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Measures of Asymmetry
2. Skewness – definition


Skewness is the degree of asymmetry in a dataset. Negative skew means that the left tail of the histogram is longer. Other names for negative skew include the familiar "skewed left" and "left-tailed".

Positive skew means that the right tail of the histogram is longer. We also know this type of skewness as "skewed right" or "right-tailed".

For an n-element set, skewness is given by the formula

Skewness=\dfrac{\frac{1}{n}((x_1-\overline{x})^3+(x_2 - \overline{x})^3 + \ldots + (x_n - \overline{x})^3)}{\sqrt{(\frac{1}{n-1}((x_1-\overline{x})^2+\ldots+(x_n-\overline{x})^2))^3}}

where \overline{x} is the arithmetic mean. If skewness is less than -1 or greater than +1, the histogram is highly skewed. If the skewness value falls between -1 and -\frac{1}{2} or between +\frac{1}{2} and +1, the histogram is moderately skewed. If skewness is between -\frac{1}{2} and +\frac{1}{2}, the histogram is close to symmetric.