peakedness

Peakedness is a statistical term that describes how sharply a probability distribution rises to its central peak and falls toward its tails. It conveys the degree to which most of the probability mass is concentrated near the center.

In formal terms, peakedness is typically assessed by kurtosis, a standardized moment that reflects both the height of the central peak and the heaviness of the tails. A distribution with higher kurtosis has greater peakedness (leptokurtic), while lower kurtosis indicates a flatter peak (platykurtic); a normal distribution is described as mesokurtic with a kurtosis close to 3. Excess kurtosis, defined as kurtosis minus 3, is commonly used to compare a distribution with the normal reference.

Kurtosis is defined as E[(X − μ)⁴] / σ⁴, where μ is the mean and σ² the variance. In practice, sample kurtosis estimates are used, and sometimes Fisher’s or Pearson’s definitions are applied. Estimates can be biased in small samples, and they are sensitive to outliers, which can distort assessments of peakedness.

Peakedness is distinct from skewness, which measures asymmetry rather than central sharpness. While kurtosis primarily captures tail behavior and the height of the peak, it does not uniquely determine all aspects of a distribution’s shape.

Common examples include the Laplace distribution, which is leptokurtic and exhibits high peakedness, and the uniform distribution, which is platykurtic with a flatter peak. The normal distribution remains the reference for mesokurtic peakedness.

See also: kurtosis, skewness, leptokurtic, mesokurtic, platykurtic.