PEP 450 -- Adding A Statistics Module To The Standard Library
PEP: 450 Adding A Statistics Module To The Standard Library \$Revision\$ \$Date\$ Steven D'Aprano Final Standards Track text/plain 01-Aug-2013 3.4 13-Sep-2013

### Abstract

```    This PEP proposes the addition of a module for common statistics functions
such as mean, median, variance and standard deviation to the Python

```

### Rationale

```    The proposed statistics module is motivated by the "batteries included"
philosophy towards the Python standard library.  Raymond Hettinger and
other senior developers have requested a quality statistics library that
falls somewhere in between high-end statistics libraries and ad hoc
code.[1]  Statistical functions such as mean, standard deviation and others
are obvious and useful batteries, familiar to any Secondary School student.
Even cheap scientific calculators typically include multiple statistical
functions such as:

- mean
- population and sample variance
- population and sample standard deviation
- linear regression
- correlation coefficient

Graphing calculators aimed at Secondary School students typically
include all of the above, plus some or all of:

- median
- mode
- functions for calculating the probability of random variables
from the normal, t, chi-squared, and F distributions
- inference on the mean

and others[2].  Likewise spreadsheet applications such as Microsoft Excel,
LibreOffice and Gnumeric include rich collections of statistical
functions[3].

In contrast, Python currently has no standard way to calculate even the
simplest and most obvious statistical functions such as mean.  For those
who need statistical functions in Python, there are two obvious solutions:

- install numpy and/or scipy[4];

- or use a Do It Yourself solution.

Numpy is perhaps the most full-featured solution, but it has a few

- It may be overkill for many purposes.  The documentation for numpy even
warns

"It can be hard to know what functions are available in
numpy.  This is not a complete list, but it does cover
most of them."[5]

and then goes on to list over 270 functions, only a small number of
which are related to statistics.

- Numpy is aimed at those doing heavy numerical work, and may be
intimidating to those who don't have a background in computational
mathematics and computer science.  For example, numpy.mean takes four
arguments:

mean(a, axis=None, dtype=None, out=None)

although fortunately for the beginner or casual numpy user, three are
optional and numpy.mean does the right thing in simple cases:

>>>  numpy.mean([1, 2, 3, 4])
2.5

- For many people, installing numpy may be difficult or impossible.  For
example, people in corporate environments may have to go through a
difficult, time-consuming process before being permitted to install
third-party software.  For the casual Python user, having to learn about
installing third-party packages in order to average a list of numbers is
unfortunate.

This leads to option number 2, DIY statistics functions.  At first glance,
this appears to be an attractive option, due to the apparent simplicity of
common statistical functions.  For example:

def mean(data):
return sum(data)/len(data)

def variance(data):
# Use the Computational Formula for Variance.
n = len(data)
ss = sum(x**2 for x in data) - (sum(data)**2)/n
return ss/(n-1)

def standard_deviation(data):
return math.sqrt(variance(data))

The above appears to be correct with a casual test:

>>> data = [1, 2, 4, 5, 8]
>>> variance(data)
7.5

But adding a constant to every data point should not change the variance:

>>> data = [x+1e12 for x in data]
>>> variance(data)
0.0

And variance should *never* be negative:

>>> variance(data*100)
-1239429440.1282566

By contrast, the proposed reference implementation gets the exactly correct
answer 7.5 for the first two examples, and a reasonably close answer for
the third: 6.012. numpy does no better[6].

Even simple statistical calculations contain traps for the unwary, starting
with the Computational Formula itself.  Despite the name, it is numerically
unstable and can be extremely inaccurate, as can be seen above.  It is
completely unsuitable for computation by computer[7].  This problem plagues
users of many programming language, not just Python[8], as coders reinvent
the same numerically inaccurate code over and over again[9], or advise
others to do so[10].

It isn't just the variance and standard deviation. Even the mean is not
quite as straightforward as it might appear.  The above implementation
seems too simple to have problems, but it does:

- The built-in sum can lose accuracy when dealing with floats of wildly
differing magnitude.  Consequently, the above naive mean fails this
"torture test":

assert mean([1e30, 1, 3, -1e30]) == 1

returning 0 instead of 1, a purely computational error of 100%.

- Using math.fsum inside mean will make it more accurate with float data,
but it also has the side-effect of converting any arguments to float
even when unnecessary.  E.g. we should expect the mean of a list of
Fractions to be a Fraction, not a float.

While the above mean implementation does not fail quite as catastrophically
as the naive variance does, a standard library function can do much better
than the DIY versions.

The example above involves an especially bad set of data, but even for
more realistic data sets accuracy is important.  The first step in
interpreting variation in data (including dealing with ill-conditioned
data) is often to standardize it to a series with variance 1 (and often
mean 0).  This standardization requires accurate computation of the mean
and variance of the raw series.  Naive computation of mean and variance
can lose precision very quickly.  Because precision bounds accuracy, it is
important to use the most precise algorithms for computing mean and
variance that are practical, or the results of standardization are
themselves useless.

