Basic Statistics¶
This file contains basic descriptive functions. Included are the mean, median, mode, moving average, standard deviation, and the variance. When calling a function on data, there are checks for functions already defined for that data type.
The mean()
function returns the arithmetic mean (the sum of all the members
of a list, divided by the number of members). Further revisions may include
the geometric and harmonic mean. The median()
function returns the number
separating the higher half of a sample from the lower half. The mode()
returns the most common occurring member of a sample, plus the number of times
it occurs. If entries occur equally common, the smallest of a list of the most
common entries is returned. The moving_average()
is a finite impulse
response filter, creating a series of averages using a user-defined number of
subsets of the full data set. The std()
and the variance()
return a
measurement of how far data points tend to be from the arithmetic mean.
Functions are available in the namespace stats
, i.e. you can use them by
typing stats.mean
, stats.median
, etc.
REMARK: If all the data you are working with are floating point
numbers, you may find stats.TimeSeries
helpful, since it is
extremely fast and offers many of the same descriptive statistics as
in the module.
AUTHOR:
Andrew Hou (11/06/2009)
- sage.stats.basic_stats.mean(v)[source]¶
Return the mean of the elements of \(v\).
We define the mean of the empty list to be the (symbolic) NaN, following the convention of MATLAB, Scipy, and R.
This function is deprecated. Use
numpy.mean()
ornumpy.nanmean()
instead.INPUT:
v
– list of numbers
OUTPUT: a number
EXAMPLES:
sage: mean([pi, e]) # needs sage.symbolic doctest:warning... DeprecationWarning: sage.stats.basic_stats.mean is deprecated; use numpy.mean or numpy.nanmean instead See https://github.com/sagemath/sage/issues/29662 for details. 1/2*pi + 1/2*e sage: mean([]) # needs sage.symbolic NaN sage: mean([I, sqrt(2), 3/5]) # needs sage.symbolic 1/3*sqrt(2) + 1/3*I + 1/5 sage: mean([RIF(1.0103,1.0103), RIF(2)]) # needs sage.rings.real_interval_field 1.5051500000000000? sage: mean(range(4)) 3/2 sage: v = stats.TimeSeries([1..100]) # needs numpy sage: mean(v) # needs numpy 50.5
>>> from sage.all import * >>> mean([pi, e]) # needs sage.symbolic doctest:warning... DeprecationWarning: sage.stats.basic_stats.mean is deprecated; use numpy.mean or numpy.nanmean instead See https://github.com/sagemath/sage/issues/29662 for details. 1/2*pi + 1/2*e >>> mean([]) # needs sage.symbolic NaN >>> mean([I, sqrt(Integer(2)), Integer(3)/Integer(5)]) # needs sage.symbolic 1/3*sqrt(2) + 1/3*I + 1/5 >>> mean([RIF(RealNumber('1.0103'),RealNumber('1.0103')), RIF(Integer(2))]) # needs sage.rings.real_interval_field 1.5051500000000000? >>> mean(range(Integer(4))) 3/2 >>> v = stats.TimeSeries((ellipsis_range(Integer(1),Ellipsis,Integer(100)))) # needs numpy >>> mean(v) # needs numpy 50.5
- sage.stats.basic_stats.median(v)[source]¶
Return the median (middle value) of the elements of \(v\).
If \(v\) is empty, we define the median to be NaN, which is consistent with NumPy (note that R returns NULL). If \(v\) is comprised of strings,
TypeError
occurs. For elements other than numbers, the median is a result ofsorted()
.This function is deprecated. Use
numpy.median()
ornumpy.nanmedian()
instead.INPUT:
v
– list
OUTPUT: median element of \(v\)
EXAMPLES:
sage: median([1,2,3,4,5]) doctest:warning... DeprecationWarning: sage.stats.basic_stats.median is deprecated; use numpy.median or numpy.nanmedian instead See https://github.com/sagemath/sage/issues/29662 for details. 3 sage: median([e, pi]) # needs sage.symbolic 1/2*pi + 1/2*e sage: median(['sage', 'linux', 'python']) 'python' sage: median([]) # needs sage.symbolic NaN sage: class MyClass: ....: def median(self): ....: return 1 sage: stats.median(MyClass()) 1
>>> from sage.all import * >>> median([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) doctest:warning... DeprecationWarning: sage.stats.basic_stats.median is deprecated; use numpy.median or numpy.nanmedian instead See https://github.com/sagemath/sage/issues/29662 for details. 3 >>> median([e, pi]) # needs sage.symbolic 1/2*pi + 1/2*e >>> median(['sage', 'linux', 'python']) 'python' >>> median([]) # needs sage.symbolic NaN >>> class MyClass: ... def median(self): ... return Integer(1) >>> stats.median(MyClass()) 1
- sage.stats.basic_stats.mode(v)[source]¶
Return the mode of \(v\).
