Skewness and Kurtosis¶
Descriptive statistics quantify the shape of a distribution and assess how closely it resembles a normal distribution. Two key measures for evaluating normality are skewness and kurtosis. These metrics describe the asymmetry and peakedness of the data distribution, respectively.
Skewness¶
Skewness measures the asymmetry of the distribution. For a perfectly normal distribution, skewness is 0. A positive skewness indicates a long right tail (data skewed to the right), while a negative skewness indicates a long left tail (data skewed to the left).
The formula for skewness is:
where
- \(n\) is the number of data points,
- \(x_i\) is each data point,
- \(\bar{x}\) is the sample mean,
- \(s\) is the sample standard deviation.
If the skewness is close to zero, the population distribution will likely be symmetric, indicating normality. Significant deviations from zero suggest non-normality.
import numpy as np
from scipy import stats
np.random.seed(0)
# Generate a sample dataset
data = np.random.normal(0, 3, 1000)
# data = np.random.exponential(1, 1000)
# Calculate skewness
skewness_value = stats.skew(data)
print(f"Skewness: {skewness_value:.4f}")
Kurtosis¶
Kurtosis describes the "tailedness" of the distribution. A normal distribution has a kurtosis value of 3 (also called mesokurtic). Kurtosis values above 3 indicate a distribution with heavy tails (leptokurtic), while values below 3 indicate lighter tails (platykurtic).
The formula for kurtosis is:
and the formula for excess kurtosis is:
The subtraction of 3 ensures that a normal distribution has an excess kurtosis of 0 (for easier comparison). The scipy.stats.kurtosis function computes this excess kurtosis, not the raw kurtosis.
A kurtosis value (computed by scipy.stats.kurtosis) near zero suggests a normal distribution. Larger values indicate heavier tails, while smaller values suggest lighter tails than normal.
import numpy as np
from scipy import stats
np.random.seed(0)
# Generate a sample dataset
# data = np.random.normal(0, 1, 1000)
data = np.random.exponential(1, 1000)
# Calculate skewness
skewness_value = stats.skew(data)
print(f"Skewness: {skewness_value:.4f}")
# Calculate kurtosis
kurtosis_value = stats.kurtosis(data)
print(f"Kurtosis: {kurtosis_value:.4}")