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  "body": "\n## Summary\n\nThe Hodges–Lehmann estimator estimates the centre of a symmetric distribution by the [[median|07Y2E7P9NP]] of all pairwise averages of the observations.\nIt is the estimator that corresponds to the [[Wilcoxon signed-rank test|38SETDT2HN]], and it inherits its good properties.\nIt resists outliers much better than the mean, and for normal data it is almost as precise, with asymptotic relative efficiency $3/\\pi \\approx 0.955$.\nA two-sample version estimates a shift between two populations by the median of all pairwise differences.\n\n## Explanations\n\n### L1 human\n\nTake every pair of data points, including each point paired with itself, and write down the midpoint of the pair.\nThe median of all these midpoints is the Hodges–Lehmann estimate.\nFor the data $1, 2, 9$ the midpoints are $1, 1.5, 2, 5, 5.5, 9$, and their median is $3.5$.\nThis lies between the median $2$, which ignores how far $9$ is, and the mean $4$, which is pulled by it.\n",
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