The mode is the list of the most frequently occurring elements in \(v\). If \(n\) is the most times that any element occurs in \(v\), then the mode is the list of elements of \(v\) that occur \(n\) times. The list is sorted if possible.
This function is deprecated. Use
scipy.stats.mode()
orstatistics.mode()
instead.Note
The elements of \(v\) must be hashable.
INPUT:
v
– list
OUTPUT: list (sorted if possible)
EXAMPLES:
sage: v = [1,2,4,1,6,2,6,7,1] sage: mode(v) doctest:warning... DeprecationWarning: sage.stats.basic_stats.mode is deprecated; use scipy.stats.mode or statistics.mode instead See https://github.com/sagemath/sage/issues/29662 for details. [1] sage: v.count(1) 3 sage: mode([]) [] sage: mode([1,2,3,4,5]) [1, 2, 3, 4, 5] sage: mode([3,1,2,1,2,3]) [1, 2, 3] sage: mode([0, 2, 7, 7, 13, 20, 2, 13]) [2, 7, 13] sage: mode(['sage', 'four', 'I', 'three', 'sage', 'pi']) ['sage'] sage: class MyClass: ....: def mode(self): ....: return [1] sage: stats.mode(MyClass()) [1]
>>> from sage.all import * >>> v = [Integer(1),Integer(2),Integer(4),Integer(1),Integer(6),Integer(2),Integer(6),Integer(7),Integer(1)] >>> mode(v) doctest:warning... DeprecationWarning: sage.stats.basic_stats.mode is deprecated; use scipy.stats.mode or statistics.mode instead See https://github.com/sagemath/sage/issues/29662 for details. [1] >>> v.count(Integer(1)) 3 >>> mode([]) [] >>> mode([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) [1, 2, 3, 4, 5] >>> mode([Integer(3),Integer(1),Integer(2),Integer(1),Integer(2),Integer(3)]) [1, 2, 3] >>> mode([Integer(0), Integer(2), Integer(7), Integer(7), Integer(13), Integer(20), Integer(2), Integer(13)]) [2, 7, 13] >>> mode(['sage', 'four', 'I', 'three', 'sage', 'pi']) ['sage'] >>> class MyClass: ... def mode(self): ... return [Integer(1)] >>> stats.mode(MyClass()) [1]
- sage.stats.basic_stats.moving_average(v, n)[source]¶
Return the moving average of a list \(v\).
The moving average of a list is often used to smooth out noisy data.
If \(v\) is empty, we define the entries of the moving average to be NaN.
This method is deprecated. Use
pandas.Series.rolling()
instead.INPUT:
v
– listn
– the number of values used in computing each average
OUTPUT: list of length
len(v)-n+1
, since we do not fabric any valuesEXAMPLES:
sage: moving_average([1..10], 1) doctest:warning... DeprecationWarning: sage.stats.basic_stats.moving_average is deprecated; use pandas.Series.rolling instead See https://github.com/sagemath/sage/issues/29662 for details. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] sage: moving_average([1..10], 4) [5/2, 7/2, 9/2, 11/2, 13/2, 15/2, 17/2] sage: moving_average([], 1) [] sage: moving_average([pi, e, I, sqrt(2), 3/5], 2) # needs sage.symbolic [1/2*pi + 1/2*e, 1/2*e + 1/2*I, 1/2*sqrt(2) + 1/2*I, 1/2*sqrt(2) + 3/10]
>>> from sage.all import * >>> moving_average((ellipsis_range(Integer(1),Ellipsis,Integer(10))), Integer(1)) doctest:warning... DeprecationWarning: sage.stats.basic_stats.moving_average is deprecated; use pandas.Series.rolling instead See https://github.com/sagemath/sage/issues/29662 for details. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] >>> moving_average((ellipsis_range(Integer(1),Ellipsis,Integer(10))), Integer(4)) [5/2, 7/2, 9/2, 11/2, 13/2, 15/2, 17/2] >>> moving_average([], Integer(1)) [] >>> moving_average([pi, e, I, sqrt(Integer(2)), Integer(3)/Integer(5)], Integer(2)) # needs sage.symbolic [1/2*pi + 1/2*e, 1/2*e + 1/2*I, 1/2*sqrt(2) + 1/2*I, 1/2*sqrt(2) + 3/10]
We check if the input is a time series, and if so use the optimized
simple_moving_average()
method, but with (slightly different) meaning as defined above (the point is that thesimple_moving_average()
on time series returns \(n\) values:sage: a = stats.TimeSeries([1..10]) # needs numpy sage: stats.moving_average(a, 3) # needs numpy [2.0000, 3.0000, 4.0000, 5.0000, 6.0000, 7.0000, 8.0000, 9.0000] sage: stats.moving_average(list(a), 3) # needs numpy [2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0]
>>> from sage.all import * >>> a = stats.TimeSeries((ellipsis_range(Integer(1),Ellipsis,Integer(10)))) # needs numpy >>> stats.moving_average(a, Integer(3)) # needs numpy [2.0000, 3.0000, 4.0000, 5.0000, 6.0000, 7.0000, 8.0000, 9.0000] >>> stats.moving_average(list(a), Integer(3)) # needs numpy [2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0]
- sage.stats.basic_stats.std(v, bias=False)[source]¶
Return the standard deviation of the elements of \(v\).
