1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
|
:mod:`statistics` --- Mathematical statistics functions
=======================================================
.. module:: statistics
:synopsis: mathematical statistics functions
.. moduleauthor:: Steven D'Aprano <steve+python@pearwood.info>
.. sectionauthor:: Steven D'Aprano <steve+python@pearwood.info>
.. versionadded:: 3.4
**Source code:** :source:`Lib/statistics.py`
.. testsetup:: *
from statistics import *
__name__ = '<doctest>'
--------------
This module provides functions for calculating mathematical statistics of
numeric (:class:`Real`-valued) data.
.. note::
Unless explicitly noted otherwise, these functions support :class:`int`,
:class:`float`, :class:`decimal.Decimal` and :class:`fractions.Fraction`.
Behaviour with other types (whether in the numeric tower or not) is
currently unsupported. Mixed types are also undefined and
implementation-dependent. If your input data consists of mixed types,
you may be able to use :func:`map` to ensure a consistent result, e.g.
``map(float, input_data)``.
Averages and measures of central location
-----------------------------------------
These functions calculate an average or typical value from a population
or sample.
======================= =============================================
:func:`mean` Arithmetic mean ("average") of data.
:func:`fmean` Fast, floating point arithmetic mean.
:func:`harmonic_mean` Harmonic mean of data.
:func:`median` Median (middle value) of data.
:func:`median_low` Low median of data.
:func:`median_high` High median of data.
:func:`median_grouped` Median, or 50th percentile, of grouped data.
:func:`mode` Mode (most common value) of discrete data.
======================= =============================================
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.
======================= =============================================
Function details
----------------
Note: The functions do not require the data given to them to be sorted.
However, for reading convenience, most of the examples show sorted sequences.
.. function:: mean(data)
Return the sample arithmetic mean of *data* which can be a sequence or iterator.
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:
.. doctest::
>>> mean([1, 2, 3, 4, 4])
2.8
>>> mean([-1.0, 2.5, 3.25, 5.75])
2.625
>>> from fractions import Fraction as F
>>> mean([F(3, 7), F(1, 21), F(5, 3), F(1, 3)])
Fraction(13, 21)
>>> from decimal import Decimal as D
>>> mean([D("0.5"), D("0.75"), D("0.625"), D("0.375")])
Decimal('0.5625')
.. note::
The mean is strongly affected 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.
.. function:: fmean(data)
Convert *data* to floats and compute the arithmetic mean.
This runs faster than the :func:`mean` function and it always returns a
:class:`float`. The result is highly accurate but not as perfect as
:func:`mean`. If the input dataset is empty, raises a
:exc:`StatisticsError`.
.. doctest::
>>> fmean([3.5, 4.0, 5.25])
4.25
.. versionadded:: 3.8
.. function:: harmonic_mean(data)
Return the harmonic mean of *data*, a sequence or iterator of
real-valued numbers.
The harmonic mean, sometimes called the subcontrary mean, is the
reciprocal of the arithmetic :func:`mean` of the reciprocals of the
data. For example, the harmonic mean of three values *a*, *b* and *c*
will be equivalent to ``3/(1/a + 1/b + 1/c)``.
The harmonic mean is a type of average, a measure of the central
location of the data. It is often appropriate when averaging quantities
which are rates or ratios, for example speeds. For example:
Suppose an investor purchases an equal value of shares in each of
three companies, with P/E (price/earning) ratios of 2.5, 3 and 10.
What is the average P/E ratio for the investor's portfolio?
.. doctest::
>>> harmonic_mean([2.5, 3, 10]) # For an equal investment portfolio.
3.6
Using the arithmetic mean would give an average of about 5.167, which
is too high.
:exc:`StatisticsError` is raised if *data* is empty, or any element
is less than zero.
.. versionadded:: 3.6
.. function:: median(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.
*data* can be a sequence or iterator.
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:
.. 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.
If your data is ordinal (supports order operations) but not numeric (doesn't
support addition), you should use :func:`median_low` or :func:`median_high`
instead.
.. seealso:: :func:`median_low`, :func:`median_high`, :func:`median_grouped`
.. function:: median_low(data)
Return the low median of numeric data. If *data* is empty,
:exc:`StatisticsError` is raised. *data* can be a sequence or iterator.
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::
>>> median_low([1, 3, 5])
3
>>> 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.
.. function:: median_high(data)
Return the high median of data. If *data* is empty, :exc:`StatisticsError`
is raised. *data* can be a sequence or iterator.
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::
>>> median_high([1, 3, 5])
3
>>> 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.
.. function:: median_grouped(data, interval=1)
Return the median of grouped continuous data, calculated as the 50th
percentile, using interpolation. If *data* is empty, :exc:`StatisticsError`
is raised. *data* can be a sequence or iterator.
