{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "TFCWC3N41W"
    }
  ],
  "body": "\n## Summary\n\nA pivotal quantity is a function of the data and the unknown parameter whose law does not depend on the parameter.\nIt is the standard tool to build a [[confidence interval|0S6T9KQNC0]]: one fixes an interval of probability $1 - \\alpha$ for the pivot and solves for the parameter.\nMany classical intervals, such as the Student t interval for a normal mean, come from a pivot.\n\n## Explanations\n\n### L1 human\n\nA statistic cannot contain the unknown parameter, but a pivot may.\nThe point is that the pivot has a known law whatever the true parameter is.\nFor example, if $X$ is normal with unknown mean $\\mu$ and variance $1$, then $X - \\mu$ is standard normal for every $\\mu$.\nSo $X - \\mu$ lies in $[-1.96, 1.96]$ with probability about $0.95$, and this gives the interval $[X - 1.96, X + 1.96]$ for $\\mu$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "62F25",
      "62E15"
    ],
    "name": "Pivotal quantity",
    "same_as": {
      "wikidata": "Q7199665",
      "wikipedia": "Pivotal quantity"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Pivotal_quantity"
      }
    ],
    "tag": "PXFYDT3M52"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/PXFYDT3M52/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/PXFYDT3M52.md",
  "msc": [
    "62F25",
    "62E15"
  ],
  "name": "Pivotal quantity",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PXF/PXFYDT3M52.md",
  "status": null,
  "tag": "PXFYDT3M52",
  "uri": "m2://PXFYDT3M52",
  "url": "https://tamnd.github.io/math2/wiki/pivotal-quantity/",
  "writing": {
    "tag": "PXFYDT3M52",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}