{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $U = \\set{1, \\dots, N}$ be a finite population of $N$ units, and let $1 \\le n \\le N$.\nLet $S \\subseteq U$ be a random subset, defined on a [[probability space|0NFPJVJJHX]].\nThen $S$ is a *simple random sample* of size $n$ from $U$ if for every subset $s \\subseteq U$ with $\\abs{s} = n$,\n$$P(S = s) = \\binom{N}{n}^{-1}.$$\nHere $\\binom{N}{n}$ is the [[binomial coefficient|0GBZM0MT0F]], the number of [[combinations|098PHYBTV9]] of $n$ units out of $N$.\n\n## Remarks\n\nEquivalently, one draws $n$ units one after the other, each uniformly at random among the units not yet drawn.\nEach unit is in $S$ with probability $n/N$.\nFor values $y_1, \\dots, y_N$ attached to the units, the sample mean $\\bar y_S = \\frac{1}{n}\\sum_{i \\in S} y_i$ is unbiased for the population mean $\\bar Y$.\nIts variance is $\\big(1 - \\frac{n}{N}\\big) \\frac{S_y^2}{n}$, where $S_y^2 = \\frac{1}{N-1} \\sum_{i=1}^{N} (y_i - \\bar Y)^2$.\nSampling *with replacement*, where $n$ independent uniform draws are made, is a different design.\n\nExample: $N = 4$ and $n = 2$, with $S$ uniform on the $\\binom{4}{2} = 6$ pairs.\nEach pair has probability $1/6$, as required, and unit $1$ lies in $3$ of the pairs, so it is chosen with probability $1/2 = n/N$.\n\nNon-example: with $N = 4$, choose $r \\in \\set{1, 2}$ uniformly and take $S = \\set{r, r + 2}$.\nEvery unit is chosen with probability $1/2$, but the pair $\\set{1, 2}$ has probability $0$ instead of $1/6$.\nSo equal inclusion probabilities are not enough for a simple random sample.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "Y0PTCJBPEA",
    "kind": "definition",
    "msc": [
      "62D05"
    ],
    "name": "Simple random sample",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Simple_random_sample"
      }
    ],
    "tag": "6B05T6XHY0",
    "uses": [
      "098PHYBTV9",
      "0GBZM0MT0F",
      "0NFPJVJJHX"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/6B05T6XHY0/",
  "links": [
    {
      "field": "defines",
      "target": "Y0PTCJBPEA"
    },
    {
      "field": "uses",
      "target": "098PHYBTV9"
    },
    {
      "field": "uses",
      "target": "0GBZM0MT0F"
    },
    {
      "field": "uses",
      "target": "0NFPJVJJHX"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/6B05T6XHY0.md",
  "msc": [
    "62D05"
  ],
  "name": "Simple random sample",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6B0/6B05T6XHY0.md",
  "status": null,
  "tag": "6B05T6XHY0",
  "uri": "m2://6B05T6XHY0",
  "url": "https://tamnd.github.io/math2/wiki/simple-random-sample-definition/",
  "writing": {
    "tag": "6B05T6XHY0",
    "kind": "definition",
    "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
      }
    ]
  }
}