Outlier Detection Explanation Calculator

The three rules offered here differ mainly in which location and scale estimates they trust. The interquartile range rule places fences at Q1 minus k times IQR and Q3 plus k times IQR, with k of 1.5 marking mild outliers and 3.0 extreme ones; under a normal distribution the 1.5 fences sit at roughly 2.7 standard deviations and flag about 0.7 percent of observations. The z-score rule uses the sample mean and standard deviation directly, so a threshold of 2 flags about 4.6 percent of normally distributed data and a threshold of 3 flags about 0.27 percent. The median absolute deviation rule multiplies the MAD by 1.4826, the reciprocal of the standard normal 75th percentile, so that it estimates the same quantity as the standard deviation under normality, then flags points beyond three scaled MADs. The decisive property is the breakdown point, the fraction of contamination an estimator tolerates before it can be driven arbitrarily far. The mean and standard deviation have a breakdown point of zero, which creates masking: a single extreme value inflates the standard deviation enough to hide itself. The effect is severe in small samples, since the largest possible absolute z-score in a sample of n is capped outright. With the population form used here, which divides the squared deviations by n, that cap is the square root of n minus 1 - with ten observations no point can exceed 3.000, so a threshold of 3 can never flag anything, and the cap only rises above 3 from eleven observations onward. The sample form, dividing by n minus 1, is tighter still at (n minus 1) over the square root of n, or 2.85 at ten observations. The median and MAD have a 50 percent breakdown point. All three rules are univariate and cannot detect points that are unusual only in combination.

Rules for discarding discordant observations came out of nineteenth-century astronomy, where a single misread transit could distort an orbit. Benjamin Peirce published his criterion for rejecting doubtful observations in 1852, and William Chauvenet set out the simpler criterion that carries his name in his 1863 Manual of Spherical and Practical Astronomy. These were controversial from the start; many astronomers, including those in the Gaussian tradition, objected that mechanical rejection discards real information. Formal hypothesis tests followed in the twentieth century, notably Frank Grubbs's maximum normed residual test published in the Annals of Mathematical Statistics in 1950, which is still used for detecting a single outlier in approximately normal data. The distribution-free approach used by the box plot came from John Tukey, whose 1977 book Exploratory Data Analysis introduced the box-and-whisker display with fences based on the hinges and the 1.5 multiplier, chosen deliberately so the rule would not depend on a normality assumption. In parallel, robust statistics matured as its own field: Tukey's 1960 essay on contaminated distributions, Peter Huber's 1964 paper on M-estimation, and Frank Hampel's 1974 work on the influence function, which popularized both the breakdown point and the median absolute deviation as a scale estimator. Christophe Leys and coauthors argued in the Journal of Experimental Social Psychology in 2013 that behavioural researchers should abandon standard-deviation rules in favour of the median and MAD, which is the procedure the MAD option implements.