ISBN_978-4-7806-0852-6_Edition1_Printing4_p175_20_6.ipynb

-isbn_978-4-7806-0852-6_edition1_printing4_p175_20_6.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": " ISBN_978-4-7806-0852-6_Edition1_Printing4_p175_20_6.ipynb",
      "provenance": [],
      "authorship_tag": "ABX9TyNZ6lCjt2kaMR0vO3OntYVL",
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/gist/axjack/c9e0274acab5760051eade6a0cabf4cc/-isbn_978-4-7806-0852-6_edition1_printing4_p175_20_6.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "BNI_te9MmYU-"
      },
      "source": [
        "# 書籍情報\n",
        "  - 『統計検定準1級対応　統計学実践ワークブック』 日本統計学会編，学術図書出版社\n",
        "  - 第1版第4刷\n",
        "  - p.175の式(20.6)\n",
        "\n",
        "# 問い合わせ箇所\n",
        "式(20.6)\n",
        "\n",
        "$$\n",
        "S_{[k]} = \\sum_{i=1}^N \\sum_{j=1}^2 (\\bar{y}_{[k]j} - \\bar{y} )^2 = \\frac{N}{4}(\\bar{y}_{[k]1} - \\bar{y}_{[k]2})^2\n",
        "$$\n",
        "\n",
        "の真ん中の式から一番右の式は導出できない。\n",
        "\n",
        "# どうあるべきか\n",
        "\n",
        "$$\n",
        "S_{[k]} = \\sum_{i=1}^{N/2} \\sum_{j=1}^2 (\\bar{y}_{[k]j} - \\bar{y} )^2 = \\frac{N}{4}(\\bar{y}_{[k]1} - \\bar{y}_{[k]2})^2\n",
        "$$\n",
        "\n",
        "もしくは、\n",
        "\n",
        "$$\n",
        "S_{[k]} = \\frac{1}{2}\\sum_{i=1}^{N} \\sum_{j=1}^2 (\\bar{y}_{[k]j} - \\bar{y} )^2 = \\frac{N}{4}(\\bar{y}_{[k]1} - \\bar{y}_{[k]2})^2\n",
        "$$\n",
        "\n",
        "ではないかと考えます。\n",
        "\n",
        "# 問い合わせに至った理由\n",
        "\n",
        "応答 $ y_i $ の総平均を、\n",
        "$$\\bar{y} = \\frac{1}{N}\\sum_{i=1}^N y_i \\tag{0}$$\n",
        "\n",
        "とし、以下が成り立つとする。\n",
        "\n",
        "$$\n",
        "  \\frac{N}{2} \\bar{y}_{[k]1} +   \\frac{N}{2} \\bar{y}_{[k]2} = \\sum_{i=1}^N y_i \\tag{1}\n",
        "$$\n",
        "\n",
        "このとき、\n",
        "$$\n",
        "  \\frac{1}{2} \\Biggl( \\bar{y}_{[k]1} + \\bar{y}_{[k]2} \\Biggr)  = \\frac{1}{N}\\sum_{i=1}^N y_i  = \\bar{y} \\tag{2}\n",
        "$$\n",
        "\n",
        "である。すると、p.175 (20.6)は、\n",
        "\n",
        "$$\n",
        "\\begin{aligned}\n",
        "S_{[k]} &= \\sum_{i=1}^N \\sum_{j=1}^2 (\\bar{y}_{[k]j} - \\bar{y} )^2 \\\\\n",
        "        &= \\sum_{i=1}^N \\Biggl[\n",
        "            (\\bar{y}_{[k]1} - \\bar{y} )^2 \n",
        "            +\n",
        "            (\\bar{y}_{[k]2} - \\bar{y} )^2 \n",
        "            \\Biggr]\\\\\n",
        "        &= \\sum_{i=1}^N \\Biggl[\n",
        "            (\\bar{y}_{[k]1} -  \\frac{1}{2} \\Bigl( \\bar{y}_{[k]1} + \\bar{y}_{[k]2} \\Bigr) )^2 \n",
        "            +\n",
        "            (\\bar{y}_{[k]2} - \\frac{1}{2} \\Bigl( \\bar{y}_{[k]1} + \\bar{y}_{[k]2} \\Bigr)  )^2 \n",
        "            \\Biggr]\\\\\n",
        "        &= \\sum_{i=1}^N \\Biggl[\n",
        "            (\\frac{1}{2} \\Bigl( \\bar{y}_{[k]1} - \\bar{y}_{[k]2} \\Bigr) )^2 \n",
        "            +\n",
        "           (-\\frac{1}{2} \\Bigl( \\bar{y}_{[k]1} - \\bar{y}_{[k]2} \\Bigr) )^2 \n",
        "            \\Biggr]\\\\\n",
        "        &= \\sum_{i=1}^N \\Biggl[\n",
        "            \\frac{2}{2^2} \\Bigl( \\bar{y}_{[k]1} - \\bar{y}_{[k]2} \\Bigr)^2 \\Biggr]\\\\\n",
        "        &= \n",
        "            \\frac{N}{2} \\Bigl( \\bar{y}_{[k]1} - \\bar{y}_{[k]2} \\Bigr)^2 \\\\\n",
        "\\end{aligned}\n",
        "$$\n",
        "\n",
        "となって、式(20.6)のようにイコールでは結べない、と考えました。\n",
        "\n",
        "以上。\n"
      ]
    }
  ]
}

問い合わせ結果

• 2022-01-06, 出版社より式(20.6)は誤りであったと連絡あり。
• 正誤表に反映された。
• https://www.gakujutsu.co.jp/text/isbn978-4-7806-0852-6/