16.3. Math Statistics

  • statistics module

16.3.1. Mean

Table 16.1. Mean

Function

Description

statistics.mean()

Arithmetic mean ('average') of data

statistics.fmean()

faster, floating point variant of statistics.mean(), since Python 3.8

statistics.harmonic_mean()

Harmonic mean of data

statistics.geometric_mean()

since Python 3.8

Arithmetic mean ('average') of data:

from statistics import mean


mean([1, 2, 3, 4, 4])
# 2.8
mean([-1.0, 2.5, 3.25, 5.75])
# 2.625

Harmonic mean of data:

from statistics import harmonic_mean


harmonic_mean([2.5, 3, 10])
# 3.6

16.3.2. Median

Table 16.2. Median

Function

Description

statistics.median()

Median (middle value) of data

statistics.median_low()

Low median of data

statistics.median_high()

High median of data

statistics.median_grouped()

Median, or 50th percentile, of grouped data

Median (middle value) of data:

from statistics import median


median([1, 3, 5])
# 3
median([1, 3, 5, 7])
# 4.0
  • 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.

Low median of data:

from statistics import median_low


median_low([1, 3, 5])
# 3
median_low([1, 3, 5, 7])
# 3
  • 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.

High median of data:

from statistics import median_high


median_high([1, 3, 5])
# 3
median_high([1, 3, 5, 7])
# 5
  • Median of grouped continuous data.

  • Calculated using interpolation as the 50th percentile.

Median, or 50th percentile, of grouped data:

from statistics import median_grouped


median_grouped([52, 52, 53, 54])
# 52.5
median_grouped([1, 3, 3, 5, 7], interval=1)
# 3.25
median_grouped([1, 3, 3, 5, 7], interval=2)
# 3.5

16.3.3. Mode

Table 16.3. Mode

Function

Description

statistics.mode()

Mode (most common value) of discrete data

statistics.multimode()

returns a list of the most common values, since Python 3.8

statistics.quantiles()

divides data or a distribution in to equiprobable intervals (e.g. quartiles, deciles, or percentiles), since Python 3.8

Mode (most common value) of discrete data:

from statistics import mode


mode([1, 1, 2, 3, 3, 3, 3, 4])
# 3
mode(["red", "blue", "blue", "red", "green", "red", "red"])
# 'red'

16.3.4. Distribution

Table 16.4. Distribution

Function

Description

statistics.NormalDist

tool for creating and manipulating normal distributions of a random variable

16.3.5. Standard Deviation

Table 16.5. Standard Deviation

Function

Description

statistics.pstdev()

Population standard deviation of data

statistics.stdev()

Sample standard deviation of data

Sample standard deviation of data:

from statistics import stdev


stdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75])
# 1.0810874155219827
  • Population standard deviation

  • Is the square root of the population variance

Population standard deviation:

from statistics import pstdev


pstdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75])
# 0.986893273527251

16.3.6. Variance

Table 16.6. Variance

Function

Description

statistics.pvariance()

Population variance of data

statistics.variance()

Sample variance of data

Sample variance of data:

from statistics import variance


variance([2.75, 1.75, 1.25, 0.25, 0.5, 1.25, 3.5])
# 1.3720238095238095

Population variance of data:

from statistics import pvariance


pvariance([0.0, 0.25, 0.25, 1.25, 1.5, 1.75, 2.75, 3.25])
# 1.25

16.3.7. Examples

temperature_feb = NormalDist.from_samples([4, 12, -3, 2, 7, 14])

temperature_feb.mean
# 6.0
temperature_feb.stdev
# 6.356099432828281

# Chance of being under 3 degrees
temperature_feb.cdf(3)  # 0.3184678262814532

# Relative chance of being 7 degrees versus 10 degrees
temperature_feb.pdf(7) / temperature_feb.pdf(10)  # 1.2039930378537762


el_niño = NormalDist(4, 2.5)

# Add in a climate effect
temperature_feb += el_niño

temperature_feb
# NormalDist(mu=10.0, sigma=6.830080526611674)

# Convert to Fahrenheit
temperature_feb * (9/5) + 32
# NormalDist(mu=50.0, sigma=12.294144947901014)

# Generate random samples
temperature_feb.samples(3)
# [7.672102882379219, 12.000027119750287, 4.647488369766392]

16.3.8. Assignments

Code 16.5. Solution
"""
* Assignment: Math Statistics Stats
* Complexity: easy
* Lines of code: 30 lines
* Time: 21 min

English:
    1. For columns:
        a. Sepal length,
        b. Sepal width,
        c. Petal length,
        d. Petal width.
    2. Print calculated values:
        a. mean,
        b. median,
        c. standard deviation,
        d. variance.
    3. Use `statistics` module from Python standard library
    4. Run doctests - all must succeed

