Skewness and kurtosis are essential measures to describe the shape of a data distribution. While both are moments (*) of the
distribution, they serve different purposes and capture unique aspects of data behavior.
One of the main differences lies in the power to which the deviations from the mean are raised: skewness uses the third power, while kurtosis uses
the fourth power.
Skewness measures the asymmetry of a distribution around its mean. It indicates whether the data is skewed to the left (with a longer left tail) or
to the right (with a longer right tail). It is calculated by taking the deviations of each data point from the mean, raising them to the third power, and then averaging these "cubed"
deviations.
Skewness Formula: (1/N) * Σ((xi - μ)^3 / σ^3)
- N: the number of data points
- xi: each data point
- μ: the mean of the data
- σ: the standard deviation
The third power accounts for the direction of the skew. Positive deviations (greater than the mean) remain positive when cubed, and negative
deviations (less than the mean) remain negative. This directional aspect is crucial as it indicates whether a distribution leans to the right (positive skewness) or to the left (negative
skewness).
By raising deviations to the third power, skewness emphasizes larger deviations. This ensures that significant deviations, especially those in the
tails of the distribution, have a stronger influence on the skewness measure.
Kurtosis, on the other hand, measures the "sharpness" of the peak of the distribution and the weight of its tails. It indicates whether a
distribution has more or fewer extreme values (heavy or light tails) compared to a normal distribution. Kurtosis is calculated by raising the deviations of data points to the fourth power and
then averaging these values.
Kurtosis Formula: (1/N) * Σ((xi - μ)^4 / σ^4)
The fourth power is used because it amplifies the effect of extreme deviations much more than the third power. Since kurtosis focuses on the
presence of outliers or extreme values, raising deviations to the fourth power ensures that these extreme values have a disproportionately large effect on the kurtosis measure.
Unlike skewness, kurtosis does not consider the direction (left or right) of deviations. The fourth power makes all deviations positive, focusing
solely on their magnitude.
(*) Moments are numerical values that describe the
shape of a distribution, calculated by raising deviations from a reference value, typically the mean, to different powers.
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