Consider the distribution on the figure. The bars on the right side of the distribution taper differently than the bars on the left side. These tapering sides are called tails, and they provide a visual means for determining which of the two kinds of skewness a distribution has:
If there is zero skewness (i.e., the distribution is symmetric) then the mean = median. (If, in addition, the distribution is unimodal, then the mean = median = mode.)
"Many textbooks," a recent article points out, "teach a rule of thumb stating that the mean is right of the median under right skew, and left of the median under left skew. [But] this rule fails with surprising frequency. It can fail in multimodal distributions, or in distributions where one tail is long but the other is heavy. Most commonly, though, the rule fails in discrete distributions where the areas to the left and right of the median are not equal. Such distributions not only contradict the textbook relationship between mean, median, and skew, they also contradict the textbook interpretation of the median."
The skewness of a random variable X is the third standardized moment, denoted γ1 and defined as
where μ3 is the third moment about the mean μ, and σ is the standard deviation. The last equality expresses skewness in terms of the ratio of the third cumulant κ3 and the 1.5th power of the second cumulant κ2. This is analogous to the definition of kurtosis as the fourth cumulant normalized by the square of the second cumulant.
The skewness is also sometimes denoted Skew[X]. Older textbooks used to denote the skewness as , which was rather inconvenient since skewness can be negative.
The formula expressing skewness in terms of the non-central moment E[X3] is
For a sample of n values the sample skewness is
Given samples from a population, the equation for the sample skewness g1 above is a biased estimator of the population skewness. The usual estimator of population skewness is
where k3 is the unique symmetric unbiased estimator of the third cumulant and k2 is the symmetric unbiased estimator of the second cumulant. Unfortunately G1 is, nevertheless, generally biased. Its expected value can even have the opposite sign from the true skewness; compare unbiased estimation of standard deviation.
If Y is the sum of n independent random variables, all with the same distribution as X, then it can be shown that Skew[Y] = Skew[X] / √n.
Skewness has benefits in many areas. Many simplistic models assume normal distribution; i.e., data are symmetric about the mean. The normal distribution has a skewness of zero. But in reality, data points may not be perfectly symmetric. So, an understanding of the skewness of the dataset indicates whether deviations from the mean are going to be positive or negative.
Pearson's first skewness coefficient , defined by
as well as Pearson's second skewness coefficient, defined by
There is no guarantee that these will be the same sign as each other or as the ordinary definition of skewness.
Skewness refers to asymmetry (or "tapering") in the distribution of sample data:
In a skewed (unbalanced, lopsided) distribution, the mean is farther out in the long tail than is the median. If there is no skewness or the distribution is symmetric like the bell-shaped normal curve then the mean = median = mode.