Descriptive Statistics & IQR Calculator

Most of these statistics are moments of a distribution, meaning averages of powers of the deviations from the mean. The first moment gives location, the second gives variance, the third standardized moment gives skewness and the fourth gives kurtosis. Dividing the sum of squared deviations by n minus 1 rather than n applies Bessel's correction, which removes the downward bias that arises because deviations are taken from the sample mean rather than the unknown population mean; the population forms shown alongside divide by n and are appropriate only when the data are the entire population. The higher moments use the adjusted Fisher-Pearson coefficients, the same definitions used by spreadsheet SKEW and KURT functions, and kurtosis is reported in excess form so that a normal distribution scores 0. Interpretation rests on a few working thresholds. Skewness within plus or minus 0.5 is treated as approximately symmetric, 0.5 to 1 as moderate and beyond 1 as strong. Positive excess kurtosis indicates heavier tails and more outlier risk than a normal curve. The coefficient of variation expresses the standard deviation as a percentage of the mean and is only meaningful for ratio-scale data with a positive mean. The interquartile range, Q3 minus Q1, is the robust counterpart to standard deviation, and Tukey's fences at 1.5 times the IQR beyond each quartile are the standard outlier screen. One caveat: quartiles have no single agreed definition, and the nine methods catalogued by Hyndman and Fan in 1996 can give different Q1 and Q3 on small samples, so results may differ slightly from another package.

Summarizing data by its average and spread became a scientific method in the nineteenth century. The Belgian astronomer and statistician Adolphe Quetelet applied the arithmetic mean and the normal curve to human measurements in the 1830s, arguing that population traits clustered around an average in the way astronomical observation errors did. Francis Galton, working on heredity in the 1870s and 1880s, found the mean unhelpful for skewed data and promoted order-based summaries instead, introducing quartiles and percentiles and popularizing the median as a measure of position. Karl Pearson supplied most of the vocabulary still in use. He coined the term standard deviation in 1893, replacing older expressions such as mean error, and developed the method of moments in his Contributions to the Mathematical Theory of Evolution in the mid-1890s, where he formalized skewness as a measure of asymmetry; he introduced kurtosis as a name for tail and peak behavior around 1905. The n minus 1 divisor is named after the astronomer Friedrich Bessel, who had encountered the same correction earlier in error analysis. The robust half of this summary is more recent: John Tukey's Exploratory Data Analysis of 1977 packaged the five-number summary, the box plot and the 1.5 times IQR fence rule as a deliberate alternative to moment-based description for data with outliers.