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Commit eb2aeecc authored by Georg Brandl's avatar Georg Brandl
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Reformat statistics.rst and remove unnecessary headings for each function.

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Doc/library/statistics.rst
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......@@ -35,21 +35,34 @@ or sample.
:func:`mode` Mode (most common value) of discrete data.
======================= =============================================
:func:`mean`
~~~~~~~~~~~~
Measures of spread
------------------
These functions calculate a measure of how much the population or sample
tends to deviate from the typical or average values.
======================= =============================================
:func:`pstdev` Population standard deviation of data.
:func:`pvariance` Population variance of data.
:func:`stdev` Sample standard deviation of data.
:func:`variance` Sample variance of data.
======================= =============================================
The :func:`mean` function calculates the arithmetic mean, commonly known
as the average, of its iterable argument:
Function details
----------------
.. function:: mean(data)
Return the sample arithmetic mean of *data*, a sequence or iterator
of real-valued numbers.
Return the sample arithmetic mean of *data*, a sequence or iterator of
real-valued numbers.
The arithmetic mean is the sum of the data divided by the number of data
points. It is commonly called "the average", although it is only one of many
different mathematical averages. It is a measure of the central location of
the data.
The arithmetic mean is the sum of the data divided by the number of
data points. It is commonly called "the average", although it is only
one of many different mathematical averages. It is a measure of the
central location of the data.
If *data* is empty, :exc:`StatisticsError` will be raised.
Some examples of use:
......@@ -70,75 +83,56 @@ as the average, of its iterable argument:
.. note::
The mean is strongly effected by outliers and is not a robust
estimator for central location: the mean is not necessarily a
typical example of the data points. For more robust, although less
efficient, measures of central location, see :func:`median` and
:func:`mode`. (In this case, "efficient" refers to statistical
efficiency rather than computational efficiency.)
The sample mean gives an unbiased estimate of the true population
mean, which means that, taken on average over all the possible
samples, ``mean(sample)`` converges on the true mean of the entire
population. If *data* represents the entire population rather than
a sample, then ``mean(data)`` is equivalent to calculating the true
population mean μ.
The mean is strongly effected by outliers and is not a robust estimator
for central location: the mean is not necessarily a typical example of the
data points. For more robust, although less efficient, measures of
central location, see :func:`median` and :func:`mode`. (In this case,
"efficient" refers to statistical efficiency rather than computational
efficiency.)
If ``data`` is empty, :exc:`StatisticsError` will be raised.
The sample mean gives an unbiased estimate of the true population mean,
which means that, taken on average over all the possible samples,
``mean(sample)`` converges on the true mean of the entire population. If
*data* represents the entire population rather than a sample, then
``mean(data)`` is equivalent to calculating the true population mean μ.
:func:`median`
~~~~~~~~~~~~~~
The :func:`median` function calculates the median, or middle, data point,
using the common "mean of middle two" method.
.. seealso::
:func:`median_low`
:func:`median_high`
:func:`median_grouped`
.. function:: median(data)
Return the median (middle value) of numeric data.
Return the median (middle value) of numeric data, using the common "mean of
middle two" method. If *data* is empty, :exc:`StatisticsError` is raised.
The median is a robust measure of central location, and is less affected
by the presence of outliers in your data. When the number of data points
is odd, the middle data point is returned:
The median is a robust measure of central location, and is less affected by
the presence of outliers in your data. When the number of data points is
odd, the middle data point is returned:
.. doctest::
>>> median([1, 3, 5])
3
When the number of data points is even, the median is interpolated by
taking the average of the two middle values:
When the number of data points is even, the median is interpolated by taking
the average of the two middle values:
.. doctest::
>>> median([1, 3, 5, 7])
4.0
This is suited for when your data is discrete, and you don't mind that
the median may not be an actual data point.
This is suited for when your data is discrete, and you don't mind that the
median may not be an actual data point.
If data is empty, :exc:`StatisticsError` is raised.
.. seealso:: :func:`median_low`, :func:`median_high`, :func:`median_grouped`
:func:`median_low`
~~~~~~~~~~~~~~~~~~
The :func:`median_low` function calculates the low median without
interpolation.
.. function:: median_low(data)
Return the low median of numeric data.
