an estimator of the population MAD, which in this case is 1, whereas the population variance does not exist. Efficiency These robust estimators typically have inferior statistical efficiency compared to conventional estimators for data drawn from a distribution without outliers (such as a normal distribution), but have superior efficiency for data drawn from a mixture distribution or from a heavy-tailed distribution, for which non-robust measures such as the standard deviation should not be used. For example, for data drawn from the normal distribution, the MAD is 37% as efficient as the sample standard deviation, while the Rousseeuw–Croux estimator Qn is 88% as efficient as the sample standard deviation. Absolute pairwise differences Rousseeuw and Croux propose alternatives to the MAD, motivated by two weaknesses of it: It is inefficient (37% efficiency) at Gaussian distributions. it computes a symmetric statistic about a location estimate, thus not dealing with skewness. They propose two alternative statistics based on pairwise differences: Sn and Qn, defined as: where is a constant depending on . These can be computed in O(n log n) time and O(n) space. Neither of these requires location estimation, as they are based only on differences between values. They are both more efficient than the MAD under a Gaussian distribution: Sn is 58% efficient, while Qn is 82% efficient. For a sample from a normal distribution, Sn is approximately unbiased for the population standard deviation even down to very modest sample sizes (<1% bias for n = 10). For a large sample from a normal distribution, 2.219144465985075864722Qn is approximately unbiased for the population standard deviation. For small or moderate samples, the expected value of Qn under a normal distribution depends markedly on the sample size, so finite-sample correction factors (obtained from a table or from simulations) are used to calibrate the scale of Qn. The biweight midvariance Like Sn and Qn, the biweight midvariance aims to be robust without sacrificing too much efficiency. It is defined as where I is the indicator function, Q is the sample median of the Xi, and Its square root is a robust estimator of scale, since data points are downweighted as their distance from the median increases, with points more than 9 MAD units from the median having no influence at all. Extensions propose a robust depth-based estimator for location and scale simultaneously. They propose a new measure named the Student median. Confidence intervals A robust confidence