Question

why is there so much debate on the effectiveness/use of kurtosis?

Answer 1

If you look at the (standard) measure of Kurtosis, . This is nothing but the average of the standardised data (which is raised to the power of 4) of the data/random variable whatever the case may be. So, if you think of those observations which are very close to the mean, they will contribute very little to the value ok Kurtosis. Hence, Kurtosis basically tells you how extreme are the outliers in the dataset.

The debate arises mainly when people try to infer a notion of 'peakedness' from Kurtosis. Kurtosis doesn't really tell you anything meaningful about the peak. But it does tell you about the presence of outliers. There are some more wrong interpretations of Kurtosis but this is the most popular one.