A names attribute is also returned if names = TRUE. Missing values (NAs) and values that are not a number (NaNs), are passed through to the result.
References
Hyndman, R. J. and Fan, Y (1996), "Sample Quantiles in Statistical Packages," The American Statistician, 50, 361-365.
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.
See Also
approx, diff, function, median, ppoints, qqplot, rank.
Examples

    quantile(1:9)
    # 0%  25%  50%  75% 100%
    #  1    3    5    7    9
    quantile(1:9, probs=seq(0,1,.1))
    #  0%  10%  20%  30%  40%  50%  60%  70%  80%  90% 100%
    # 1.0  1.8  2.6  3.4  4.2  5.0  5.8  6.6  7.4  8.2  9.0
    quantile(1:9, probs=c(0.1,0.3,NA,NA))
    # 10% 30%
    # 1.8 3.4  NA  NA

Package stats version 6.1.5-18
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