```

### Comparison To Other Languages/Packages

```    The proposed statistics library is not intended to be a competitor to such
third-party libraries as numpy/scipy, or of proprietary full-featured
statistics packages aimed at professional statisticians such as Minitab,
SAS and Matlab.  It is aimed at the level of graphing and scientific
calculators.

Most programming languages have little or no built-in support for
statistics functions.  Some exceptions:

R
R (and its proprietary cousin, S) is a programming language designed
for statistics work. It is extremely popular with statisticians and
is extremely feature-rich[11].

C#

The C# LINQ package includes extension methods to calculate the
average of enumerables[12].

Ruby

Ruby does not ship with a standard statistics module, despite some
apparent demand[13].  Statsample appears to be a feature-rich third-
party library, aiming to compete with R[14].

PHP

PHP has an extremely feature-rich (although mostly undocumented) set

Delphi

Delphi includes standard statistical functions including Mean, Sum,
Variance, TotalVariance, MomentSkewKurtosis in its Math library[16].

GNU Scientific Library

The GNU Scientific Library includes standard statistical functions,
percentiles, median and others[17].  One innovation I have borrowed
from the GSL is to allow the caller to optionally specify the pre-
calculated mean of the sample (or an a priori known population mean)
when calculating the variance and standard deviation[18].

```

### Design Decisions Of The Module

```    My intention is to start small and grow the library as needed, rather than
try to include everything from the start. Consequently, the current
reference implementation includes only a small number of functions: mean,
variance, standard deviation, median, mode. (See the reference
implementation for a full list.)

I have aimed for the following design features:

- Correctness over speed.  It is easier to speed up a correct but slow
function than to correct a fast but buggy one.

- Concentrate on data in sequences, allowing two-passes over the data,
rather than potentially compromise on accuracy for the sake of a one-pass
algorithm.  Functions expect data will be passed as a list or other
sequence; if given an iterator, they may internally convert to a list.

- Functions should, as much as possible, honour any type of numeric data.
E.g. the mean of a list of Decimals should be a Decimal, not a float.
When this is not possible, treat float as the "lowest common data type".

- Although functions support data sets of floats, Decimals or Fractions,
there is no guarantee that *mixed* data sets will be supported. (But on
the other hand, they aren't explicitly rejected either.)

- Plenty of documentation, aimed at readers who understand the basic
concepts but may not know (for example) which variance they should use
(population or sample?). Mathematicians and statisticians have a terrible
habit of being inconsistent with both notation and terminology[19], and
having spent many hours making sense of the contradictory/confusing
definitions in use, it is only fair that I do my best to clarify rather
than obfuscate the topic.

- But avoid going into tedious[20] mathematical detail.

