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Keywords:

  • Standard deviation;
  • Measures of dispersion;
  • Estimation;
  • Robustness;
  • Outliers;
  • Contaminated distributions;
  • Heavy tailed distributions

Abstract

Contaminated observations (e.g. outliers) and heavy tails in the underlying distribution influence the standard deviation as a measure of dispersion even more than, e.g., the mean. Other measures of dispersion, namely absolute deviation, (α, β)-trimmed standard deviation, interquartile range and median absolute deviation (MAD) are defined for population, their properties — especially robustness — are explained and estimators are given, discussed and computed for a medical example. It is investigated how these measures of dispersion can be used to estimate a scale parameter of the underlying distribution more robustly. In numerical comparisons and simulations the robustness of these measures is demonstrated for heavy tailed distributions and contaminated distributions. Among other proposals it is recommended to use the (α, β)-trimmed standard deviation and transform it to the ordinary standard deviation for easier interpretation, if possible.