We define the standard deviation of the empty list to be NaN, following the convention of MATLAB, Scipy, and R.
This function is deprecated. Use
numpy.std()
ornumpy.nanstd()
instead.INPUT:
v
– list of numbersbias
– boolean (default:False
); ifFalse
, divide bylen(v) - 1
instead oflen(v)
to give a less biased estimator (sample) for the standard deviation.
OUTPUT: a number
EXAMPLES:
sage: # needs sage.symbolic sage: std([1..6], bias=True) doctest:warning... DeprecationWarning: sage.stats.basic_stats.std is deprecated; use numpy.std or numpy.nanstd instead See https://github.com/sagemath/sage/issues/29662 for details. doctest:warning... DeprecationWarning: sage.stats.basic_stats.variance is deprecated; use numpy.var or numpy.nanvar instead See https://github.com/sagemath/sage/issues/29662 for details. doctest:warning... DeprecationWarning: sage.stats.basic_stats.mean is deprecated; use numpy.mean or numpy.nanmean instead See https://github.com/sagemath/sage/issues/29662 for details. 1/2*sqrt(35/3) sage: std([1..6], bias=False) sqrt(7/2) sage: std([e, pi]) sqrt(1/2)*abs(pi - e) sage: std([]) NaN sage: std([I, sqrt(2), 3/5]) 1/15*sqrt(1/2)*sqrt((10*sqrt(2) - 5*I - 3)^2 + (5*sqrt(2) - 10*I + 3)^2 + (5*sqrt(2) + 5*I - 6)^2) sage: std([RIF(1.0103, 1.0103), RIF(2)]) 0.6998235813403261? sage: # needs numpy sage: import numpy sage: if int(numpy.version.short_version[0]) > 1: ....: numpy.set_printoptions(legacy="1.25") sage: x = numpy.array([1,2,3,4,5]) sage: std(x, bias=False) 1.5811388300841898 sage: x = stats.TimeSeries([1..100]) sage: std(x) 29.011491975882016
>>> from sage.all import * >>> # needs sage.symbolic >>> std((ellipsis_range(Integer(1),Ellipsis,Integer(6))), bias=True) doctest:warning... DeprecationWarning: sage.stats.basic_stats.std is deprecated; use numpy.std or numpy.nanstd instead See https://github.com/sagemath/sage/issues/29662 for details. doctest:warning... DeprecationWarning: sage.stats.basic_stats.variance is deprecated; use numpy.var or numpy.nanvar instead See https://github.com/sagemath/sage/issues/29662 for details. doctest:warning... DeprecationWarning: sage.stats.basic_stats.mean is deprecated; use numpy.mean or numpy.nanmean instead See https://github.com/sagemath/sage/issues/29662 for details. 1/2*sqrt(35/3) >>> std((ellipsis_range(Integer(1),Ellipsis,Integer(6))), bias=False) sqrt(7/2) >>> std([e, pi]) sqrt(1/2)*abs(pi - e) >>> std([]) NaN >>> std([I, sqrt(Integer(2)), Integer(3)/Integer(5)]) 1/15*sqrt(1/2)*sqrt((10*sqrt(2) - 5*I - 3)^2 + (5*sqrt(2) - 10*I + 3)^2 + (5*sqrt(2) + 5*I - 6)^2) >>> std([RIF(RealNumber('1.0103'), RealNumber('1.0103')), RIF(Integer(2))]) 0.6998235813403261? >>> # needs numpy >>> import numpy >>> if int(numpy.version.short_version[Integer(0)]) > Integer(1): ... numpy.set_printoptions(legacy="1.25") >>> x = numpy.array([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) >>> std(x, bias=False) 1.5811388300841898 >>> x = stats.TimeSeries((ellipsis_range(Integer(1),Ellipsis,Integer(100)))) >>> std(x) 29.011491975882016
- sage.stats.basic_stats.variance(v, bias=False)[source]¶
Return the variance of the elements of \(v\).
