Which measure represents the sharpness of the peak of a statistical…

Concept

Measures of distribution shape describe features other than the location of the centre. Skewness describes directional asymmetry, while kurtosis is based on the fourth standardized moment and is associated with tail weight and, in elementary descriptions, peakedness.

Mean and median are measures of central tendency; they identify the centre of a distribution rather than its shape.

Application

The question asks for the measure traditionally used for the sharpness or peakedness of a distribution. That shape characteristic is represented by kurtosis.

Contrast

• Kurtosis: a shape measure related to the fourth standardized moment, tail weight and conventional peakedness.

• Skewness: a shape measure of left-right asymmetry.

• Mean: the arithmetic-average location of observations.

• Median: the middle-position location after observations are ordered.

Cross-check

Two symmetric distributions can both have zero skewness yet have different tail weights and peak profiles; kurtosis distinguishes that aspect. Therefore, the required measure is kurtosis.