Polish:
    1. Dla kolumn:
        a. Sepal length,
        b. Sepal width,
        c. Petal length,
        d. Petal width.
    2. Wypisz wyliczone wartości:
        a. średnią,
        b. medianę,
        c. odchylenie standardowe,
        d. wariancję.
    3. Użyj modułu `statistics` z biblioteki standardowej Python
    4. Uruchom doctesty - wszystkie muszą się powieść

Tests:
    >>> import sys; sys.tracebacklimit = 0

    >>> stats(sepal_length)
    {'mean': 5.833333333333333, 'stdev': 0.9084785816591018, 'median': 5.7, 'variance': 0.8253333333333333}
    >>> stats(sepal_width)
    {'mean': 3.0619047619047617, 'stdev': 0.36670995415476587, 'median': 3.0, 'variance': 0.1344761904761905}
    >>> stats(petal_length)
    {'mean': 3.8523809523809525, 'stdev': 1.8602739173624534, 'median': 4.5, 'variance': 3.4606190476190477}
    >>> stats(petal_width)
    {'mean': 1.2333333333333334, 'stdev': 0.7741662181555931, 'median': 1.4, 'variance': 0.5993333333333334}
"""

from statistics import mean, stdev, variance, median


DATA = [('Sepal length', 'Sepal width', 'Petal length', 'Petal width', 'Species'),
        (5.8, 2.7, 5.1, 1.9, 'virginica'),
        (5.1, 3.5, 1.4, 0.2, 'setosa'),
        (5.7, 2.8, 4.1, 1.3, 'versicolor'),
        (6.3, 2.9, 5.6, 1.8, 'virginica'),
        (6.4, 3.2, 4.5, 1.5, 'versicolor'),
        (4.7, 3.2, 1.3, 0.2, 'setosa'),
        (7.0, 3.2, 4.7, 1.4, 'versicolor'),
        (7.6, 3.0, 6.6, 2.1, 'virginica'),
        (4.9, 3.0, 1.4, 0.2, 'setosa'),
        (4.9, 2.5, 4.5, 1.7, 'virginica'),
        (7.1, 3.0, 5.9, 2.1, 'virginica'),
        (4.6, 3.4, 1.4, 0.3, 'setosa'),
        (5.4, 3.9, 1.7, 0.4, 'setosa'),
        (5.7, 2.8, 4.5, 1.3, 'versicolor'),
        (5.0, 3.6, 1.4, 0.3, 'setosa'),
        (5.5, 2.3, 4.0, 1.3, 'versicolor'),
        (6.5, 3.0, 5.8, 2.2, 'virginica'),
        (6.5, 2.8, 4.6, 1.5, 'versicolor'),
        (6.3, 3.3, 6.0, 2.5, 'virginica'),
        (6.9, 3.1, 4.9, 1.5, 'versicolor'),
        (4.6, 3.1, 1.5, 0.2, 'setosa')]


Code 16.6. Solution
"""
* Assignment: Math Statistics Iris
* Complexity: easy
* Lines of code: 30 lines
* Time: 21 min

English:
    1. Create dict `result: dict[str, dict]`
    2. For each species calculate for numerical values:
        a. mean,
        b. median,
        c. standard deviation,
        d. variance.
    3. Save data to `result` dict
    4. Non-functional requirements:
        a. Use `statistics` module from Python standard library
    5. Run doctests - all must succeed

Polish:
    1. Stwórz słownik `result: dict[str, dict]`
    2. Dla każdego gatunku wylicz dla wartości numerycznych:
        a. średnią,
        b. medianę,
        c. odchylenie standardowe,
        d. wariancję.
    3. Dane zapisz w słowniku `result`
    4. Wymagania niefunkcjonalne:
        a. Użyj modułu `statistics` z biblioteki standardowej Python
    5. Uruchom doctesty - wszystkie muszą się powieść

