Statistical Inference: A Short Course

3.10 Kurtosis

The concept of kurtosis refers to how flat or rounded the peak of a distribution happens to be. While there are a variety of ways to measure the degree of kurtosis in a data set, one approach, which is quite straightforward to apply, is the so-called coefficient of kurtosis:

(3.17)

where the quartile deviation (q.d.) is determined from Equation (3.13). The benchmark or point of reference for determining the degree of kurtosis for a data set is the normal distribution (it is continuous, bell-shaped, and symmetrical). In this regard:

a. For a normal curve, k ≈ 0.263.

b. For a flat curve, k ≈ 0.

c. For a sharply peaked curve, k ≈ 0.5.

Typically k is computed when one has a fairly large data set.

An alternative to Equation (3.17) is to calculate standard kurtosis as the average of the Z-scores raised to the fourth power or

(3.18)

For a normal distribution (again used as our benchmark), k₄ = 3. Then our estimate of the degree of kurtosis is k₄ − 3 (called the excess). So if k₄ − 3 > 0 (positive kurtosis), the data set has a peak that is sharper than that of a normal distribution; and if k₃ − 3 < 0 (negative kurtosis), the data set has a peak that is flatter than that of a normal distribution. In this latter instance, the distribution might have elongated tails, thus ...