Statistics

The aim of the descriptive statistics is to provide

• measures of central tendency for the signal samples including mean, weighted mean, winsor mean, median, mode and midrange.

• measures of variability or dispersion for the signal samples including minimum, maximum, range, average absolute deviation, median absolute deviation, variance, standard deviation, coefficient of variation, skew and kurtosis.

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Number of Sample

You can compute the number of the signal samples with the built-in function len().

>>> # number of signal samples
>>> len(Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]))
6

Sum of the Samples

You can compute the sum \(x_{sum} = \sum\limits_{i=0}^{N}{x_i}\) of the signal samples \(N\) by calling the method sum().

>>> # sum of the signal samples
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).sum()
10.0

You can compute the sum \(x_{sum}\) of the signal samples with the built-in function sum().

>>> # sum of the signal samples
>>> sum(Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]))
10.0

Count Occurrences in the Samples

You can count the occurrences of a sample \(x\) in the signal samples \(N\) by calling the method count().

>>> # count occurrences in the signal samples
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).count(1.5)
1

Sort the Samples

You can sort the signal samples in ascending order by calling the method sort().

A new Trace instance labeled with the performed transformation 'sort' is returned.

>>> Trace('Signal', [1.3, 1.4, 1.5, 1.6, 1.7, 2.5]).sort()
Trace(label='Signal:sort', samples=[1.3, 1.4, 1.5, 1.6, 1.7, 2.5])

You can sort the signal samples in ascending order with the built-in function sorted().

>>> sorted(Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]))
[1.3, 1.4, 1.5, 1.6, 1.7, 2.5]

Winsorize the Samples

You can winsorize the signal samples by calling the method winsorize(), either with the percentage limits of the number of samples shall be “trimmed” for each side of the ascending sorted signal samples, or with the percentage limit of the number of samples shall be “trimmed” from both sides.

>>> # winsorized signal samples with 20% of the largest samples trimmed
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).winsorize((None, 0.2))
Trace(label='Signal:winsorize', samples=[1.5, 1.6, 1.4, 1.7, 1.3, 1.7])

>>> # winsorized signal samples with 20% of the smallest samples trimmed
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).winsorize((0.2, None))
Trace(label='Signal:winsorize', samples=[1.5, 1.6, 1.4, 2.5, 1.4, 1.7])

>>> # winsorized signal samples with 20% of smallest and largest samples trimmed
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).winsorize(0.2)
Trace(label='Signal:winsorize', samples=[1.5, 1.6, 1.4, 1.7, 1.4, 1.7])

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Mean of the Samples

Note

The mean \(\overline{x}\) is the 1st common moment \(\mu_1\) in the descriptive statistics.

You can compute the arithmetic mean \(\overline{x} = \frac{1}{N} \cdot \sum\limits_{i=0}^{N}{x_i}\) of the signal samples \(N\) by calling the method mean().

>>> # arithmetic mean of the signal samples
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).mean()
1.6666666666666667

You can compute the arithmetic mean \(\overline{x}\) of the signal samples \(N\) with the function mean() of the statistics module.

>>> import statistics as stats
>>> # arithmetic mean of the signal samples
>>> stats.mean(list(Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7])))
1.6666666666666667

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Weighted Mean of the Samples

You can compute the linear weighted mean \(\overline{x_w} = \frac{2}{N \cdot (N + 1)} \cdot \sum\limits_{i=0}^{N}{(i+1) \cdot x_i}\) of the signal samples \(N\) by calling the method weighted_mean().

>>> # linear weighted mean of the signal samples
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).weighted_mean()
1.6952380952380948

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Winsor Mean of the Samples

You can compute the winsor mean \(\overline{x}_{w\alpha}\) of the signal samples \(N\) by calling the method winsor_mean(), either with the percentage limits of the number of samples shall be “trimmed” for each side of the ascending sorted signal samples, or with the percentage limit of the number of samples shall be “trimmed” from both sides.

>>> # mean of the signal samples with 20% of the largest samples trimmed
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).winsor_mean((None, 0.2))
1.5333333333333332

>>> # mean of the signal samples with 20% of the smallest samples trimmed
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).winsor_mean((0.2, None))
1.6833333333333333

>>> # mean of the signal samples with 20% of smallest and largest samples trimmed
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).winsor_mean(0.2)
1.55

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Median of the Samples

You can compute the median \(\overline{x}_{med}\) of the signal samples \(N\) by calling the method median().

>>> # median of the signal samples
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).median()
1.55

You can compute the median \(\overline{x}_{med}\) of the signal samples \(N\) with the function median() of the statistics module.

>>> import statistics as stats
>>> # median of the signal samples
>>> stats.median(Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]))
1.55

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Mode of the Samples

You can compute the mode \(\overline{x}_{mod}\) of the signal samples \(N\) by calling the method mode().

>>> # mode of the signal samples
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).mode()
1.5

You can compute the mode \(\overline{x}_{mod}\) of the signal samples \(N\) with the function mode() of the statistics module.

>>> import statistics as stats
>>> # mode of the signal samples
>>> stats.mode(Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]))
1.5

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Root mean square of the Samples

You can compute the root mean square \(x_{rms} = \sqrt{\frac{1}{N} \cdot \sum\limits_{i=0}^{N}{x_i^2}}\) of the signal samples \(N\) by calling the method rms().

>>> # median of the signal samples
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).rms()
1.7126976771553504

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Minimum Sample

You can get the minimum \(x_{min}\) within the signal samples \(N\) by calling the method min().

>>> # minimum signal value in the signal samples
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).min()
1.3

You can get the minimum \(x_{min}\) within the signal samples \(N\) with the built-in function min().