```

### API

```    The initial version of the library will provide univariate (single
variable) statistics functions.  The general API will be based on a
functional model ``function(data, ...) -> result``, where ``data``
is a mandatory iterable of (usually) numeric data.

The author expects that lists will be the most common data type used,
but any iterable type should be acceptable.  Where necessary, functions
may convert to lists internally.  Where possible, functions are
expected to conserve the type of the data values, for example, the mean
of a list of Decimals should be a Decimal rather than float.

Calculating mean, median and mode

The ``mean``, ``median*`` and ``mode`` functions take a single
mandatory argument and return the appropriate statistic, e.g.:

>>> mean([1, 2, 3])
2.0

Functions provided are:

* mean(data) -> arithmetic mean of data.

* median(data) -> median (middle value) of data, taking the
average of the two middle values when there are an even
number of values.

* median_high(data) -> high median of data, taking the
larger of the two middle values when the number of items
is even.

* median_low(data) -> low median of data, taking the smaller
of the two middle values when the number of items is even.

* median_grouped(data, interval=1) -> 50th percentile of
grouped data, using interpolation.

* mode(data) -> most common data point.

``mode`` is the sole exception to the rule that the data argument
must be numeric.  It will also accept an iterable of nominal data,
such as strings.

Calculating variance and standard deviation

In order to be similar to scientific calculators, the statistics
module will include separate functions for population and sample
variance and standard deviation.  All four functions have similar
signatures, with a single mandatory argument, an iterable of
numeric data, e.g.:

>>> variance([1, 2, 2, 2, 3])
0.5

All four functions also accept a second, optional, argument, the
mean of the data.  This is modelled on a similar API provided by
the GNU Scientific Library[18].  There are three use-cases for
using this argument, in no particular order:

1)  The value of the mean is known *a priori*.

2)  You have already calculated the mean, and wish to avoid
calculating it again.

3)  You wish to (ab)use the variance functions to calculate
the second moment about some given point other than the
mean.

In each case, it is the caller's responsibility to ensure that
given argument is meaningful.

Functions provided are:

* variance(data, xbar=None) -> sample variance of data,
optionally using xbar as the sample mean.

* stdev(data, xbar=None) -> sample standard deviation of
data, optionally using xbar as the sample mean.

* pvariance(data, mu=None) -> population variance of data,
optionally using mu as the population mean.

* pstdev(data, mu=None) -> population standard deviation of
data, optionally using mu as the population mean.

Other functions

There is one other public function:

* sum(data, start=0) -> high-precision sum of numeric data.

```

### Specification

```    As the proposed reference implementation is in pure Python,
other Python implementations can easily make use of the module
unchanged, or adapt it as they see fit.

```

### What Should Be The Name Of The Module?

```    This will be a top-level module "statistics".

There was some interest in turning math into a package, and making this a
sub-module of math, but the general consensus eventually agreed on a
top-level module.  Other potential but rejected names included "stats" (too
much risk of confusion with existing "stat" module), and "statslib"
(described as "too C-like").

```

### Discussion And Resolved Issues

```    This proposal has been previously discussed here[21].

A number of design issues were resolved during the discussion on
Python-Ideas and the initial code review.  There was a lot of concern
library, see the FAQs below for more details.  In addition, the
initial implementation of ``sum`` suffered from some rounding issues
and other design problems when dealing with  Decimals.  Oscar
Benjamin's assistance in resolving this was invaluable.

Another issue was the handling of data in the form of iterators.  The
first implementation of variance silently swapped between a one- and
two-pass algorithm, depending on whether the data was in the form of
an iterator or sequence.  This proved to be a design mistake, as the
calculated variance could differ slightly depending on the algorithm
used, and ``variance`` etc. were changed to internally generate a list
and always use the more accurate two-pass implementation.

which were implemented as attributes on the ``median`` callable, e.g.
``median``, ``median.low``, ``median.high`` etc.  Although there is
at least one existing use of this style in the standard library, in
``unittest.mock``, the code reviewers felt that this was too unusual
for the standard library.  Consequently, the design has been changed
to a more traditional design of separate functions with a pseudo-
namespace naming convention, ``median_low``, ``median_high``, etc.

Another issue that was of concern to code reviewers was the existence
of a function calculating the sample mode of continuous data, with
some people questioning the choice of algorithm, and whether it was
a sufficiently common need to be included.  So it was dropped from
the API, and ``mode`` now implements only the basic schoolbook
algorithm based on counting unique values.

Another significant point of discussion was calculating statistics of
timedelta objects.  Although the statistics module will not directly
support timedelta objects, it is possible to support this use-case by
converting them to numbers first using the ``timedelta.total_seconds``
method.

