Skewness

A measure describing a distribution's asymmetry around its mean, positive skewness meaning a longer tail toward higher values.

Source: Pearson 1895

Skewness describes the asymmetry of a distribution around its centre, indicating whether it leans to one side. Positive skewness means a long tail stretching toward higher values, as with incomes or costs where a few large figures pull the tail out to the right, while negative skewness has the tail toward lower values. Unlike kurtosis, which concerns the weight of the tails, skewness is about lopsidedness, and it matters because a skewed distribution makes the mean an unreliable summary. The lopsidedness of a distribution is what it captures. Kirkwood and Sterne (2003) describe this.

Source: Kirkwood & Sterne 2003

Skewness is interpreted by its sign and magnitude: positive skewness indicates a longer right tail, so the mean is typically pulled above the median, while negative skewness indicates a longer left tail, with the mean below the median; values near zero indicate near symmetry. So skewness is interpreted as showing the direction and extent of a distribution's asymmetry, with the sign giving the direction of the longer tail and the magnitude the degree, which matters because skewness affects the choice of summary statistics and methods, since for skewed data the median is often preferred over the mean and non-parametric methods over parametric ones.

Source: Pearson 1895

Skewness describes the asymmetry of a distribution, whether it leans to one side with a longer tail in one direction, while kurtosis describes the tailedness, how heavy or light the tails are and the propensity for extreme values. Skewness concerns lopsidedness; kurtosis concerns the weight of the tails. So skewness and kurtosis are distinct aspects of distributional shape, with skewness based on the third moment and kurtosis on the fourth, and both complement measures of centre and spread, since fully describing a distribution considers its central tendency, dispersion, asymmetry through skewness, and tailedness through kurtosis.

Source: Pearson 1895

Skewness matters because it affects which summary statistics and statistical methods are appropriate: for skewed data, the mean is pulled toward the longer tail and can be misleading, so the median is often preferred, and parametric methods assuming symmetry or normality may be invalid, favouring non-parametric alternatives or transformations. So skewness matters for describing data accurately and choosing suitable analyses, since ignoring skewness can lead to misleading summaries and invalid inference, which is why the skewness of data is examined, and why skewed distributions are often summarised with the median and interquartile range and analysed with methods robust to asymmetry.

Source: Pearson 1895

Skewness is measured from the third standardised moment of the distribution, the average of the cubed deviations from the mean divided by the standard deviation cubed, giving a value whose sign indicates the direction of asymmetry and whose magnitude indicates its degree, with zero for a symmetric distribution. So skewness is measured through the third moment, and it can also be assessed by comparing the mean and median or by graphical means such as histograms, though the moment-based measure gives a single number, which is estimated from sample data, with such estimates being sensitive to outliers and sample size, considerations taken into account when interpreting them.

Source: Casella & Berger 2002