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. For example, in the distribution of adult residents across US households, the skew is to the right. However, since the majority of cases is less than or equal to the mode, which is also the median, the mean sits in the heavier left tail. As a result, the rule of thumb that the mean is right of the median under right skew failed. Definition Fisher's moment coefficient of skewness The skewness of a random variable X is the third standardized moment , defined as: where μ is the mean, σ is the standard deviation, E is the expectation operator, μ3 is the third central moment, and κt are the t-th cumulants. It is sometimes referred to as Pearson's moment coefficient of skewness, or simply the moment coefficient of skewness, but should not be confused with Pearson's other skewness statistics (see below). The last equality expresses skewness in terms of the ratio of the third cumulant κ3 to 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]. If σ is finite, μ is finite too and skewness can be expressed in terms of the non-central moment E[X3] by expanding the previous formula, Examples Skewness can be infinite, as when where the third cumulants are infinite, or as when where the third cumulant is undefined. Examples of distributions with finite skewness include the following. A normal distribution and any other symmetric distribution with finite third moment has a skewness of 0 A half-normal distribution has a skewness just below 1 An exponential distribution has a skewness of 2 A lognormal distribution can have a skewness of any positive value, depending on its parameters Sample skewness For a sample of n values, two natural estimators of the population skewness are and where is the sample mean, s is the sample standard deviation, m2 is the (biased) sample second central moment, and m3 is the sample third central moment. is a method of moments estimator. Another common definition of the sample skewness is where is the unique symmetric unbiased estimator of the third cumulant and is the symmetric unbiased