```

```    Q: Shouldn't this module spend time on PyPI before being considered for
the standard library?

A: Older versions of this module have been available on PyPI[22] since
2010. Being much simpler than numpy, it does not require many years of
external development.

Q: Does the standard library really need yet another version of ``sum``?

A: This proved to be the most controversial part of the reference
implementation.  In one sense, clearly three sums is two too many.  But
in another sense, yes.  The reasons why the two existing versions are
unsuitable are described here[23] but the short summary is:

- the built-in sum can lose precision with floats;

- the built-in sum accepts any non-numeric data type that supports
the + operator, apart from strings and bytes;

- math.fsum is high-precision, but coerces all arguments to float.

There was some interest in "fixing" one or the other of the existing
sums. If this occurs before 3.4 feature-freeze, the decision to keep
statistics.sum can be re-considered.

Q: Will this module be backported to older versions of Python?

A: The module currently targets 3.3, and I will make it available on PyPI
for 3.3 for the foreseeable future. Backporting to older versions of
the 3.x series is likely (but not yet decided). Backporting to 2.7 is
less likely but not ruled out.

Q: Is this supposed to replace numpy?

A: No. While it is likely to grow over the years (see open issues below)
it is not aimed to replace, or even compete directly with, numpy. Numpy
is a full-featured numeric library aimed at professionals, the nuclear
reactor of numeric libraries in the Python ecosystem. This is just a
battery, as in "batteries included", and is aimed at an intermediate
level somewhere between "use numpy" and "roll your own version".

```

### Future Work

```    - At this stage, I am unsure of the best API for multivariate statistical
functions such as linear regression, correlation coefficient, and
covariance. Possible APIs include:

* Separate arguments for x and y data:
function([x0, x1, ...], [y0, y1, ...])

* A single argument for (x, y) data:
function([(x0, y0), (x1, y1), ...])

This API is preferred by GvR[24].

* Selecting arbitrary columns from a 2D array:
function([[a0, x0, y0, z0], [a1, x1, y1, z1], ...], x=1, y=2)

* Some combination of the above.

In the absence of a consensus of preferred API for multivariate stats,
I will defer including such multivariate functions until Python 3.5.

- Likewise, functions for calculating probability of random variables and
inference testing (e.g. Student's t-test) will be deferred until 3.5.

- There is considerable interest in including one-pass functions that can
calculate multiple statistics from data in iterator form, without having
to convert to a list. The experimental "stats" package on PyPI includes
co-routine versions of statistics functions. Including these will be
deferred to 3.5.

```

### References

```    [1] http://mail.python.org/pipermail/python-dev/2010-October/104721.html

[2] http://support.casio.com/pdf/004/CP330PLUSver310_Soft_E.pdf

[3] Gnumeric:
https://projects.gnome.org/gnumeric/functions.shtml

LibreOffice:
https://help.libreoffice.org/Calc/Statistical_Functions_Part_One
https://help.libreoffice.org/Calc/Statistical_Functions_Part_Two
https://help.libreoffice.org/Calc/Statistical_Functions_Part_Three
https://help.libreoffice.org/Calc/Statistical_Functions_Part_Four
https://help.libreoffice.org/Calc/Statistical_Functions_Part_Five

[4] Scipy: http://scipy-central.org/
Numpy: http://www.numpy.org/

[5] http://wiki.scipy.org/Numpy_Functions_by_Category

[6] Tested with numpy 1.6.1 and Python 2.7.

[7] http://www.johndcook.com/blog/2008/09/26/comparing-three-methods-of-computing-standard-deviation/

[8] http://rosettacode.org/wiki/Standard_deviation

[9] https://bitbucket.org/larsyencken/simplestats/src/c42e048a6625/src/basic.py

[10] http://stackoverflow.com/questions/2341340/calculate-mean-and-variance-with-one-iteration

[11] http://www.r-project.org/

[12] http://msdn.microsoft.com/en-us/library/system.linq.enumerable.average.aspx

[13] https://www.bcg.wisc.edu/webteam/support/ruby/standard_deviation

[14] http://ruby-statsample.rubyforge.org/

[15] http://www.php.net/manual/en/ref.stats.php

[16] http://www.ayton.id.au/gary/it/Delphi/D_maths.htm#Delphi%20Statistical%20functions.

[17] http://www.gnu.org/software/gsl/manual/html_node/Statistics.html

[18] http://www.gnu.org/software/gsl/manual/html_node/Mean-and-standard-deviation-and-variance.html

[19] http://mathworld.wolfram.com/Skewness.html

[20] At least, tedious to those who don't like this sort of thing.

[21] http://mail.python.org/pipermail/python-ideas/2011-September/011524.html

[22] https://pypi.python.org/pypi/stats/

[23] http://mail.python.org/pipermail/python-ideas/2013-August/022630.html

[24] https://mail.python.org/pipermail/python-dev/2013-September/128429.html

```

```    This document has been placed in the public domain.