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 occuring 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 finance.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)

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.

INPUT:

  • \(v\) – a list of numbers

OUTPUT:

  • a number

EXAMPLES:

sage: mean([pi, e])
1/2*pi + 1/2*e
sage: mean([])
NaN
sage: mean([I, sqrt(2), 3/5])
1/3*sqrt(2) + 1/3*I + 1/5
sage: mean([RIF(1.0103,1.0103), RIF(2)])
1.5051500000000000?
sage: mean(range(4))
3/2
sage: v = finance.TimeSeries([1..100])
sage: mean(v)
50.5
sage.stats.basic_stats.median(v)

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 of sorted().

INPUT:

  • \(v\) – a list

OUTPUT:

  • median element of \(v\)

EXAMPLES:

sage: median([1,2,3,4,5])
3
sage: median([e, pi])
1/2*pi + 1/2*e
sage: median(['sage', 'linux', 'python'])
'python'
sage: median([])
NaN
sage: class MyClass:
...      def median(self):
...         return 1
sage: stats.median(MyClass())
1
sage.stats.basic_stats.mode(v)

Return the mode of \(v\). The mode is the sorted list of the most frequently occuring elements in \(v\). If \(n\) is the most times that any element occurs in \(v\), then the mode is the sorted list of elements of \(v\) that occur \(n\) times.

NOTE: The elements of \(v\) must be hashable and comparable.

INPUT:

  • \(v\) – a list

OUTPUT:

  • a list

EXAMPLES:

sage: v = [1,2,4,1,6,2,6,7,1]
sage: mode(v)
[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(['sage', 4, I, 3/5, 'sage', pi])
['sage']
sage: class MyClass:
...     def mode(self):
...         return [1]
sage: stats.mode(MyClass())
[1]
sage.stats.basic_stats.moving_average(v, n)

Provides 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.

INPUT:

  • \(v\) – a list
  • \(n\) – the number of values used in computing each average.

OUTPUT:

  • a list of length len(v)-n+1, since we do not fabric any values

EXAMPLES:

sage: moving_average([1..10], 1)
[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)
[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 the simple_moving_average on time series returns \(n\) values:

sage: a = finance.TimeSeries([1..10])
sage: stats.moving_average(a, 3)
[2.0000, 3.0000, 4.0000, 5.0000, 6.0000, 7.0000, 8.0000, 9.0000]
sage: stats.moving_average(list(a), 3)
[2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0]
sage.stats.basic_stats.std(v, bias=False)

Returns 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.

INPUT:

  • \(v\) – a list of numbers

  • bias – bool (default: False); if False, divide by

    len(v) - 1 instead of len(v) to give a less biased estimator (sample) for the standard deviation.

OUTPUT:

  • a number

EXAMPLES:

sage: std([1..6], bias=True)
1/2*sqrt(35/3)
sage: std([1..6], bias=False)
sqrt(7/2)
sage: std([e, pi])
sqrt(1/2)*sqrt((pi - e)^2)
sage: std([])
NaN
sage: std([I, sqrt(2), 3/5])
sqrt(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: std([RIF(1.0103, 1.0103), RIF(2)])
0.6998235813403261?
sage: import numpy
sage: x = numpy.array([1,2,3,4,5])
sage: std(x, bias=False)
1.5811388300841898
sage: x = finance.TimeSeries([1..100])
sage: std(x)
29.011491975882016
sage.stats.basic_stats.variance(v, bias=False)

Returns 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.

INPUT:

  • \(v\) – a list of numbers

  • bias – bool (default: False); if False, divide by

    len(v) - 1 instead of len(v) to give a less biased estimator (sample) for the standard deviation.

OUTPUT:

  • a number

EXAMPLES:

sage: variance([1..6])
7/2
sage: variance([1..6], bias=True)
35/12
sage: variance([e, pi])
1/2*(pi - e)^2
sage: variance([])
NaN
sage: variance([I, sqrt(2), 3/5])
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
sage: x = numpy.array([1,2,3,4,5])
sage: variance(x, bias=False)
2.5
sage: x = finance.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 = [2L,4L]
...     def __len__(self):
...         return len(self.__list)
...     def __iter__(self):
...         return self.__list.__iter__()
...     def mean(self):
...         return 3L
sage: R = SillyPythonList()
sage: variance(R)
2
sage: variance(R, bias=True)
1

TESTS:

The performance issue from #10019 is solved:

sage: variance([1] * 2^18)
0

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