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Kurtosis

https://www.statisticshowto.com/probability-and-statistics/statistics-definitions/kurtosis-leptokurtic-platykurtic/

https://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm

https://en.wikipedia.org/wiki/Kurtosis

The kurtosis is a measure of the "tailedness" of the probability distribution. https://en.wikipedia.org/wiki/Kurtosis

• A positive value tells that the distribution has a heavy tail (outlier) compared to the normal distribution (which means that there is a lot of data in the tail).
• A negative value means that the distribution has a light tail (which means that there is little data in the tail).
• This heaviness or lightness in the tails usually means that your data looks flatter (or less flat) compared to the normal distribution.
• The standard normal distribution has a kurtosis of 3, so if your values are close to that then your graph’s tails are nearly normal.

The Kurtosis is related to the tails of the distribution, not its peak; hence, the sometimes-seen characterization of kurtosis as "peakedness" is incorrect. https://en.wikipedia.org/wiki/Kurtosis

• The kurtosis of any univariate normal distribution is 3.
• Distributions with kurtosis less than 3 are said to be platykurtic. An example of a platykurtic distribution is the uniform distribution, which does not produce outliers.
• Distributions with kurtosis greater than 3 are said to be leptokurtic. An example of a leptokurtic distribution is the Laplace distribution, which has tails that asymptotically approach zero more slowly than a Gaussian, and therefore produces more outliers than the normal distribution.
• It is also common practice to use an adjusted version of Pearson's kurtosis, the excess kurtosis, which is the kurtosis minus 3, to provide the comparison to the standard normal distribution. Some authors use "kurtosis" by itself to refer to the excess kurtosis. https://en.wikipedia.org/wiki/Kurtosis

In Python: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.kurtosis.html

scipy.stats.kurtosis(a, axis=0, fisher=True, bias=True, nan_policy='propagate')

Compute the kurtosis (Fisher or Pearson) of a dataset.

import numpy as np
from scipy.stats import kurtosis

data = norm.rvs(size=1000, random_state=3)
data2 = np.random.randn(1000)

kurtosis(data2)

from scipy.stats import kurtosis
import matplotlib.pyplot as plt
import scipy.stats as stats

x = np.linspace(-5, 5, 100)
ax = plt.subplot()
distnames = ['laplace', 'norm', 'uniform']

for distname in distnames:
    if distname == 'uniform':
        dist = getattr(stats, distname)(loc=-2, scale=4)
    else:
        dist = getattr(stats, distname)
    data = dist.rvs(size=1000)
    kur = kurtosis(data, fisher=True)
    y = dist.pdf(x)
    ax.plot(x, y, label="{}, {}".format(distname, round(kur, 3)))
    ax.legend()

Recreated from https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.kurtosis.html