Skewness and Kurtosis simply explained…

Skewness and kurtosis are essential measures that help describe the shape of a data distribution. While both are moments * of the distribution, they serve different purposes and utilize different mathematical approaches to capture unique aspects of data behavior.

A particularly interesting point of differentiation between these two metrics is the power to which the deviations from the mean are raised: skewness uses the third power, while kurtosis uses the fourth power. But why is this the case?

Skewness measures the asymmetry of a distribution around its mean. In other words, it tells us whether the data is skewed to the left (with a longer left tail) or to the right (with a longer right tail). To capture this asymmetry, skewness 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.

The use of the third power allows skewness to account for the direction of the skew. Positive deviations (data points greater than the mean) remain positive when cubed, and negative deviations (data points less than the mean) remain negative. This directional aspect is crucial because it lets skewness indicate whether a distribution leans to the right (positive skewness) or to the left (negative skewness).

By cubing the deviations, skewness emphasizes larger deviations more than smaller ones. This amplification is important because it ensures that significant deviations—especially those in the tails—are given more weight in determining the skewness of the distribution.

Kurtosis, on the other hand, measures the "tailedness" or the sharpness of the peak of the distribution and the heaviness of its tails. It tells us whether a distribution has more extreme values (heavy tails) or fewer extreme values (light tails) compared to a normal distribution. Kurtosis is calculated by raising the deviations of data points from the mean to the fourth power and then averaging these values.

The fourth power is chosen because it magnifies the impact of extreme deviations much more than the third power would. Since kurtosis is concerned with the presence of outliers or extreme values, raising the deviations to the fourth power ensures that these extremes have a disproportionately large effect on the kurtosis measure.

Unlike skewness, kurtosis is not concerned with the direction (left or right) of the deviations. The fourth power makes all deviations positive, focusing purely on the magnitude rather than the sign.

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Moments are numerical values that describe the shape of a distribution, calculated by raising deviations from a reference value, usually the mean, to different powers.