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  "body": "\n## Definition\n\nLet $p \\ge 2$ and let $S^{p-1} = \\set{x \\in \\R^p : \\abs{x} = 1}$ be the unit [[sphere|T62369RFH6]].\n*Directional statistics* is the statistics of data $x_1, \\ldots, x_n$ with values in $S^{p-1}$.\nMore generally the data may lie in a compact manifold such as a rotation group.\nFor $p = 2$ it is called circular statistics.\nIts basic [[statistics|0Q5CG2NEQ8]] come from the resultant vector $R = x_1 + \\cdots + x_n$.\n\n1. The *mean resultant length* is $\\bar R = \\abs{R}/n$, a number in $[0, 1]$.\n2. If $R \\ne 0$, the *mean direction* is $R / \\abs{R} \\in S^{p-1}$.\n3. For $p = 2$, the *circular variance* is $1 - \\bar R$.\n\n## Remarks\n\nExample: the angles $350°$ and $10°$ give the unit vectors $(\\cos 10°, -\\sin 10°)$ and $(\\cos 10°, \\sin 10°)$.\nTheir sum is $R = (2 \\cos 10°, 0)$, so the mean direction is $0°$ and $\\bar R = \\cos 10° \\approx 0.985$.\nNon-example: for the angles $0°$ and $180°$, $R = 0$, so the mean direction is not defined; then $\\bar R = 0$ and the circular variance is $1$.\nThe value $\\bar R$ is close to $1$ when the directions are concentrated, and close to $0$ when they are spread out or balanced.\nThe mean direction is the point $m \\in S^{p-1}$ that maximises $\\sum_i m^T x_i$, so it does not depend on the choice of a zero angle.\n",
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