stats.e

Preparing the data

Consequently in order to use these functions you need to arrange your data in an array, within a sequence. Sometimes, however, your data may be in the form of a frequency distribution - pairs of values and frequencies. In this case you can easily use Euphoria's 'repeat' function and the concatenation operator '&' to create the right form of input.

N.B. No provision is made in this library for 'missing values'. If you wish to use these functions then you must remove such data from your sequence.

Most functions operate on integer or real data values without problem. Some functions, however, have two forms:

• one for integer data, and
• one for real data.

The function frequency can handle string-type data as well as numeric data.

Moments

A number of the functions in this library are defined using moments. Mainly for this reason all the functions are based on 'n' (the number of observations) and not on 'n-1' (for sample-bound estimates). If you wish to record such estimates then you can easily make the simple calculation adjustment before reporting your results.

Library specification

Version: 1.2.1

Interface

Globally-defined includes

• sort.e - for function sort

Routines:

All these functions take, as the parameter, a sequence consisting of atoms in the form of a one-dimensional array. A vector type has been defined with the library to enable type-checking to take place when each function is called.

• [function] average(vector this) - calculates the first moment, centred on zero.
  The return value is the (real) arithmetic average of the values in the vector.
• [function] count(vector this) - synonym for Euphoria's length function
  The return value is the (real) number of values in the vector.
• [function] kurtosis(vector this) - calculates the kurtosis of the distribution in the vector. This is based the fourth moment about the arithmetic mean. The kurtosis of the distribution is defined as the 'standardised' fourth moment, that is, the fourth moment divided by the fourth power of the standard deviation.
  Because the kurtosis of the normal distribution is 3, some authors subtract the 3 before reporting the value. If you wish to use this convention then simply subtract three from the return value of this function.
  The return value is the (real) kurtosis of the values in the vector.
• [function] maximum(vector this) - calculates the maximum value in the vector.
  The return value is the (real) maximum.
• [function] median(vector this) - calculates the median value in the vector.
  If there are an even number then we take the average of the two "middle" ones.
  The return value is the (real) median.
• [function] minimum(vector this) - calculates the minimum value in the vector.
  The return value is the (real) minimum.
• [function] quartile(vector this) Calculates the qth (1st, 2nd or 3rd) quartile of the vector.
  If there are an even number then we take the average of the two adjacent ones.
  The return value is the (real) quartile.
• [function] range(vector this) - calculates the difference between the maximum and minimum value in the vector.
  The return value is the (real) range.
• [function] skewness(vector this) - calculates the skewness of the distribution in the vector. This is based the third /moment about the arithmetic mean. The skewness of the distribution is defined as the 'standardised' third moment, that is, the third moment divided by the cube of the standard deviation.
  The return value is the (real) skewness coefficient of the values in the vector.
  Assumes no missing values.
• [function] sum(vector this) - calculates the sum of the values in the called vector.
  The return value is the (real) sum of values in the vector.
• [function] variance(vector this) - calculates the second moment of the distribution, about the arithmetic mean.
  The return value is the (real) variance of the values in the vector.

This function take, as the parameter, a sequence consisting of objects in the form of a vector.

• [function] frequency(sequence this) - determines the frequency distribution of the vector.
  The return value is a tree containing counts.
  Each node contains the value, its count and the pointers to its neighbours.
  Will handle any kind of values in the vector.
• [procedure] printTree(sequence node) - displays the counts in a tree structure.
• [procedure] zeroiseRoot() - zeroises a tree structure (if more than one frequency Tree is to be analysed.

N.B. All these functions take, as parameters, two sequences consisting of atoms in the form of a vector. A vector type has been defined with the library to enable type-checking to take place when each function is called. The data are analysed pair-wise: that is, the first value in vector1 is associated with the first value in vector2, etc.

• [function] anova(vector this, vector class) - calculates the one-way analysis of variance of the array, given the class groupings.
  The return value is a sequence containing the following arrays:
  • the sum of squares due to methods (classification) and error
  • the corresponding degrees of freedom
  The total sum of squares is the sum of the values in the first array and the total degrees of freedom are the sum of the values in the second array.
• [function] correlation(vector this, vector that) - calculates the correlation coefficient of the paired values in the two vectors.
  The return value is the (real) correlation of the values in the vectors.
• [function] covariance(vector this, vector that) - calculates the covariance of the paired values in the two vectors.
  The return value is the (real) covariance of the values in the vectors.
• [function] regression(vector this,vector that) - calculates the simple linear regression coefficient of the values in this given the values in that.
  The return value is the (real) simple linear regression coefficient.
• [function] t_test(vector this,vector that) - calculates the t-test for the difference between the means of the two vectors.
  The return value is a sequence giving:
  • the (real) t statistic, with sign
  • a sequence containing the degrees of freedom (1,df)
  The statistical significance of the result can be looked up in the tables in the library F.e.

Instances where there are separate functions for integer and real data.

	Distributions	Modal value[s]
Integer	distribution	mode
Real	real_distribution	real_mode

Integer functions

• [function] distribution(vector this) - determines the frequency distribution of the vector.
  The base unit is relative to the minimum.
  The return value is a sequence with counts. It consists of the base unit, followed by all the counts.
• [function] mode(vector this) - determines the modal value(s) of the frequency distribution of the vector.
  The return value is a sequence with one or more values.

Real functions

• [function] real_distribution(vector this,integer interval) - determines the frequency distribution of the vector.
  The base unit is relative to the minimum.
  The intervals used in the distribution are determined by the second parameter - starting with the integer immediately less than the minimum value.
  The return value is a sequence with counts. Its value consists of the base unit, followed by all the counts.
  NB. The base will always be an integer.
• [function] real_mode(vector this,integer interval) - determines the modal value(s) of the frequency distribution of the vector.
  The return value is a sequence with one or more values.
  Assumes that the values in the array will be grouped into intervals - of size 'interval'.