.. _skewness-kurtosis:

Skewness and Kurtosis
======================

Skewness and kurtosis are standardised central moments that characterise
the *shape* of a distribution independently of its location and scale.
ToFUL computes both automatically when the moment reference is set to
**Mean (a = μ)** and at least four orders are computed.

.. contents:: On this page
   :local:
   :depth: 2

----

Skewness
--------

Definition
~~~~~~~~~~

Skewness is the normalised third central moment:

.. math::

   \gamma_1 = \frac{\mu_3}{\sigma^3} = \frac{E[(X-\mu)^3]}{(E[(X-\mu)^2])^{3/2}}

It is dimensionless and invariant under linear transformations
:math:`Y = aX + b` (with :math:`a > 0`).

Interpretation
~~~~~~~~~~~~~~

+---------------------+-----------------------------------------------------+
| Skewness value      | Shape                                               |
+=====================+=====================================================+
| γ₁ < 0              | Left-skewed (negative skew). The left tail is       |
|                     | longer; the mass is concentrated on the right.      |
+---------------------+-----------------------------------------------------+
| γ₁ ≈ 0              | Approximately symmetric.                            |
+---------------------+-----------------------------------------------------+
| γ₁ > 0              | Right-skewed (positive skew). The right tail is     |
|                     | longer; the mass is concentrated on the left.       |
+---------------------+-----------------------------------------------------+

.. note::

   "Concentrated on the right with a long left tail" is sometimes
   counter-intuitive. Remember: the tail points in the direction of the
   skew, not the bulk of the distribution.

Examples
~~~~~~~~

+------------------------------+------------+---------------------------------+
| Distribution                 | Skewness   | Intuition                       |
+==============================+============+=================================+
| Normal(mu, sigma)            | 0          | Perfectly symmetric             |
+------------------------------+------------+---------------------------------+
| Exponential(lambda)          | 2          | Strong right skew               |
+------------------------------+------------+---------------------------------+
| Uniform(a, b)                | 0          | Symmetric                       |
+------------------------------+------------+---------------------------------+
| Geometric(p=0.3)             | ~2.03      | Right-skewed                    |
+------------------------------+------------+---------------------------------+
| Beta(2, 5)                   | ~0.596     | Mild right skew                 |
+------------------------------+------------+---------------------------------+
| Beta(5, 2)                   | ~-0.596    | Mild left skew (mirror of above)|
+------------------------------+------------+---------------------------------+

----

Kurtosis
--------

Definition
~~~~~~~~~~

Kurtosis is the normalised fourth central moment:

.. math::

   \gamma_2 = \frac{\mu_4}{\sigma^4} = \frac{E[(X-\mu)^4]}{(E[(X-\mu)^2])^{2}}

**Excess kurtosis** subtracts 3 (the kurtosis of the Normal distribution)
to make the Normal a natural reference point:

.. math::

   \kappa = \gamma_2 - 3 = \frac{\mu_4}{\sigma^4} - 3

Interpretation
~~~~~~~~~~~~~~

Kurtosis measures tail heaviness — how much probability mass sits in the
tails relative to a Normal distribution with the same mean and variance.

+----------------------+-----------------------------------------------------+
| Excess kurtosis      | Shape classification                                |
+======================+=====================================================+
| κ < 0                | **Platykurtic** — lighter tails than Normal.        |
|                      | Distribution is "flatter" with less extreme values. |
|                      | Example: Uniform distribution (κ = −1.2).           |
+----------------------+-----------------------------------------------------+
| κ ≈ 0                | **Mesokurtic** — tails like the Normal distribution.|
|                      | Example: Normal (κ = 0 by definition).              |
+----------------------+-----------------------------------------------------+
| κ > 0                | **Leptokurtic** — heavier tails than Normal.        |
|                      | More probability in the tails; more extreme values. |
|                      | Example: Exponential (κ = 6), Laplace (κ = 3).      |
+----------------------+-----------------------------------------------------+

.. important::

   Kurtosis is often described as measuring "peakedness", but this is
   misleading. A distribution can have a high peak and low kurtosis, or
   a flat peak and high kurtosis. Tail weight is the more accurate
   interpretation.

Examples
~~~~~~~~

+-------------------------------+----------+----------------+
| Distribution                  | Kurtosis | Excess kurtosis|
+===============================+==========+================+
| Normal(μ, σ)                  | 3        | 0              |
+-------------------------------+----------+----------------+
| Exponential(λ)                | 9        | 6              |
+-------------------------------+----------+----------------+
| Uniform(a, b)                 | 1.8      | −1.2           |
+-------------------------------+----------+----------------+
| Laplace(μ, b)                 | 6        | 3              |
+-------------------------------+----------+----------------+
| Geometric(p=0.3)              | ≈ 9.61   | ≈ 6.61         |
+-------------------------------+----------+----------------+

----

Numerical Considerations
-------------------------

Both skewness and kurtosis involve dividing by a power of the standard
deviation. If the standard deviation is very small (near-constant
distribution), the division amplifies numerical errors significantly.
ToFUL guards against this by only computing skewness and kurtosis when
:math:`\sigma > 10^{-15}`.

For heavy-tailed distributions like Pareto or Cauchy, the integrals
defining :math:`\mu_3` and :math:`\mu_4` may not converge. In these
cases the values shown are the partial results from truncated integration
and should not be interpreted as the true population skewness/kurtosis.
The Convergence tab will indicate non-convergence.

See also
--------

* :doc:`statistical-moments` — definitions of central moments
* :doc:`raw-vs-central` — why central moments are used here
* :doc:`/user-guide/interpreting-results` — reading the Statistics tab