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Colin Kuskie

# NAME

Statistics::Descriptive - Module of basic descriptive statistical functions.

# SYNOPSIS

``````  use Statistics::Descriptive;
\$stat = Statistics::Descriptive::Full->new();
\$var  = \$stat->variance();
\$tm   = \$stat->trimmed_mean(.25);
\$Statistics::Descriptive::Tolerance = 1e-10;``````

# DESCRIPTION

This module provides basic functions used in descriptive statistics. It has an object oriented design and supports two different types of data storage and calculation objects: sparse and full. With the sparse method, none of the data is stored and only a few statistical measures are available. Using the full method, the entire data set is retained and additional functions are available.

Whenever a division by zero may occur, the denominator is checked to be greater than the value `\$Statistics::Descriptive::Tolerance`, which defaults to 0.0. You may want to change this value to some small positive value such as 1e-24 in order to obtain error messages in case of very small denominators.

Many of the methods (both Sparse and Full) cache values so that subsequent calls with the same arguments are faster.

# METHODS

## Sparse Methods

\$stat = Statistics::Descriptive::Sparse->new();

Create a new sparse statistics object.

Adds data to the statistics variable. The cached statistical values are updated automatically.

\$stat->count();

Returns the number of data items.

\$stat->mean();

Returns the mean of the data.

\$stat->sum();

Returns the sum of the data.

\$stat->variance();

Returns the variance of the data. Division by n-1 is used.

\$stat->standard_deviation();

Returns the standard deviation of the data. Division by n-1 is used.

\$stat->min();

Returns the minimum value of the data set.

\$stat->mindex();

Returns the index of the minimum value of the data set.

\$stat->max();

Returns the maximum value of the data set.

\$stat->maxdex();

Returns the index of the maximum value of the data set.

\$stat->sample_range();

Returns the sample range (max - min) of the data set.

## Full Methods

Similar to the Sparse Methods above, any Full Method that is called caches the current result so that it doesn't have to be recalculated. In some cases, several values can be cached at the same time.

\$stat = Statistics::Descriptive::Full->new();

Create a new statistics object that inherits from Statistics::Descriptive::Sparse so that it contains all the methods described above.

Adds data to the statistics variable. All of the sparse statistical values are updated and cached. Cached values from Full methods are deleted since they are no longer valid.

Note: Calling add_data with an empty array will delete all of your Full method cached values! Cached values for the sparse methods are not changed

\$stat->get_data();

Returns a copy of the data array.

\$stat->sort_data();

Sort the stored data and update the mindex and maxdex methods. This method uses perl's internal sort.

\$stat->presorted(1);
\$stat->presorted();

If called with a non-zero argument, this method sets a flag that says the data is already sorted and need not be sorted again. Since some of the methods in this class require sorted data, this saves some time. If you supply sorted data to the object, call this method to prevent the data from being sorted again. The flag is cleared whenever add_data is called. Calling the method without an argument returns the value of the flag.

\$x = \$stat->percentile(25);
(\$x, \$index) = \$stat->percentile(25);

Sorts the data and returns the value that corresponds to the percentile as defined in RFC2330:

• For example, given the 6 measurements:

-2, 7, 7, 4, 18, -5

Then F(-8) = 0, F(-5) = 1/6, F(-5.0001) = 0, F(-4.999) = 1/6, F(7) = 5/6, F(18) = 1, F(239) = 1.

Note that we can recover the different measured values and how many times each occurred from F(x) -- no information regarding the range in values is lost. Summarizing measurements using histograms, on the other hand, in general loses information about the different values observed, so the EDF is preferred.

Using either the EDF or a histogram, however, we do lose information regarding the order in which the values were observed. Whether this loss is potentially significant will depend on the metric being measured.

We will use the term "percentile" to refer to the smallest value of x for which F(x) >= a given percentage. So the 50th percentile of the example above is 4, since F(4) = 3/6 = 50%; the 25th percentile is -2, since F(-5) = 1/6 < 25%, and F(-2) = 2/6 >= 25%; the 100th percentile is 18; and the 0th percentile is -infinity, as is the 15th percentile.

Care must be taken when using percentiles to summarize a sample, because they can lend an unwarranted appearance of more precision than is really available. Any such summary must include the sample size N, because any percentile difference finer than 1/N is below the resolution of the sample.

(Taken from: RFC2330 - Framework for IP Performance Metrics, Section 11.3. Defining Statistical Distributions. RFC2330 is available from: http://www.cis.ohio-state.edu/htbin/rfc/rfc2330.html.)

If the percentile method is called in a list context then it will also return the index of the percentile.

\$stat->median();

Sorts the data and returns the median value of the data.

\$stat->harmonic_mean();

Returns the harmonic mean of the data. Since the mean is undefined if any of the data are zero or if the sum of the reciprocals is zero, it will return undef for both of those cases.

\$stat->geometric_mean();

Returns the geometric mean of the data.

\$stat->mode();

Returns the mode of the data.

\$stat->trimmed_mean(ltrim[,utrim]);

`trimmed_mean(ltrim)` returns the mean with a fraction `ltrim` of entries at each end dropped. `trimmed_mean(ltrim,utrim)` returns the mean after a fraction `ltrim` has been removed from the lower end of the data and a fraction `utrim` has been removed from the upper end of the data. This method sorts the data before beginning to analyze it.

All calls to trimmed_mean() are cached so that they don't have to be calculated a second time.

\$stat->frequency_distribution(\$partitions);
\$stat->frequency_distribution(\@bins);
\$stat->frequency_distribution();

`frequency_distribution(\$partitions)` slices the data into `\$partition` sets (where \$partition is greater than 1) and counts the number of items that fall into each partition. It returns an associative array where the keys are the numerical values of the partitions used. The minimum value of the data set is not a key and the maximum value of the data set is always a key. The number of entries for a particular partition key are the number of items which are greater than the previous partition key and less then or equal to the current partition key. As an example,

