estimator of the second cumulant (i.e. the sample variance). This adjusted Fisher–Pearson standardized moment coefficient is the version found in Excel and several statistical packages including Minitab, SAS and SPSS. Under the assumption that the underlying random variable is normally distributed, it can be shown that all three ratios , and are unbiased and consistent estimators of the population skewness , with , i.e., their distributions converge to a normal distribution with mean 0 and variance 6 (Fisher, 1930). The variance of the sample skewness is thus approximately for sufficiently large samples. More precisely, in a random sample of size n from a normal distribution, In normal samples, has the smaller variance of the three estimators, with For non-normal distributions, , and are generally biased estimators of the population skewness ; their expected values can even have the opposite sign from the true skewness. For instance, a mixed distribution consisting of very thin Gaussians centred at −99, 0.5, and 2 with weights 0.01, 0.66, and 0.33 has a skewness of about −9.77, but in a sample of 3 has an expected value of about 0.32, since usually all three samples are in the positive-valued part of the distribution, which is skewed the other way. Applications Skewness is a descriptive statistic that can be used in conjunction with the histogram and the normal quantile plot to characterize the data or distribution. Skewness indicates the direction and relative magnitude of a distribution's deviation from the normal distribution. With pronounced skewness, standard statistical inference procedures such as a confidence interval for a mean will be not only incorrect, in the sense that the true coverage level will differ from the nominal (e.g., 95%) level, but they will also result in unequal error probabilities on each side. Skewness can be used to obtain approximate probabilities and quantiles of distributions (such as value at risk in finance) via the Cornish-Fisher expansion. Many 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. D'Agostino's K-squared test is a goodness-of-fit normality test based on sample skewness and sample kurtosis. Other measures of skewness Other measures of skewness have been used, including simpler calculations suggested by Karl