a measure of the peakedness of the distribution

Kurtosis is a measure of the peakedness of the distribution, and the high kurtosis values from the data set imply the data are non-normal and result in a sharply peaked distribution.

a measure of tail density

Kurtosis is a measure of tail density and this is the interpretation that matters from a risk perspective.

the fourth standardized moment of a random variable

The kurtosis is the fourth standardized moment of a random variable.

the shape of the probability distribution at the tails of the tails

Skewness measures the asymmetry of the probability distribution around the mean, similarly kurtosis defines the shape of the probability distribution at the tails of the tails .

a statistical measure that is used to describe the distribution

Like skewness, kurtosis is a statistical measure that is used to describe the distribution.

a measure of whether the distribution is heavy tailed or light tailed

Kurtosis is a measure of whether the distribution is heavy tailed or light tailed; “excess kurtosis” is kurtosis relative to the normal distribution (with a kurtosis of 3).

a measure of the flattening of the distribution

Kurtosis is a measure of the flattening of the distribution, and for a normal distribution amounts to 0.

a statistical measure , that's used to describe the distribution, or skewness, of observed data around the meansometimes referred to as the volatility of volatility

Kurtosis is a statistical measure that's used to describe the distribution, or skewness, of observed data around the mean, sometimes referred to as the volatility of volatility.

a measure of peakiness

Kurtosis, on the other hand, is a measure of peakiness.

a measure of the heaviness of the tails of a distribution

k. Kurtosis – Kurtosis is a measure of the heaviness of the tails of a distribution.

a measure of how outlier-prone a distribution is

Kurtosis is a measure of how outlier-prone a distribution is.

as the volatility of volatility

Kurtosis is sometimes referred to as the volatility of volatility.

to the degree to which the distribution is peaked

While skewness refers to the lack of symmetry, kurtosis is refers to the degree to which the distribution is peaked.