``````   \$stat->add_data(1,1.5,2,2.5,3,3.5,4);
%f = \$stat->frequency_distribution(2);
for (sort {\$a <=> \$b} keys %f) {
print "key = \$_, count = \$f{\$_}\n";
}``````

prints

``````   key = 2.5, count = 4
key = 4, count = 3``````

since there are four items less than or equal to 2.5, and 3 items greater than 2.5 and less than 4.

`frequency_distribution(\@bins)` provides the bins that are to be used for the distribution. This allows for non-uniform distributions as well as trimmed or sample distributions to be found. `@bins` must be monotonic and contain at least one element. Note that unless the set of bins contains the range that the total counts returned will be less than the sample size.

Calling `frequency_distribution()` with no arguments returns the last distribution calculated, if such exists.

\$stat->least_squares_fit();
\$stat->least_squares_fit(@x);

`least_squares_fit()` performs a least squares fit on the data, assuming a domain of `@x` or a default of 1..\$stat->count(). It returns an array of four elements `(\$q, \$m, \$r, \$rms)` where

`\$q and \$m`

satisfy the equation C(\$y = \$m*\$x + \$q).

`\$r`

is the Pearson linear correlation cofficient.

`\$rms`

is the root-mean-square error.

If case of error or division by zero, the empty list is returned.

The array that is returned can be "coerced" into a hash structure by doing the following:

``````  my %hash = ();
@hash{'q', 'm', 'r', 'err'} = \$stat->least_squares_fit();``````

Because calling `least_squares_fit()` with no arguments defaults to using the current range, there is no caching of the results.

# REPORTING ERRORS

I read my email frequently, but since adopting this module I've added 2 children and 1 dog to my family, so please be patient about my response times. When reporting errors, please include the following to help me out:

• Your version of perl. This can be obtained by typing perl `-v` at the command line.

• Which version of Statistics::Descriptive you're using. As you can see below, I do make mistakes. Unfortunately for me, right now there are thousands of CD's with the version of this module with the bugs in it. Fortunately for you, I'm a very patient module maintainer.

• Details about what the error is. Try to narrow down the scope of the problem and send me code that I can run to verify and track it down.

# AUTHOR

Colin Kuskie

My email address can be found at http://www.perl.com under Who's Who or at: http://search.cpan.org/author/COLINK/.

# REFERENCES

RFC2330, Framework for IP Performance Metrics

The Art of Computer Programming, Volume 2, Donald Knuth.

Handbook of Mathematica Functions, Milton Abramowitz and Irene Stegun.

Probability and Statistics for Engineering and the Sciences, Jay Devore.

Copyright (c) 1997,1998 Colin Kuskie. All rights reserved. This program is free software; you can redistribute it and/or modify it under the same terms as Perl itself.

Copyright (c) 1998 Andrea Spinelli. All rights reserved. This program is free software; you can redistribute it and/or modify it under the same terms as Perl itself.

Copyright (c) 1994,1995 Jason Kastner. All rights reserved. This program is free software; you can redistribute it and/or modify it under the same terms as Perl itself.

# REVISION HISTORY

v2.3

Rolled into November 1998

Code provided by Andrea Spinelli to prevent division by zero and to make consistent return values for undefined behavior. Andrea also provided a test bench for the module.

A bug fix for the calculation of frequency distributions. Thanks to Nick Tolli for alerting this to me.

Added 4 lines of code to Makefile.PL to make it easier for the ActiveState installation tool to use. Changes work fine in perl5.004_04, haven't tested them under perl5.005xx yet.

v2.2

Rolled into March 1998.

Fixed problem with sending 0's and -1's as data. The old 0 : true ? false thing. Use defined to fix.

Provided a fix for AUTOLOAD/DESTROY/Carp bug. Very strange.

v2.1

August 1997

Fixed errors in statistics algorithms caused by changing the interface.

v2.0

August 1997

Fixed errors in removing cached values (they weren't being removed!) and added sort_data and presorted methods.

June 1997

Transferred ownership of the module from Jason to Colin.

Rewrote OO interface, modified function distribution, added mindex, maxdex.

v1.1

April 1995

v1.0

March 1995

Released to comp.lang.perl and placed on archive sites.

v.20

December 1994

Complete rewrite after extensive and invaluable e-mail correspondence with Anno Siegel.

v.10

December 1994

Initital concept, released to perl5-porters list.

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