C Library Functions  - bds (3)

NAME
bds(3f) - [M_math:statistics] Basic Statistical Measures

CONTENTS

Synopsis
Description
Options
Returns
Definitions
Examples

SYNOPSIS

subroutine bds(x,n,stat) 

    real,intent(in)    :: x(n)
    integer,intent(in) :: n
    real,intent(out)   :: STAT(13)

DESCRIPTION

Given a vector of real values calculate common basic statistical measures for the array.

OPTIONS
x REAL input vector of values to determine statistical properties for
n size of input vector

RETURNS
stat array of statistical measurements calculated
1. mean
2. second moment about the mean
3. third moment about the mean
4. fourth moment about the mean
5. variance
6. standard deviation
7. skewness
8. kurtosis
9. sum
10. largest value
11. smallest value
12. location of largest value
13. location of smallest value

DEFINITIONS

    MEAN

A type of average, calculated by dividing the sum of a set of values by the number of values. 

             mean = Sum(Xi)/N

    MEDIAN

A type of average, found by arranging the values in order and then selecting the one in the middle. If the total number of values in the sample is even, then the median is the mean of the two middle numbers.

    MODE

The most frequent value in a group of values.

    VARIANCE

The average of the square of the distance of each data point from the mean 

             variance = Sum((Xi-mean)^2))/N
for a population, or more commonly, for a sample the unbiased value is 

             variance = Sum((Xi-mean)^2))/(N-1)

    STANDARD DEVIATION

The standard deviation is the square root of the variance. 

             sd = sqrt(variance)
It is the most commonly used measure of spread.

    SKEWNESS

Skewness is a measure of symmetry, or more precisely, the lack of symmetry. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point. 

             skewness = Sum{(X(i)-mean)^3} /((N-1)*SD^3)
Where SD is the standard deviation, and N is the number of samples. Some sources will use N instead of N-1 or they might present the formula in a slightly different mathematically equivalent format.

The skewness of symmetric data is zero

    KURTOSIS

Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. That is, data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak. A uniform distribution would be the extreme case. 

             kurtosis = ( SUM{(X(i)-mean)^4} ) / ((N-1)*SD^4) -3
The standard normal distribution has a kurtosis of zero. Positive kurtosis indicates a "peaked" distribution and negative kurtosis indicates a "flat" distribution.

Although often called kurtosis, historically the above expression is for "excess kurtosis" because three is subtracted from the value. The purpose of this is to give the normal distribution a kurtosis of 0. In recent times, the term "excess kurtosis" is often simply called "kurtosis", so consider that whether to subtract 3 or not is merely a convention, not a right or wrong answer. When using a particular program, you just need to be aware of which convention.

Again, another frequent difference is whether they use N in the denominator or the bias corrected N-1.

The formulas for skewness and kurtosis are treated in 

             Sokal, R. R., & Rohlf, F. J. (1995).
             Biometry: The principles and practice of statistics in biological
             research. (3rd ed.) New York: W. H. Freeman, pp. 114-115.
Similar formulas are given in 

             Zar, J. H., Biostatistical analysis (3rd ed)., Prentice
             Hall, 1996.
which refers to "machine formulas" and cites 

             Bennett and Franklin, Statistical analysis in chemistry and
             the chemical industry. NY: Wiley, 1954, at p. 81.

EXAMPLES

Nemo Release 3.1 bds (3) February 23, 2025
Generated by manServer 1.08 from 636edbb6-0562-464e-9519-d51c4f7b71cc using man macros.