These functions do various statistical computations on single vectors. Given a numeric prefix argument, they actually pop n objects from the stack and combine them into a data vector. Each object may be either a number or a vector; if a vector, any sub-vectors inside it are “flattened” as if by v a 0; see Manipulating Vectors. By default one object is popped, which (in order to be useful) is usually a vector.
If an argument is a variable name, and the value stored in that variable is a vector, then the stored vector is used. This method has the advantage that if your data vector is large, you can avoid the slow process of manipulating it directly on the stack.
These functions are left in symbolic form if any of their arguments are not numbers or vectors, e.g., if an argument is a formula, or a non-vector variable. However, formulas embedded within vector arguments are accepted; the result is a symbolic representation of the computation, based on the assumption that the formula does not itself represent a vector. All varieties of numbers such as error forms and interval forms are acceptable.
Some of the functions in this section also accept a single error form or interval as an argument. They then describe a property of the normal or uniform (respectively) statistical distribution described by the argument. The arguments are interpreted in the same way as the M argument of the random number function k r. In particular, an interval with integer limits is considered an integer distribution, so that ‘[2 .. 6)’ is the same as ‘[2 .. 5]’. An interval with at least one floating-point limit is a continuous distribution: ‘[2.0 .. 6.0)’ is not the same as ‘[2.0 .. 5.0]’!
The u #
(calc-vector-count) [vcount] command
computes the number of data values represented by the inputs. For
example, ‘vcount(1, [2, 3], [[4,
5], [], x, y])’ returns 7. If the argument is
a single vector with no sub-vectors, this simply computes the
length of the vector.
The u
+ (calc-vector-sum) [vsum]
command computes the sum of the data values. The u *
(calc-vector-prod) [vprod] command
computes the product of the data values. If the input is a single
flat vector, these are the same as V R + and V R
* (see Reducing and
Mapping).
The u X
(calc-vector-max) [vmax] command
computes the maximum of the data values, and the u N
(calc-vector-min) [vmin] command
computes the minimum. If the argument is an interval, this finds
the minimum or maximum value in the interval. (Note that
‘vmax([2..6)) =
5’ as described above.) If the argument is an
error form, this returns plus or minus infinity.
The u M
(calc-vector-mean) [vmean] command
computes the average (arithmetic mean) of the data values. If the
inputs are error forms
‘x +/- s’,
this is the weighted mean of the ‘x’ values with weights
‘1 / s^2’. If
the inputs are not error forms, this is simply the sum of the
values divided by the count of the values.
Note that a plain number can be considered an error form with error ‘s = 0’. If the input to u M is a mixture of plain numbers and error forms, the result is the mean of the plain numbers, ignoring all values with non-zero errors. (By the above definitions it's clear that a plain number effectively has an infinite weight, next to which an error form with a finite weight is completely negligible.)
This function also works for distributions (error forms or intervals). The mean of an error form `a +/- b' is simply ‘a’. The mean of an interval is the mean of the minimum and maximum values of the interval.
The I u
M (calc-vector-mean-error)
[vmeane] command computes the mean of the data
points expressed as an error form. This includes the estimated
error associated with the mean. If the inputs are error forms,
the error is the square root of the reciprocal of the sum of the
reciprocals of the squares of the input errors. (I.e., the
variance is the reciprocal of the sum of the reciprocals of the
variances.) If the inputs are plain numbers, the error is equal
to the standard deviation of the values divided by the square
root of the number of values. (This works out to be equivalent to
calculating the standard deviation and then assuming each value's
error is equal to this standard deviation.)
The H u M
(calc-vector-median) [vmedian] command
computes the median of the data values. The values are first
sorted into numerical order; the median is the middle value after
sorting. (If the number of data values is even, the median is
taken to be the average of the two middle values.) The median
function is different from the other functions in this section in
that the arguments must all be real numbers; variables are not
accepted even when nested inside vectors. (Otherwise it is not
possible to sort the data values.) If any of the input values are
error forms, their error parts are ignored.
The median function also accepts distributions. For both normal (error form) and uniform (interval) distributions, the median is the same as the mean.
The
H I u M (calc-vector-harmonic-mean)
[vhmean] command computes the harmonic mean of the
data values. This is defined as the reciprocal of the arithmetic
mean of the reciprocals of the values.
The u G
(calc-vector-geometric-mean) [vgmean]
command computes the geometric mean of the data values. This is
the nth root of the product of the values. This is
also equal to the exp of the arithmetic mean of the
logarithms of the data values.
The H u
G [agmean] command computes the
“arithmetic-geometric mean” of two numbers taken from
the stack. This is computed by replacing the two numbers with
their arithmetic mean and geometric mean, then repeating until
the two values converge.
Another commonly used mean, the RMS (root-mean-square), can be computed for a vector of numbers simply by using the A command.
The u S
(calc-vector-sdev) [vsdev] command
computes the standard
deviation of the data values. If the values are error forms, the
errors are used as weights just as for u M. This is
the sample standard deviation, whose value is the square
root of the sum of the squares of the differences between the
values and the mean of the ‘N’ values, divided by
‘N-1’.
This function also applies to distributions. The standard deviation of a single error form is simply the error part. The standard deviation of a continuous interval happens to equal the difference between the limits, divided by ‘sqrt(12)’. The standard deviation of an integer interval is the same as the standard deviation of a vector of those integers.
The I u S
(calc-vector-pop-sdev) [vpsdev] command
computes the population standard deviation. It is
defined by the same formula as above but dividing by
‘N’ instead of
by ‘N-1’. The
population standard deviation is used when the input represents
the entire set of data values in the distribution; the sample
standard deviation is used when the input represents a sample of
the set of all data values, so that the mean computed from the
input is itself only an estimate of the true mean.
For error forms and continuous intervals, vpsdev
works exactly like vsdev. For integer intervals, it
computes the population standard deviation of the equivalent
vector of integers.
The H u S
(calc-vector-variance) [vvar] and
H I u S (calc-vector-pop-variance)
[vpvar] commands compute the variance of the data
values. The variance is the
square of the standard deviation, i.e., the sum of the squares
of the deviations of the data values from the mean. (This
definition also applies when the argument is a distribution.)
The
vflat algebraic function returns a vector of its
arguments, interpreted in the same way as the other functions in
this section. For example, ‘vflat(1, [2, [3, 4]], 5)’ returns
‘[1, 2, 3, 4,
5]’.