.. 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:
.. 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:
.. doctest::
>>> median_grouped([1, 3, 3, 5, 7], interval=1)
3.25
>>> median_grouped([1, 3, 3, 5, 7], interval=2)
3.5
This function does not check whether the data points are at least
*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.
.. seealso::
* "Statistics for the Behavioral Sciences", Frederick J Gravetter and
Larry B Wallnau (8th Edition).
* The `SSMEDIAN
<https://help.gnome.org/users/gnumeric/stable/gnumeric.html#gnumeric-function-SSMEDIAN>`_
function in the Gnome Gnumeric spreadsheet, including `this discussion
<https://mail.gnome.org/archives/gnumeric-list/2011-April/msg00018.html>`_.
.. function:: mode(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:
.. doctest::
>>> mode([1, 1, 2, 3, 3, 3, 3, 4])
3
The mode is unique in that it is the only statistic which also applies
to nominal (non-numeric) data:
.. doctest::
>>> mode(["red", "blue", "blue", "red", "green", "red", "red"])
'red'
.. function:: pstdev(data, mu=None)
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
.. function:: pvariance(data, mu=None)
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
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.
Raises :exc:`StatisticsError` if *data* is empty.
Examples:
.. doctest::
>>> data = [0.0, 0.25, 0.25, 1.25, 1.5, 1.75, 2.75, 3.25]
>>> 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:
.. doctest::
>>> mu = mean(data)
>>> 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.
Decimals and Fractions are supported:
.. doctest::
>>> from decimal import Decimal as D
>>> pvariance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")])
Decimal('24.815')
>>> from fractions import Fraction as F
>>> pvariance([F(1, 4), F(5, 4), F(1, 2)])
Fraction(13, 72)
.. 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.
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.
.. function:: stdev(data, xbar=None)
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
.. function:: variance(data, xbar=None)
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
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`.
Raises :exc:`StatisticsError` if *data* has fewer than two values.
Examples:
.. doctest::
>>> data = [2.75, 1.75, 1.25, 0.25, 0.5, 1.25, 3.5]
>>> 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:
.. doctest::
>>> m = mean(data)
>>> 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
impossible results.
Decimal and Fraction values are supported:
.. doctest::
>>> from decimal import Decimal as D
>>> variance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")])
Decimal('31.01875')
>>> from fractions import Fraction as F
>>> variance([F(1, 6), F(1, 2), F(5, 3)])
Fraction(67, 108)
.. 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.
Exceptions
----------
A single exception is defined:
.. exception:: StatisticsError
Subclass of :exc:`ValueError` for statistics-related exceptions.
:class:`NormalDist` objects
===========================
A :class:`NormalDist` is a a composite class that treats the mean and standard
deviation of data measurements as a single entity. It is a tool for creating
and manipulating normal distributions of a random variable.
Normal distributions arise from the `Central Limit Theorem
<https://en.wikipedia.org/wiki/Central_limit_theorem>`_ and have a wide range
of applications in statistics, including simulations and hypothesis testing.
.. class:: NormalDist(mu=0.0, sigma=1.0)
Returns a new *NormalDist* object where *mu* represents the `arithmetic
mean <https://en.wikipedia.org/wiki/Arithmetic_mean>`_ and *sigma*
represents the `standard deviation
<https://en.wikipedia.org/wiki/Standard_deviation>`_.
If *sigma* is negative, raises :exc:`StatisticsError`.
.. attribute:: mean
A read-only property representing the `arithmetic mean
<https://en.wikipedia.org/wiki/Arithmetic_mean>`_ of a normal
distribution.
.. attribute:: stdev
A read-only property representing the `standard deviation
<https://en.wikipedia.org/wiki/Standard_deviation>`_ of a normal
distribution.
.. attribute:: variance
A read-only property representing the `variance
<https://en.wikipedia.org/wiki/Variance>`_ of a normal
distribution. Equal to the square of the standard deviation.
.. classmethod:: NormalDist.from_samples(data)
Class method that makes a normal distribution instance
from sample data. The *data* can be any :term:`iterable`
and should consist of values that can be converted to type
:class:`float`.
If *data* does not contain at least two elements, raises
:exc:`StatisticsError` because it takes at least one point to estimate
a central value and at least two points to estimate dispersion.
.. method:: NormalDist.samples(n, seed=None)
Generates *n* random samples for a given mean and standard deviation.
Returns a :class:`list` of :class:`float` values.
If *seed* is given, creates a new instance of the underlying random
number generator. This is useful for creating reproducible results,
even in a multi-threading context.
.. method:: NormalDist.pdf(x)
Using a `probability density function (pdf)
<https://en.wikipedia.org/wiki/Probability_density_function>`_,
compute the relative likelihood that a random sample *X* will be near
the given value *x*. Mathematically, it is the ratio ``P(x <= X <
x+dx) / dx``.
