Discussion
If kurtosis is defined as \(\left(\frac{m_{4}}{\sigma^{4}} - 3\right)\), where \(m_{4}\) is the 4th order central moment and \(\sigma\) is the standard deviation, then its value is positive for
Answer: A
The value is positive for a leptokurtic distribution.
The ratio \(m_{4}/\sigma^{4}\) (often written \(\beta_{2}\)) equals 3 for a normal distribution, so subtracting 3 measures excess kurtosis relative to the normal curve. A leptokurtic distribution is more peaked, with heavier tails, so \(\beta_{2} \gt 3\) and the measure is positive.
A mesokurtic distribution has \(\beta_{2} = 3\), and the normal distribution is the standard mesokurtic case, so both give exactly zero. A platykurtic distribution is flatter than the normal, with \(\beta_{2} \lt 3\), so its value is negative.
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