skewness#

pycafee.normalitycheck.gaussian.Gaussian.skewness(self, x_exp, alfa=None, details=None)#

This function applies the skewness test on a sample dataset using the method described by [1].

Parameters

x_expnumpy array

One dimension numpy array with at least 4 sample data.

alfafloat, optional

The level of significance (ɑ). Default is None which results in 0.05 (ɑ = 5%).

detailsstr, optional

The details parameter determines the amount of information presented about the hypothesis test.

If details = "short" (or None, e.g, the default), a simplified version of the test result is returned.
If details = "full", a detailed version of the hypothesis test result is returned.
if details = "binary", the conclusion will be 1 (\(H_0\) is rejected) or 0 (\(H_0\) is accepted).

Returns

resulttuple with

statisticfloat

The test statistic.

criticallist of two floats

The critical values for the adopted significance level, where:

critical[0] is the lower critical value (always negative);
critical[1] is the upper critical value (always positive);

alphafloat

The adopted level of significance.

conclusionstr or int

The test conclusion (e.g, Normal/ not Normal).

See also

pycafee.normalitycheck.shapirowilk.ShapiroWilk.fit
kurtosis

Notes

Skewness (\(S\)) is estimated using the following equation:

\[S=\frac{n}{(n-1)(n-2)}\sum_{i=1}^n\left(\frac{x_i - \overline{x}}{s}\right)^{3}\]

The standard deviation of skewness (\(S_{std}\)) is estimated using the following equation:

\[S_{std} = \sqrt{\frac{6n(n-1)}{(n-2)(n+1)(n+3)}}\]

and the test statistic (\(z_{S}\)) is estimated through the following equation:

\[z_{S} = \frac{S-0}{S_{std}}\]

The test hypotheses are as follows:

☕

\(H_0:\) the skewness is Normal

\(H_1:\) the skewness is Not Normal

The comparison is performed between the calculated test statistic and the critical values (at alpha significance level) as follows:

if critical.Lower <= statistic <= critical.Upper:
    the skewness is Normal
else:
    the skewness is Not Normal

The lower critical value is obtained with alfa/2 and the upper critical value is obtained with 1 - alfa/2 significance level (e.g., two side distribution), using stats.norm.ppf() [2].

References

1: CORDER, G. W.; FOREMAN, D. I. NONPARAMETRIC STATISTICS A Step-by-Step Approach. 2. ed. New Jersey: John Wiley & Sons, 2014, p. 19, eq 2.7.
2: SCIPY. stats.norm.ppf. Available at: https://docs.scipy.org. Access on: 10 May. 2022.

Examples

>>> from pycafee.normalitycheck.gaussian import Gaussian
>>> import numpy as np
>>> x = np.array([90, 72, 90, 64, 95, 89, 74, 88, 100, 77, 57, 35, 100, 64, 95, 65, 80, 84, 90, 100, 76])
>>> teste = Gaussian()
>>> result, conclusion = teste.skewness(x, details='full')
>>> print(result)
SkewnessResult(Statistics=-2.0304772981338326, Critical=[-1.9599639845400545, 1.959963984540054], Alpha=0.05)
>>> print(conclusion)
Since the calculated statistic for the skewness (-2.03) is a value lower than the lower critical value (-1.959), we have evidence to reject the null hypothesis, and we can say that the distribution does not have skewness similar to the skewness of a Normal distribution (with 95.0% confidence).

>>> from pycafee.normalitycheck.gaussian import Gaussian
>>> import numpy as np
>>> x = np.array([5.1, 4.9, 4.7, 4.6, 5.0, 5.4, 4.6, 5.0, 4.4, 4.9])
>>> teste = Gaussian(language='pt-br')
>>> result, conclusion = teste.skewness(x)
>>> print(result)
ResultadoAssimetria(Estatistica=0.32955498519357374, Critico=[-1.9599639845400545, 1.959963984540054], Alfa=0.05)
>>> print(conclusion)
A Assimetria é Normal (95.0% de confiança)

kurtosis

DotPlot