Tests:
    >>> import sys; sys.tracebacklimit = 0

    >>> result  # doctest: +NORMALIZE_WHITESPACE
    {'virginica': {'Sepal length': {'values': [5.8, 6.3, 7.6, 4.9, 7.1, 6.5, 6.3],
                                    'mean': 6.357142857142857,
                                    'median': 6.3,
                                    'stdev': 0.871506631944823,
                                    'variance': 0.7595238095238092},
                   'Sepal width': {'values': [2.7, 2.9, 3.0, 2.5, 3.0, 3.0, 3.3],
                                   'mean': 2.914285714285714,
                                   'median': 3.0,
                                   'stdev': 0.25448360411214066,
                                   'variance': 0.06476190476190473},
                   'Petal length': {'values': [5.1, 5.6, 6.6, 4.5, 5.9, 5.8, 6.0],
                                    'mean': 5.642857142857142,
                                    'median': 5.8,
                                    'stdev': 0.6754187413675136,
                                    'variance': 0.45619047619047615},
                   'Petal width': {'values': [1.9, 1.8, 2.1, 1.7, 2.1, 2.2, 2.5],
                                   'mean': 2.0428571428571427,
                                   'median': 2.1,
                                   'stdev': 0.26992062325273125,
                                   'variance': 0.07285714285714287}},
     'setosa': {'Sepal length': {'values': [5.1, 4.7, 4.9, 4.6, 5.4, 5.0, 4.6],
                                 'mean': 4.9,
                                 'median': 4.9,
                                 'stdev': 0.2943920288775951,
                                 'variance': 0.08666666666666677},
                'Sepal width': {'values': [3.5, 3.2, 3.0, 3.4, 3.9, 3.6, 3.1],
                                'mean': 3.3857142857142857,
                                'median': 3.4,
                                'stdev': 0.31320159337914943,
                                'variance': 0.09809523809523807},
                'Petal length': {'values': [1.4, 1.3, 1.4, 1.4, 1.7, 1.4, 1.5],
                                 'mean': 1.4428571428571428,
                                 'median': 1.4,
                                 'stdev': 0.12724180205607036,
                                 'variance': 0.01619047619047619},
                'Petal width': {'values': [0.2, 0.2, 0.2, 0.3, 0.4, 0.3, 0.2],
                                'mean': 0.2571428571428572,
                                'median': 0.2,
                                'stdev': 0.07867957924694431,
                                'variance': 0.006190476190476191}},
       'versicolor': {'Sepal length': {'values': [5.7, 6.4, 7.0, 5.7, 5.5, 6.5, 6.9],
                                       'mean': 6.242857142857143,
                                       'median': 6.4,
                                       'stdev': 0.6106202935189289,
                                       'variance': 0.3728571428571429},
                      'Sepal width': {'values': [2.8, 3.2, 3.2, 2.8, 2.3, 2.8, 3.1],
                                      'mean': 2.8857142857142857,
                                      'median': 2.8,
                                      'stdev': 0.31847852585154235,
                                      'variance': 0.10142857142857152},
                      'Petal length': {'values': [4.1, 4.5, 4.7, 4.5, 4.0, 4.6, 4.9],
                                       'mean': 4.4714285714285715,
                                       'median': 4.5,
                                       'stdev': 0.31997023671109237,
                                       'variance': 0.10238095238095248},
                      'Petal width': {'values': [1.3, 1.5, 1.4, 1.3, 1.3, 1.5, 1.5],
                                      'mean': 1.4,
                                      'median': 1.4,
                                      'stdev': 0.09999999999999998,
                                      'variance': 0.009999999999999995}}}
"""

from statistics import mean, stdev, median, variance


DATA = [('Sepal length', 'Sepal width', 'Petal length', 'Petal width', 'Species'),
        (5.8, 2.7, 5.1, 1.9, 'virginica'),
        (5.1, 3.5, 1.4, 0.2, 'setosa'),
        (5.7, 2.8, 4.1, 1.3, 'versicolor'),
        (6.3, 2.9, 5.6, 1.8, 'virginica'),
        (6.4, 3.2, 4.5, 1.5, 'versicolor'),
        (4.7, 3.2, 1.3, 0.2, 'setosa'),
        (7.0, 3.2, 4.7, 1.4, 'versicolor'),
        (7.6, 3.0, 6.6, 2.1, 'virginica'),
        (4.9, 3.0, 1.4, 0.2, 'setosa'),
        (4.9, 2.5, 4.5, 1.7, 'virginica'),
        (7.1, 3.0, 5.9, 2.1, 'virginica'),
        (4.6, 3.4, 1.4, 0.3, 'setosa'),
        (5.4, 3.9, 1.7, 0.4, 'setosa'),
        (5.7, 2.8, 4.5, 1.3, 'versicolor'),
        (5.0, 3.6, 1.4, 0.3, 'setosa'),
        (5.5, 2.3, 4.0, 1.3, 'versicolor'),
        (6.5, 3.0, 5.8, 2.2, 'virginica'),
        (6.5, 2.8, 4.6, 1.5, 'versicolor'),
        (6.3, 3.3, 6.0, 2.5, 'virginica'),
        (6.9, 3.1, 4.9, 1.5, 'versicolor'),
        (4.6, 3.1, 1.5, 0.2, 'setosa')]

result = {}