Skewtest and Kurtosistest¶

Skewtest¶

The skewness test, provided by scipy.stats.skewtest(), is a formal statistical test that evaluates whether the skewness of a dataset significantly deviates from zero, indicating whether the data is symmetric or not.

Hypotheses¶

Null Hypothesis (\(H_0\)): The data has zero skewness (is symmetrically distributed).
Alternative Hypothesis (\(H_1\)): The data has non-zero skewness (is asymmetrical).

The skewness test provides a test statistic and a \(p\)-value. If the \(p\)-value is small (typically less than 0.05), we reject the null hypothesis and conclude that the data is not symmetrically distributed.

Step-by-Step Explanation¶

Calculate the sample skewness: We first compute the sample skewness using the formula

\[ \text{Skewness} = \frac{1}{n} \sum_{i=1}^{n} \left( \frac{x_i - \bar{x}}{\sigma} \right)^3 \]

where \(n\) is the number of data points, \(x_i\) are the individual data points, \(\bar{x}\) is the mean, and \(\sigma\) is the standard deviation.
Compute the z-score: The test statistic, or z-score, is computed by dividing the observed skewness by its standard error. The z-score tells us how far the skewness deviates from the expected value (zero for a normal distribution).
Convert z-score to p-value: We compute the p-value using the cumulative distribution function (CDF) of the standard normal distribution. We compare the z-score obtained in the previous step to the standard normal distribution to get the two-tailed p-value. This p-value quantifies how likely it is to observe a skewness as extreme as the observed value under the null hypothesis of zero skewness (i.e., symmetric distribution).

Applications¶

The skewness test is practical when assessing whether the data can be analyzed using parametric statistical methods that assume symmetry (such as specific versions of the \(t\)-test or ANOVA). If we conclude that the data is skewed, it may be necessary to transform the data (e.g., using a log or Box-Cox transformation) or use non-parametric methods that do not assume symmetry.

import numpy as np
from scipy import stats

np.random.seed(0)

# Generate a sample dataset with skewness
# data = np.random.gamma(2, 2, 1000)
data = np.random.normal(2, 2, 1000)

# Calculate skewness
skewness_value = stats.skew(data)
print(f"Skewness: {skewness_value:.4f}")

# Perform skewness test
stat, p_value = stats.skewtest(data)
print(f"Skewness Test: Statistic={stat:.4f}, p-value={p_value:.4f}")

# Interpretation
alpha = 0.05
if p_value <= alpha:
    print("Reject H_0: The data is not symmetrically distributed (significant skewness).")
else:
    print("Fail to reject H_0: The data is symmetrically distributed (no significant skewness).")

Kurtosistest¶

The kurtosis test, provided by scipy.stats.kurtosistest(), is a formal statistical test that evaluates whether the excess kurtosis of a dataset significantly deviates from the kurtosis of a normal distribution.

Hypotheses¶

Null Hypothesis (\(H_0\)): The data has normal kurtosis.
Alternative Hypothesis (\(H_1\)): The data does not have normal kurtosis.

The kurtosis test provides a test statistic and a \(p\)-value. If the \(p\)-value is small (typically less than 0.05), we reject the null hypothesis and conclude that the data does not have normal kurtosis.

Step-by-Step Explanation¶

Calculate the sample excess kurtosis: We first compute the sample excess kurtosis using the formula

\[ \text{Excess Kurtosis} = \frac{1}{n} \sum_{i=1}^{n} \left( \frac{x_i - \bar{x}}{\sigma} \right)^4 - 3 \]

where \(n\) is the number of data points, \(x_i\) are the individual data points, \(\bar{x}\) is the mean, and \(\sigma\) is the standard deviation.
Compute the z-score: The test statistic, or z-score, is computed by dividing the observed excess kurtosis by its standard error. The z-score tells us how far the excess kurtosis deviates from the expected value (zero for a normal distribution).
Convert z-score to p-value: We compute the p-value using the CDF of the standard normal distribution to get the two-tailed p-value. This p-value quantifies how likely it is to observe an excess kurtosis as extreme as the observed value under the null hypothesis.

Applications¶

We use the kurtosis test when assessing the suitability of parametric methods (such as the \(t\)-test or ANOVA), which assume normal kurtosis. If the kurtosis test indicates that the data has significantly heavy or light tails, transformations (e.g., log transformation or Box-Cox transformation) or non-parametric methods may be needed.

import numpy as np
from scipy import stats

np.random.seed(0)

# Generate a sample dataset
# data = np.random.gamma(2, 2, 1000)
data = np.random.normal(2, 2, 1000)

# Calculate kurtosis
kurtosis_value = stats.kurtosis(data)
print(f"Kurtosis: {kurtosis_value:.4f}")

# Perform kurtosis test
stat, p_value = stats.kurtosistest(data)
print(f"Kurtosis Test: Statistic={stat:.4f}, p-value={p_value:.4f}")

# Interpretation
alpha = 0.05
if p_value <= alpha:
    print("Reject H_0: The data does not have normal kurtosis.")
else:
    print("Fail to reject H_0: The data has normal kurtosis.")