A normal distribution is said to be mesokurtic and has a 'normal' amount of peakedness and tails that are not too heavy nor too light.
There are several different but related formulae for computing kurtosis. The one given below is the one used by Stata.
kurtosis = (m4/m22) - 3
where m2 = Σ(X - mean)2/N
m4 = Σ(X - mean)4/N
The formula uses the second and fourth moment about the mean. Moments involve
powers of deviations, for example the second moment uses the sum of squared deviations and
the fourth moment uses the sum of the fourth powers of deviations.
The second moment about the mean, m2, is really a version of the variance related to the unbiased estimate of the variance that uses N-1 in the denominator. Thus, the index of kurtosis is based on ratio of the fourth moment about the mean divided by the variance squared. In a normal distribution the kurtosis is equal to 3. A platykurtic distribution has a value less than three, while leptokurtic distributions have values greater than three. Below, are some examples to give you an idea of the values that kurtosis can take on.
ex 1: kurtosis = 11.00 ex 2: kurtosis = 4.33 ex 3: kurtosis = 3.00 ex 4: kurtosis = 2.00 ex 5: kurtosis = 1.50 ex 5: kurtosis = 1.25 ex 7: kurtosis = 1.00The next example is from a t-tdistribution with 2 degrees of freedom. It is extremely leptokurtic.
ex 8: kurtosis = 26.65Next a t-tdistribution with 5 degrees of freedom. It is less leptokurtic than the previous example but still somewhat tail-heavy.
ex 9: kurtosis = 4.89The final example shows a normal distribution or at least as close as we can come to a normal distribution with 400 observations.
ex 10: kurtosis = 2.98
Phil Ender, 24Oct04