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### 25.2 Basic Statistical Functions

Octave also supports various helpful statistical functions.

— Function File: mahalanobis (x, y)

> Return the Mahalanobis' D-square distance between the multivariate samples x and y, which must have the same number of components (columns), but may have a different number of observations (rows).

— Function File: center (x)
— Function File: center (x, dim)

> If x is a vector, subtract its mean. If x is a matrix, do the above for each column. If the optional argument dim is given, perform the above operation along this dimension

— Function File: studentize (x, dim)

> If x is a vector, subtract its mean and divide by its standard deviation.

> If x is a matrix, do the above along the first non-singleton dimension. If the optional argument dim is given then operate along this dimension.

— Function File: c = nchoosek (n, k)

> Compute the binomial coefficient or all combinations of n. If n is a scalar then, calculate the binomial coefficient of n and k, defined as

>                /   \
>                | n |    n (n-1) (n-2) ... (n-k+1)       n!
>                |   |  = ------------------------- =  ---------
>                | k |               k!                k! (n-k)!
>                \   /

> If n is a vector generate all combinations of the elements of n, taken k at a time, one row per combination. The resulting c has size `[nchoosek (length (`n`),` k`),` k`]`.

> `nchoosek` works only for nonnegative integer arguments; use `bincoeff` for non-integer scalar arguments and for using vector arguments to compute many coefficients at once.

> See also: bincoeff.

— Function File: perms (v)

> Generate all permutations of v, one row per permutation. The result has size `factorial (`n`) *` n, where n is the length of v.

> As an example, `perms([1, 2, 3])` returns the matrix

>                 1   2   3
>                 2   1   3
>                 1   3   2
>                 2   3   1
>                 3   1   2
>                 3   2   1

— Function File: values (x)

> Return the different values in a column vector, arranged in ascending order.

> As an example, `values([1, 2, 3, 1])` returns the vector `[1, 2, 3]`.

— Function File: [t, l_x] = table (x)
— Function File: [t, l_x, l_y] = table (x, y)

> Create a contingency table t from data vectors. The l vectors are the corresponding levels.

> Currently, only 1- and 2-dimensional tables are supported.

— Function File: spearman (x, y)

> Compute Spearman's rank correlation coefficient rho for each of the variables specified by the input arguments.

> For matrices, each row is an observation and each column a variable; vectors are always observations and may be row or column vectors.

> `spearman (`x`)` is equivalent to `spearman (`x`,` x`)`.

> For two data vectors x and y, Spearman's rho is the correlation of the ranks of x and y.

> If x and y are drawn from independent distributions, rho has zero mean and variance `1 / (n - 1)`, and is asymptotically normally distributed.

— Function File: run_count (x, n)

> Count the upward runs along the first non-singleton dimension of x of length 1, 2, ..., n-1 and greater than or equal to n. If the optional argument dim is given operate along this dimension

— Function File: ranks (x, dim)

> Return the ranks of x along the first non-singleton dimension adjust for ties. If the optional argument dim is given, operate along this dimension.

— Function File: range (x)
— Function File: range (x, dim)

> If x is a vector, return the range, i.e., the difference between the maximum and the minimum, of the input data.

> If x is a matrix, do the above for each column of x.

> If the optional argument dim is supplied, work along dimension dim.

— Function File: probit (p)

> For each component of p, return the probit (the quantile of the standard normal distribution) of p.

— Function File: logit (p)

> For each component of p, return the logit of p defined as

>               logit(p) = log (p / (1-p))

— Function File: cloglog (x)

> Return the complementary log-log function of x, defined as

>               cloglog(x) = - log (- log (x))

— Function File: kendall (x, y)

> Compute Kendall's tau for each of the variables specified by the input arguments.

> For matrices, each row is an observation and each column a variable; vectors are always observations and may be row or column vectors.

> `kendall (`x`)` is equivalent to `kendall (`x`,` x`)`.

> For two data vectors x, y of common length n, Kendall's tau is the correlation of the signs of all rank differences of x and y; i.e., if both x and y have distinct entries, then

>                        1
>               tau = -------   SUM sign (q(i) - q(j)) * sign (r(i) - r(j))
>                     n (n-1)   i,j

> in which the q(i) and r(i) are the ranks of x and y, respectively.

> If x and y are drawn from independent distributions, Kendall's tau is asymptotically normal with mean 0 and variance `(2 * (2` n`+5)) / (9 *` n`* (`n`-1))`.

— Function File: iqr (x, dim)

> If x is a vector, return the interquartile range, i.e., the difference between the upper and lower quartile, of the input data.

> If x is a matrix, do the above for first non-singleton dimension of x. If the option dim argument is given, then operate along this dimension.

— Function File: cut (x, breaks)

> Create categorical data out of numerical or continuous data by cutting into intervals.

> If breaks is a scalar, the data is cut into that many equal-width intervals. If breaks is a vector of break points, the category has `length (`breaks`) - 1` groups.

> The returned value is a vector of the same size as x telling which group each point in x belongs to. Groups are labelled from 1 to the number of groups; points outside the range of breaks are labelled by `NaN`.