Return the low median of numeric data. If *data* is empty,
:exc:`StatisticsError` is raised.
The low median is always a member of the data set. When the number
of data points is odd, the middle value is returned. When it is
even, the smaller of the two middle values is returned.
The low median is always a member of the data set. When the number of data
points is odd, the middle value is returned. When it is even, the smaller of
the two middle values is returned.
.. doctest::
......@@ -147,24 +141,18 @@ interpolation.
>>> median_low([1, 3, 5, 7])
3
Use the low median when your data are discrete and you prefer the median
to be an actual data point rather than interpolated.
Use the low median when your data are discrete and you prefer the median to
be an actual data point rather than interpolated.
If data is empty, :exc:`StatisticsError` is raised.
:func:`median_high`
~~~~~~~~~~~~~~~~~~~
The :func:`median_high` function calculates the high median without
interpolation.
.. function:: median_high(data)
Return the high median of data.
Return the high median of data. If *data* is empty, :exc:`StatisticsError`
is raised.
The high median is always a member of the data set. When the number of
data points is odd, the middle value is returned. When it is even, the
larger of the two middle values is returned.
The high median is always a member of the data set. When the number of data
points is odd, the middle value is returned. When it is even, the larger of
the two middle values is returned.
.. doctest::
......@@ -173,41 +161,34 @@ interpolation.
>>> median_high([1, 3, 5, 7])
5
Use the high median when your data are discrete and you prefer the median
to be an actual data point rather than interpolated.
If data is empty, :exc:`StatisticsError` is raised.
:func:`median_grouped`
~~~~~~~~~~~~~~~~~~~~~~
Use the high median when your data are discrete and you prefer the median to
be an actual data point rather than interpolated.
The :func:`median_grouped` function calculates the median of grouped data
as the 50th percentile, using interpolation.
.. function:: median_grouped(data [, interval])
.. function:: median_grouped(data, interval=1)
Return the median of grouped continuous data, calculated as the
50th percentile.
Return the median of grouped continuous data, calculated as the 50th
percentile, using interpolation. If *data* is empty, :exc:`StatisticsError`
is raised.
.. doctest::
>>> median_grouped([52, 52, 53, 54])
52.5
In the following example, the data are rounded, so that each value
represents the midpoint of data classes, e.g. 1 is the midpoint of the
class 0.5-1.5, 2 is the midpoint of 1.5-2.5, 3 is the midpoint of
2.5-3.5, etc. With the data given, the middle value falls somewhere in
the class 3.5-4.5, and interpolation is used to estimate it:
In the following example, the data are rounded, so that each value represents
the midpoint of data classes, e.g. 1 is the midpoint of the class 0.5-1.5, 2
is the midpoint of 1.5-2.5, 3 is the midpoint of 2.5-3.5, etc. With the data
given, the middle value falls somewhere in the class 3.5-4.5, and
interpolation is used to estimate it:
.. doctest::
>>> median_grouped([1, 2, 2, 3, 4, 4, 4, 4, 4, 5])
3.7
Optional argument ``interval`` represents the class interval, and
defaults to 1. Changing the class interval naturally will change the
interpolation:
Optional argument *interval* represents the class interval, and defaults
to 1. Changing the class interval naturally will change the interpolation:
.. doctest::
......@@ -217,36 +198,34 @@ as the 50th percentile, using interpolation.
3.5
This function does not check whether the data points are at least
``interval`` apart.
*interval* apart.
.. impl-detail::
Under some circumstances, :func:`median_grouped` may coerce data
points to floats. This behaviour is likely to change in the future.
Under some circumstances, :func:`median_grouped` may coerce data points to
floats. This behaviour is likely to change in the future.
.. seealso::
* "Statistics for the Behavioral Sciences", Frederick J Gravetter
and Larry B Wallnau (8th Edition).
* "Statistics for the Behavioral Sciences", Frederick J Gravetter and
Larry B Wallnau (8th Edition).
* Calculating the `median <http://www.ualberta.ca/~opscan/median.html>`_.
* The `SSMEDIAN <https://projects.gnome.org/gnumeric/doc/gnumeric-function-SSMEDIAN.shtml>`_
function in the Gnome Gnumeric spreadsheet, including
`this discussion <https://mail.gnome.org/archives/gnumeric-list/2011-April/msg00018.html>`_.
