Absolute Kurtosis vs. Relative Kurtosis

Attribute	Absolute Kurtosis	Relative Kurtosis
Definition	Measure of the peakedness of a distribution	Measure of the peakedness of a distribution relative to a normal distribution
Formula	Fourth standardized moment	Fourth standardized moment minus 3
Interpretation	Higher values indicate heavier tails and a sharper peak	Values greater than 0 indicate heavier tails and a sharper peak compared to a normal distribution
Range	No specific range	-2 to +∞

Absolute Kurtosis

Absolute kurtosis is a measure of the "tailedness" of a probability distribution. It quantifies how heavy the tails of a distribution are compared to a normal distribution. A distribution with high absolute kurtosis has heavy tails, meaning it has more extreme values than a normal distribution. Absolute kurtosis is calculated by taking the fourth moment of a distribution and dividing it by the square of the variance.

One of the key attributes of absolute kurtosis is that it is a dimensionless quantity. This means that it is a pure number and does not depend on the units of measurement of the data. This makes it a useful measure for comparing the shape of different distributions regardless of their scales. Absolute kurtosis can be positive or negative, with positive values indicating heavier tails than a normal distribution and negative values indicating lighter tails.

Another important aspect of absolute kurtosis is that it is sensitive to outliers in the data. Outliers can have a significant impact on the kurtosis of a distribution, especially when they are extreme values. This sensitivity to outliers can make absolute kurtosis a useful tool for detecting non-normality in a dataset.

Relative Kurtosis

Relative kurtosis, on the other hand, is a normalized version of absolute kurtosis that takes into account the kurtosis of a distribution relative to a normal distribution. It is calculated by subtracting 3 from the absolute kurtosis and dividing by the standard error of kurtosis. Relative kurtosis provides a measure of how much the tails of a distribution deviate from those of a normal distribution.

One of the key attributes of relative kurtosis is that it allows for comparisons of kurtosis across different distributions. By normalizing the kurtosis values, relative kurtosis provides a standardized measure that can be used to compare the "tailedness" of distributions with different variances and means. This makes it a useful tool for comparing the shapes of distributions in a meaningful way.

Relative kurtosis is also useful for interpreting the significance of kurtosis values. A relative kurtosis of 0 indicates that the tails of a distribution are similar to those of a normal distribution. Positive values indicate heavier tails, while negative values indicate lighter tails. By providing a reference point in the form of a normal distribution, relative kurtosis helps to interpret the kurtosis values in a more intuitive way.

Comparison

• Absolute kurtosis measures the "tailedness" of a distribution, while relative kurtosis provides a normalized measure relative to a normal distribution.
• Absolute kurtosis is a dimensionless quantity, while relative kurtosis allows for comparisons across different distributions.
• Absolute kurtosis is sensitive to outliers, while relative kurtosis provides a standardized measure for interpreting kurtosis values.
• Both absolute and relative kurtosis are useful tools for assessing the shape of a distribution and detecting non-normality in data.

In conclusion, absolute kurtosis and relative kurtosis are both important measures for understanding the shape of probability distributions. Absolute kurtosis quantifies the "tailedness" of a distribution, while relative kurtosis provides a normalized measure for comparing distributions to a normal distribution. Both measures have their own unique attributes and applications, making them valuable tools for statisticians and data analysts.