Kurtosis measures a distribution's outliers, not how pointy its peak is
Many textbooks describe kurtosis as peakedness, the sharpness of a curve's summit. Statisticians now agree that description is wrong. Kurtosis is about the tails: how prone a distribution is to extreme values far from the middle. The confusion may trace back to Karl Pearson, who introduced the measure while describing it in terms of flat tops.
Its name borrows a Greek word meaning curved or arching. Pearson's standard version is the fourth standardised moment: take each value's distance from the mean in units of standard deviation, raise it to the fourth power, and average. That arithmetic explains everything. Anything close to the centre, less than a single standard deviation away, becomes a number smaller than one, and raising those to the fourth power shrinks them almost to nothing. The values that dominate are the ones far out in the tails. So high kurtosis signals outliers present in a sample, or a tendency to produce them, and says essentially nothing about the centre.
To make comparisons easy, people often use excess kurtosis, which subtracts 3 so that the normal bell curve scores zero. Distributions scoring zero are called mesokurtic. Positive scores mean leptokurtic, with fatter tails and more extreme values; the Laplace, exponential, Poisson and Student's t distributions are examples. Negative scores mean platykurtic, with fewer or milder outliers; the uniform distribution, flat across a range and zero elsewhere, is one. Despite the name, a platykurtic curve need not have a flat top.
There is a floor but no ceiling. Kurtosis can never fall below the square of the skewness plus one, a limit reached by the simple two-outcome Bernoulli distribution, but it can grow without limit and may even be infinite.
In 1986 the statistician Moors offered a tidy reinterpretation: kurtosis measures how widely values spread around the two points one standard deviation either side of the mean. It is large either when most probability hugs the mean but occasional values fly far away, or when mass piles up in the tails. Some scholars still argue that no single number captures the concept well, but the tail interpretation is now considered settled.
Source: Kurtosis