Note the relative likelihood of *x* can be greater than `1.0`. The
probability for a specific point on a continuous distribution is `0.0`,
so the :func:`pdf` is used instead. It gives the probability of a
sample occurring in a narrow range around *x* and then dividing that
probability by the width of the range (hence the word "density").
.. method:: NormalDist.cdf(x)
Using a `cumulative distribution function (cdf)
<https://en.wikipedia.org/wiki/Cumulative_distribution_function>`_,
compute the probability that a random sample *X* will be less than or
equal to *x*. Mathematically, it is written ``P(X <= x)``.
Instances of :class:`NormalDist` support addition, subtraction,
multiplication and division by a constant. These operations
are used for translation and scaling. For example:
.. doctest::
>>> temperature_february = NormalDist(5, 2.5) # Celsius
>>> temperature_february * (9/5) + 32 # Fahrenheit
NormalDist(mu=41.0, sigma=4.5)
Dividing a constant by an instance of :class:`NormalDist` is not supported.
Since normal distributions arise from additive effects of independent
variables, it is possible to `add and subtract two normally distributed
random variables
<https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables>`_
represented as instances of :class:`NormalDist`. For example:
.. doctest::
>>> birth_weights = NormalDist.from_samples([2.5, 3.1, 2.1, 2.4, 2.7, 3.5])
>>> drug_effects = NormalDist(0.4, 0.15)
>>> combined = birth_weights + drug_effects
>>> f'mean: {combined.mean :.1f} standard deviation: {combined.stdev :.1f}'
'mean: 3.1 standard deviation: 0.5'
.. versionadded:: 3.8
:class:`NormalDist` Examples and Recipes
----------------------------------------
:class:`NormalDist` readily solves classic probability problems.
For example, given `historical data for SAT exams
<https://blog.prepscholar.com/sat-standard-deviation>`_ showing that scores
are normally distributed with a mean of 1060 and standard deviation of 192,
determine the percentage of students with scores between 1100 and 1200:
.. doctest::
>>> sat = NormalDist(1060, 195)
>>> fraction = sat.cdf(1200) - sat.cdf(1100)
>>> f'{fraction * 100 :.1f}% score between 1100 and 1200'
'18.2% score between 1100 and 1200'
To estimate the distribution for a model than isn't easy to solve
analytically, :class:`NormalDist` can generate input samples for a `Monte
Carlo simulation <https://en.wikipedia.org/wiki/Monte_Carlo_method>`_ of the
model:
.. doctest::
>>> n = 100_000
>>> X = NormalDist(350, 15).samples(n)
>>> Y = NormalDist(47, 17).samples(n)
>>> Z = NormalDist(62, 6).samples(n)
>>> model_simulation = [x * y / z for x, y, z in zip(X, Y, Z)]
>>> NormalDist.from_samples(model_simulation) # doctest: +SKIP
NormalDist(mu=267.6516398754636, sigma=101.357284306067)
Normal distributions commonly arise in machine learning problems.
Wikipedia has a `nice example with a Naive Bayesian Classifier
<https://en.wikipedia.org/wiki/Naive_Bayes_classifier>`_. The challenge
is to guess a person's gender from measurements of normally distributed
features including height, weight, and foot size.
The `prior probability <https://en.wikipedia.org/wiki/Prior_probability>`_ of
being male or female is 50%:
.. doctest::
>>> prior_male = 0.5
>>> prior_female = 0.5
We also have a training dataset with measurements for eight people. These
measurements are assumed to be normally distributed, so we summarize the data
with :class:`NormalDist`:
.. doctest::
>>> height_male = NormalDist.from_samples([6, 5.92, 5.58, 5.92])
>>> height_female = NormalDist.from_samples([5, 5.5, 5.42, 5.75])
>>> weight_male = NormalDist.from_samples([180, 190, 170, 165])
>>> weight_female = NormalDist.from_samples([100, 150, 130, 150])
>>> foot_size_male = NormalDist.from_samples([12, 11, 12, 10])
>>> foot_size_female = NormalDist.from_samples([6, 8, 7, 9])
We observe a new person whose feature measurements are known but whose gender
is unknown:
.. doctest::
>>> ht = 6.0 # height
>>> wt = 130 # weight
>>> fs = 8 # foot size
The posterior is the product of the prior times each likelihood of a
feature measurement given the gender:
.. doctest::
>>> posterior_male = (prior_male * height_male.pdf(ht) *
... weight_male.pdf(wt) * foot_size_male.pdf(fs))
>>> posterior_female = (prior_female * height_female.pdf(ht) *
... weight_female.pdf(wt) * foot_size_female.pdf(fs))
The final prediction is awarded to the largest posterior -- this is known as
the `maximum a posteriori
<https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation>`_ or MAP:
.. doctest::
>>> 'male' if posterior_male > posterior_female else 'female'
'female'
..
# This modelines must appear within the last ten lines of the file.
kate: indent-width 3; remove-trailing-space on; replace-tabs on; encoding utf-8;
|