* The `SSMEDIAN
<https://projects.gnome.org/gnumeric/doc/gnumeric-function-SSMEDIAN.shtml>`_
function in the Gnome Gnumeric spreadsheet, including `this discussion
<https://mail.gnome.org/archives/gnumeric-list/2011-April/msg00018.html>`_.
If data is empty, :exc:`StatisticsError` is raised.
:func:`mode`
~~~~~~~~~~~~
The :func:`mode` function calculates the mode, or most common element, of
discrete or nominal data. The mode (when it exists) is the most typical
value, and is a robust measure of central location.
.. function:: mode(data)
Return the most common data point from discrete or nominal data.
Return the most common data point from discrete or nominal *data*. The mode
(when it exists) is the most typical value, and is a robust measure of
central location.
If *data* is empty, or if there is not exactly one most common value,
:exc:`StatisticsError` is raised.
``mode`` assumes discrete data, and returns a single value. This is the
standard treatment of the mode as commonly taught in schools:
......@@ -264,60 +243,35 @@ value, and is a robust measure of central location.
>>> mode(["red", "blue", "blue", "red", "green", "red", "red"])
'red'
If data is empty, or if there is not exactly one most common value,
:exc:`StatisticsError` is raised.
Measures of spread
------------------
These functions calculate a measure of how much the population or sample
tends to deviate from the typical or average values.
======================= =============================================
:func:`pstdev` Population standard deviation of data.
:func:`pvariance` Population variance of data.
:func:`stdev` Sample standard deviation of data.
:func:`variance` Sample variance of data.
======================= =============================================
:func:`pstdev`
~~~~~~~~~~~~~~
The :func:`pstdev` function calculates the standard deviation of a
population. The standard deviation is equivalent to the square root of
the variance.
.. function:: pstdev(data, mu=None)
.. function:: pstdev(data [, mu])
Return the square root of the population variance. See :func:`pvariance`
for arguments and other details.
Return the population standard deviation (the square root of the population
variance). See :func:`pvariance` for arguments and other details.
.. doctest::
>>> pstdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75])
0.986893273527251
:func:`pvariance`
~~~~~~~~~~~~~~~~~
The :func:`pvariance` function calculates the variance of a population.
Variance, or second moment about the mean, is a measure of the variability
(spread or dispersion) of data. A large variance indicates that the data is
spread out; a small variance indicates it is clustered closely around the
mean.
.. function:: pvariance(data [, mu])
.. function:: pvariance(data, mu=None)
Return the population variance of *data*, a non-empty iterable of
real-valued numbers.
Return the population variance of *data*, a non-empty iterable of real-valued
numbers. Variance, or second moment about the mean, is a measure of the
variability (spread or dispersion) of data. A large variance indicates that
the data is spread out; a small variance indicates it is clustered closely
around the mean.
If the optional second argument *mu* is given, it should be the mean
of *data*. If it is missing or None (the default), the mean is
If the optional second argument *mu* is given, it should be the mean of
*data*. If it is missing or ``None`` (the default), the mean is
automatically calculated.
Use this function to calculate the variance from the entire population.
To estimate the variance from a sample, the :func:`variance` function is
usually a better choice.
Use this function to calculate the variance from the entire population. To
estimate the variance from a sample, the :func:`variance` function is usually
a better choice.
Raises :exc:`StatisticsError` if *data* is empty.
Examples:
......@@ -327,8 +281,8 @@ mean.
>>> pvariance(data)
1.25
If you have already calculated the mean of your data, you can pass
it as the optional second argument *mu* to avoid recalculation:
If you have already calculated the mean of your data, you can pass it as the
optional second argument *mu* to avoid recalculation:
.. doctest::
......@@ -336,9 +290,9 @@ mean.
>>> pvariance(data, mu)
1.25
This function does not attempt to verify that you have passed the actual
mean as *mu*. Using arbitrary values for *mu* may lead to invalid or
impossible results.
This function does not attempt to verify that you have passed the actual mean
as *mu*. Using arbitrary values for *mu* may lead to invalid or impossible
results.
Decimals and Fractions are supported:
......@@ -354,53 +308,44 @@ mean.