>>> # minimum signal value in the signal samples
>>> min(Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]))
1.3

Maximum Sample

You can get the maximum \(x_{max}\) within the signal samples \(N\) by calling the method max().

>>> # maximum signal value in the signal samples
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).max()
2.5

You can get the maximum \(x_{max}\) within the signal samples \(N\) with the built-in function max().

>>> # maximum signal value in the signal samples
>>> max(Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]))
2.5

Range of the Samples

You can compute the range \(x_{max} - x_{min}\) of the signal samples \(N\) by calling the method range().

>>> # range of the signal samples
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).range()
1.2

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Mid-Range of the Samples

You can compute the midrange \(\frac{x_{max} + x_{min}}{2}\) of the signal samples \(N\) by calling the method midrange().

>>> # mid-range of the signal samples
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).midrange()
1.9

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Average Absolute Deviation of the Samples

Note

The average absolute deviation is the 1st central moment \(m_1\) in the descriptive statistics.

You can compute the absolute error or average absolute deviation \(D_{mean} = m_1 = \frac{1}{N} \cdot \sum\limits_{i=0}^{N} {\left| x_i - \overline{x} \right|}\) of the signal samples \(N\) distribution.

>>> # average absolute deviation of the distribution
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).aad()
0.2888888888888889

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Median Absolute Deviation of the Samples

You can compute the median absolute deviation \(D_{med} = \frac{1}{N} \cdot \sum\limits_{i=0}^{N} {\left| x_i - \overline{x}_{med} \right|}\) of the signal samples \(N\) distribution.

>>> # median absolute deviation of the distribution
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).mad()
0.26666666666666666

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Variance of the Samples

Note

The variance \(\sigma^2\) is the 2nd central moment \(m_2\) in the descriptive statistics.

You can compute the biased sample variance \(s^2 = \sigma^2 = m_2 = \frac{1}{N} \cdot \sum\limits_{i=0}^{N} {(x_i - \overline{x})^2}\) of the signal samples \(N\) distribution by calling the method variance().

>>> # biased sample variance
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).variance()
0.15555555555555553

You can compute the biased sample variance \(s^2\) of the signal samples \(N\) distribution with the function pvariance() of the statistics module.

>>> import statistics as stats
>>> # biased sample variance
>>> stats.pvariance(Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]))
0.15555555555555556

You can compute the unbiased sample variance \(s^2\) of the signal samples \(N - 1\) distribution with the function variance() of the statistics module.

>>> import statistics as stats
>>> # unbiased sample variance
>>> stats.variance(Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]))
0.18666666666666668

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Standard deviation of the Samples

You can compute the biased sample standard deviation \(s = \sigma= \sqrt{\sigma^2} = \sqrt{m_2}\) of the signal samples \(N\) distribution by calling the method std().

>>> # biased sample standard deviation
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).std()
0.3944053188733077

You can compute the unbiased sample standard deviation \(s\) of the signal samples \(N - 1\) distribution with the function stdev() of the statistics module.

>>> import statistics as stats
>>> # unbiased sample standard deviation
>>> stats.stdev(Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]))
0.43204937989385733

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Coefficient of Variation of the Samples

You can compute the coefficient of variation \(c_v = \frac{\sigma}{\overline{x}}\) of the signal samples \(N\) distribution by calling the method coefficient().

>>> # coefficient of variation of the distribution
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).coefficient()
0.23664319132398462

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Skew of the Samples

Note

The skew is the 3rd central moment \(m_3\) in the descriptive statistics.

You can compute the biased sample skew \({m}_3 = \sum\limits_{i=0}^{N}{(\frac{x_i - \overline{x}}{\sigma})^3}\) of the signal samples \(N\) distribution by calling the method skew().

>>> # biased sample skew
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).skew()
1.3733756639338395

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Kurtosis of the Samples

Note

The kurtosis is the 4th central moment \(m_4\) in the descriptive statistics.

You can compute the biased sample kurtosis \({m}_4 = \sum\limits_{i=0}^{N}{(\frac{x_i - \overline{x}}{\sigma})^4}\) of the signal samples \(N\) distribution by calling the method kurtosis().

>>> # biased sample kurtosis
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).kurtosis()
3.4864285714285725

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Distribution of the Samples

Histogram Plot

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Empirical Cumulative Distribution Plot

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Distribution Plot

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Quantiles of the Samples

You can compute the nth-quantile \(q_n\) of the signal samples with the function quantiles() of the statistics module.

>>> import statistics as stats
>>> # quartiles of the samples maximal value excluded
>>> stats.quantiles(Trace('Signal', range(11)))
[2.0, 5.0, 8.0]
>>> # quartiles of the samples maximal value included
>>> stats.quantiles(Trace('Signal', range(11)), method='inclusive')
[2.5, 5.0, 7.5]
>>> # deciles of the samples maximal value excluded
>>> stats.quantiles(Trace('Signal', range(11)), n=10)
[0.2, 1.4, 2.6, 3.8, 5.0, 6.2, 7.4, 8.6, 9.8]

Quartiles: Box Plot

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Quartiles: Violin Plot

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Z-Score of the Samples

You can compute for each signal sample \(x\) the zscore \(z = \frac{x - \bar{x}}{\sigma}\) by calling the method zscore().

A new Trace instance labeled with the performed transformation 'zscore' is returned.

>>> # score for each signal sample
>>> Trace('Signal', [1.5, 1.6, 1.4, 2.5, 1.3, 1.7]).zscore()
Trace(label='Signal:zscore',
      samples=[-0.4225771273642585, -0.16903085094570328, -0.6761234037828138,
               2.1128856368212916, -0.9296696802013684, 0.08451542547285136])

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Z-Score: Histogram Plot

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