Skewness and Kurtosis Calculator

Skewness and kurtosis describe aspects of a distribution that the mean and standard deviation do not capture. Skewness measures asymmetry. A distribution with a long right tail usually has positive skewness, while one with a long left tail usually has negative skewness. A value near zero indicates approximate symmetry, but it does not prove that the data follow a normal distribution. Visual plots and subject-matter context remain important. This calculator reports adjusted Fisher-Pearson sample skewness. It begins with the second and third central moments and applies a correction for finite sample size. The correction reduces a common small-sample bias and matches the sample-skewness convention used by many statistical packages. Positive values suggest that unusually large observations extend the right tail. Negative values suggest that unusually small observations extend the left tail. The magnitude should be interpreted in context rather than against a universal pass-or-fail boundary. Kurtosis measures the weight of the tails relative to the center, not simply how sharp a graph looks. The calculator reports bias-corrected excess kurtosis, which subtracts three so that an ideal normal distribution has an expected value near zero. Positive excess kurtosis is often called leptokurtic and indicates heavier tails or more extreme observations. Negative excess kurtosis is called platykurtic and indicates lighter tails or a flatter distribution. Small samples can produce volatile kurtosis estimates because fourth powers give extreme observations substantial influence. At least four observations with nonzero variation are required for the adjusted formulas. The calculator also returns the arithmetic mean and sample standard deviation to provide scale and location context. Values can be separated by commas, spaces, or semicolons, and negative values and decimals are accepted. All observations receive equal weight. Missing values should be resolved according to a documented analysis plan rather than silently replaced with zero. Skewness is useful when choosing transformations, checking model residuals, summarizing financial returns, evaluating manufacturing measurements, or describing response times. Kurtosis can highlight tail behavior that matters for risk and quality control. Neither statistic should be used alone to discard data or declare normality. Outliers may be errors, but they may also be the most important observations. Pair these statistics with a histogram, box plot, quantile plot, sample-size assessment, and knowledge of how the data were collected. This calculator offers reproducible descriptive estimates for exploration and reporting; formal inference may require standard errors, confidence intervals, or a distribution-specific model.