.. note::
When called with the entire population, this gives the population
variance σ². When called on a sample instead, this is the biased
sample variance s², also known as variance with N degrees of freedom.
When called with the entire population, this gives the population variance
σ². When called on a sample instead, this is the biased sample variance
s², also known as variance with N degrees of freedom.
If you somehow know the true population mean μ, you may use this
function to calculate the variance of a sample, giving the known
population mean as the second argument. Provided the data points are
representative (e.g. independent and identically distributed), the
result will be an unbiased estimate of the population variance.
Raises :exc:`StatisticsError` if *data* is empty.
If you somehow know the true population mean μ, you may use this function
to calculate the variance of a sample, giving the known population mean as
the second argument. Provided the data points are representative
(e.g. independent and identically distributed), the result will be an
unbiased estimate of the population variance.
:func:`stdev`
~~~~~~~~~~~~~~
The :func:`stdev` function calculates the standard deviation of a sample.
The standard deviation is equivalent to the square root of the variance.
.. function:: stdev(data, xbar=None)
.. function:: stdev(data [, xbar])
Return the square root of the sample variance. See :func:`variance` for
arguments and other details.
Return the sample standard deviation (the square root of the sample
variance). See :func:`variance` for arguments and other details.
.. doctest::
>>> stdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75])
1.0810874155219827
:func:`variance`
~~~~~~~~~~~~~~~~~
The :func:`variance` function calculates the variance of a sample. Variance,
or second moment about the mean, is a measure of the variability (spread or
dispersion) of data. A large variance indicates that the data is spread out;
a small variance indicates it is clustered closely around the mean.
.. function:: variance(data [, xbar])
.. function:: variance(data, xbar=None)
Return the sample variance of *data*, an iterable of at least two
real-valued numbers.
Return the sample variance of *data*, an iterable of at least two real-valued
numbers. Variance, or second moment about the mean, is a measure of the
variability (spread or dispersion) of data. A large variance indicates that
the data is spread out; a small variance indicates it is clustered closely
around the mean.
If the optional second argument *xbar* is given, it should be the mean
of *data*. If it is missing or None (the default), the mean is
If the optional second argument *xbar* is given, it should be the mean of
*data*. If it is missing or ``None`` (the default), the mean is
automatically calculated.
Use this function when your data is a sample from a population. To
calculate the variance from the entire population, see :func:`pvariance`.
Use this function when your data is a sample from a population. To calculate
the variance from the entire population, see :func:`pvariance`.
Raises :exc:`StatisticsError` if *data* has fewer than two values.
Examples:
......@@ -410,8 +355,8 @@ a small variance indicates it is clustered closely around the mean.
>>> variance(data)
1.3720238095238095
If you have already calculated the mean of your data, you can pass
it as the optional second argument *xbar* to avoid recalculation:
If you have already calculated the mean of your data, you can pass it as the
optional second argument *xbar* to avoid recalculation:
.. doctest::
......@@ -419,8 +364,8 @@ a small variance indicates it is clustered closely around the mean.
>>> variance(data, m)
1.3720238095238095
This function does not attempt to verify that you have passed the actual
mean as *xbar*. Using arbitrary values for *xbar* can lead to invalid or
This function does not attempt to verify that you have passed the actual mean
as *xbar*. Using arbitrary values for *xbar* can lead to invalid or
impossible results.
Decimal and Fraction values are supported:
......@@ -437,17 +382,14 @@ a small variance indicates it is clustered closely around the mean.
.. note::
This is the sample variance s² with Bessel's correction, also known
as variance with N-1 degrees of freedom. Provided that the data
points are representative (e.g. independent and identically
distributed), the result should be an unbiased estimate of the true
population variance.
If you somehow know the actual population mean μ you should pass it
to the :func:`pvariance` function as the *mu* parameter to get
the variance of a sample.
This is the sample variance s² with Bessel's correction, also known as
variance with N-1 degrees of freedom. Provided that the data points are
representative (e.g. independent and identically distributed), the result
should be an unbiased estimate of the true population variance.
Raises :exc:`StatisticsError` if *data* has fewer than two values.
If you somehow know the actual population mean μ you should pass it to the
:func:`pvariance` function as the *mu* parameter to get the variance of a
sample.
Exceptions
----------
......
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