Definition:Kurtosis
Definition
Let $X$ be a random variable with mean $\mu$ and standard deviation $\sigma$.
The kurtosis of $X$ is a measure of the concentration of $X$ about its expectation.
Definition 1
The kurtosis of $X$ is the fourth standardized moment of $X$:
- $\alpha_4 = \expect {\paren {\dfrac {X - \mu} \sigma}^4}$
where $\expect {\, \cdot \,}$ denotes expectation.
Definition 2
The kurtosis of $X$ is defined as:
- $\alpha_4 = \dfrac {\mu_4} {\paren {\mu_2}^2}$
where $\mu_i$ denotes the $i$th central moment of $X$.
Coefficient of Kurtosis
Let $X$ be a random variable with mean $\mu$ and standard deviation $\sigma$.
The coefficient of kurtosis of $X$ is the coefficient:
- $\gamma_2 = \dfrac {\mu_4} { {\mu_2}^2} - 3$
where $\mu_i$ denotes the $i$th central moment of $X$.
Notation
The kurtosis of $X$ is usually denoted $\alpha_4$.
Some sources denote the kurtosis by the symbol $\Beta_2$ or $\beta_2$.
Platykurtic
A random variable with kurtosis less than $3$ is described as platykurtic.
Mesokurtic
Let $X$ be a random variable with kurtosis equal to $3$.
Then $X$ is described as mesokurtic.
Leptokurtic
A random variable with kurtosis greater than $3$ is described as leptokurtic.
Incorrect Definition
The kurtosis of a random variable is often seen described as a measure of how steep the peak is at the mean or the mode.
However, this is incorrect.
It is more accurate to describe it as a measure of how heavy or fat the tails are.
Also see
- Results about kurtosis can be found here.
Linguistic Note
The word kurtosis derives from the Greek κυρτός (kyrtos or kurtos), meaning curved or arching.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): kurtosis
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): kurtosis