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.

Skewness 

Definition 

Skewness is the normalised third central moment:

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

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

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:

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

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

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

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 \(\sigma > 10^{-15}\).

For heavy-tailed distributions like Pareto or Cauchy, the integrals defining \(\mu_3\) and \(\mu_4\) 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.