{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a real random variable with $E[|X|^3] \u003c \\infty$, mean $\\mu$ and standard deviation $\\sigma \u003e 0$.\nThe skewness of $X$ is\n$$\\gamma_1 = E\\left[\\left(\\frac{X - \\mu}{\\sigma}\\right)^3\\right] = \\frac{\\mu_3}{\\sigma^3},$$\nwhere $\\mu_3$ is the third [[central moment|0ZT24EC90K]].\n\n## Remarks\n\nA distribution that is symmetric about its mean has skewness $0$. The converse is false.\nPositive skewness usually means a longer tail on the right. Negative skewness usually means a longer tail on the left.\nThe skewness does not change under $X \\mapsto aX + b$ with $a \u003e 0$. For $a \u003c 0$, it changes sign.\nThe exponential distribution has skewness $2$.\n\n## Explanations\n\n### L1 human\n\nSkewness tells to which side a distribution leans.\nIncomes have positive skewness: most values are moderate, and a few are very large.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0YVQKBNV2R",
    "kind": "definition",
    "msc": [
      "62E10"
    ],
    "name": "Skewness",
    "sources": [
      {
        "locator": "Fisher's moment coefficient of skewness",
        "url": "https://en.wikipedia.org/wiki/Skewness"
      }
    ],
    "tag": "0GQ3H7VW3V",
    "uses": [
      "0ZT24EC90K",
      "0VTN5RKNC5"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0GQ3H7VW3V/",
  "links": [
    {
      "field": "defines",
      "target": "0YVQKBNV2R"
    },
    {
      "field": "uses",
      "target": "0ZT24EC90K"
    },
    {
      "field": "uses",
      "target": "0VTN5RKNC5"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0GQ3H7VW3V.md",
  "msc": [
    "62E10"
  ],
  "name": "Skewness",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0GQ/0GQ3H7VW3V.md",
  "status": null,
  "tag": "0GQ3H7VW3V",
  "uri": "m2://0GQ3H7VW3V",
  "url": "https://math2.tamnd.com/wiki/skewness-definition/",
  "writing": {
    "tag": "0GQ3H7VW3V",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}