We define the variance of the empty list to be NaN, following the convention of MATLAB, Scipy, and R.
This function is deprecated. Use
numpy.var()
ornumpy.nanvar()
instead.INPUT:
v
– list of numbersbias
– boolean (default:False
); ifFalse
, divide bylen(v) - 1
instead oflen(v)
to give a less biased estimator (sample) for the standard deviation.
OUTPUT: a number
EXAMPLES:
sage: variance([1..6]) doctest:warning... DeprecationWarning: sage.stats.basic_stats.variance is deprecated; use numpy.var or numpy.nanvar instead See https://github.com/sagemath/sage/issues/29662 for details. 7/2 sage: variance([1..6], bias=True) 35/12 sage: variance([e, pi]) # needs sage.symbolic 1/2*(pi - e)^2 sage: variance([]) NaN sage: variance([I, sqrt(2), 3/5]) # needs sage.symbolic 1/450*(10*sqrt(2) - 5*I - 3)^2 + 1/450*(5*sqrt(2) - 10*I + 3)^2 + 1/450*(5*sqrt(2) + 5*I - 6)^2 sage: variance([RIF(1.0103, 1.0103), RIF(2)]) 0.4897530450000000? sage: import numpy # needs numpy sage: if int(numpy.version.short_version[0]) > 1: # needs numpy ....: numpy.set_printoptions(legacy="1.25") # needs numpy sage: x = numpy.array([1,2,3,4,5]) # needs numpy sage: variance(x, bias=False) # needs numpy 2.5 sage: x = stats.TimeSeries([1..100]) sage: variance(x) 841.6666666666666 sage: variance(x, bias=True) 833.25 sage: class MyClass: ....: def variance(self, bias=False): ....: return 1 sage: stats.variance(MyClass()) 1 sage: class SillyPythonList: ....: def __init__(self): ....: self.__list = [2, 4] ....: def __len__(self): ....: return len(self.__list) ....: def __iter__(self): ....: return self.__list.__iter__() ....: def mean(self): ....: return 3 sage: R = SillyPythonList() sage: variance(R) 2 sage: variance(R, bias=True) 1
>>> from sage.all import * >>> variance((ellipsis_range(Integer(1),Ellipsis,Integer(6)))) doctest:warning... DeprecationWarning: sage.stats.basic_stats.variance is deprecated; use numpy.var or numpy.nanvar instead See https://github.com/sagemath/sage/issues/29662 for details. 7/2 >>> variance((ellipsis_range(Integer(1),Ellipsis,Integer(6))), bias=True) 35/12 >>> variance([e, pi]) # needs sage.symbolic 1/2*(pi - e)^2 >>> variance([]) NaN >>> variance([I, sqrt(Integer(2)), Integer(3)/Integer(5)]) # needs sage.symbolic 1/450*(10*sqrt(2) - 5*I - 3)^2 + 1/450*(5*sqrt(2) - 10*I + 3)^2 + 1/450*(5*sqrt(2) + 5*I - 6)^2 >>> variance([RIF(RealNumber('1.0103'), RealNumber('1.0103')), RIF(Integer(2))]) 0.4897530450000000? >>> import numpy # needs numpy >>> if int(numpy.version.short_version[Integer(0)]) > Integer(1): # needs numpy ... numpy.set_printoptions(legacy="1.25") # needs numpy >>> x = numpy.array([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) # needs numpy >>> variance(x, bias=False) # needs numpy 2.5 >>> x = stats.TimeSeries((ellipsis_range(Integer(1),Ellipsis,Integer(100)))) >>> variance(x) 841.6666666666666 >>> variance(x, bias=True) 833.25 >>> class MyClass: ... def variance(self, bias=False): ... return Integer(1) >>> stats.variance(MyClass()) 1 >>> class SillyPythonList: ... def __init__(self): ... self.__list = [Integer(2), Integer(4)] ... def __len__(self): ... return len(self.__list) ... def __iter__(self): ... return self.__list.__iter__() ... def mean(self): ... return Integer(3) >>> R = SillyPythonList() >>> variance(R) 2 >>> variance(R, bias=True) 1