![\"Open](\"https://colab.research.google.com/assets/colab-badge.svg\"){kind=icon}
"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "_k5tMB9V9yTc"
},
"source": [
"# 분산, 표준편차 구하기(도수분포표) [공식사용][Sage Math이용]"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "RN20_e8N93t2"
},
"source": [
"### [참고]\n",
"\n",
"[Geogebra와 수학의 시각화](https://www.geogebra.org/m/gsARCQs5)"
]
},
{
"cell_type": "markdown",
"source": [
""
],
"metadata": {
"id": "1OSJ8tpNFHPO"
}
},
{
"cell_type": "markdown",
"source": [
""
],
"metadata": {
"id": "ZotJVflJFLqB"
}
},
{
"cell_type": "markdown",
"source": [
""
],
"metadata": {
"id": "ELCv3YuVFdXz"
}
},
{
"cell_type": "markdown",
"source": [
"아래는 Sage Math의 Text 파일이다. 위의 Nims 서버와 같은 내용인데, 혹시나 Nims 서버가 없어지는 경우를 대비하여 Text 형식을 아래에 붙여 놓았다.\n",
"\n",
"~~~\n",
"| \n", "$x_i$ | \n", "$f_i$ | \n", "\n", "$x_i f_i$ | \n", "
| \n", "$x_1$ | \n", "\n", "$f_1$ | \n", "\n", "$x_1 f_1$ | \n", "
| \n", "$\\vdots$ | \n", "\n", "$\\vdots$ | \n", "\n", "$\\vdots$ | \n", "
| \n", "$x_n$ | \n", "\n", "$f_n$ | \n", "\n", "$x_n f_n$ | \n", "
\n", "$ f_1 +f_2 +f_3 +\\cdots+f_n =\\displaystyle\\sum_{i=1}^n x_i f_i =N$\n", "
\n", " $\\begin{array}{rcl}\n", "\\displaystyle \\text{Mean }&:& m =\n", " \\displaystyle \\frac{\\displaystyle\\sum_{i=1}^n x_i f_i }{\\displaystyle\\sum_{i=1}^n f_i }\n", "=\\displaystyle \\frac{1}{N} \\sum_{i=1}^n x_i f_i \\\\\n", "&& \\\\\n", "\\end{array}$\n", "
\n", " $\\begin{array}{rcl}\n", "\\displaystyle \\displaystyle \\text{Variance } : \n", " \\sigma^2 &=&\\displaystyle \\frac{\\displaystyle\\sum_{i=1}^n(x_i-m)^2 f_i}{\\displaystyle\\sum_{i=1}^n f_i} \n", "=\\displaystyle \\frac{1}{N}\\sum_{i=1}^n(x_i-m)^2 f_i \n", "=\\displaystyle \\frac{1}{N}\\sum_{i=1}^n(x_i^2-2mx_i+m^2) f_i \\\\\n", " &=&\\displaystyle \\frac{1}{N}\\sum_{i=1}^n x_i^2 f_i\n", " - 2m \\times \\frac{1}{N}\\sum_{i=1}^n x_i f_i\n", " +m^2 \\times \\frac{1}{N}\\sum_{i=1}^n f_i \\\\\n", " &=&\\displaystyle \\frac{1}{N}\\sum_{i=1}^n x_i^2 f_i\n", " - 2m \\times m\n", " +m^2 \\times \\frac{1}{N} \\times N \\\\\n", " &=&\\displaystyle \\frac{1}{N}\\sum_{i=1}^n x_i^2 f_i\n", " - 2m^2 + m^2 \n", " =\\displaystyle \\frac{1}{N}\\sum_{i=1}^n x_i^2 f_i -m^2\\\\\n", "\\end{array}$\n", "\n", "
\n", "$\\begin{array}{rcl}\n", "\\text{Standard Deviation } &:& \\sigma =\\sqrt{\\sigma^2} \\\\\n", "\\end{array}$\n", "sage: list_of_class_boundaries=[50,60,70,80,90,100]\n", "sage: class_frequency=[1,9,11,7,2]\n", "sage: class_marks=[]\n", "sage: for i in range(len(list_of_class_boundaries)-1):\n", "... class_marks.insert(i,(list_of_class_boundaries[i]+list_of_class_boundaries[i+1])/2)\n", "sage: %latex\n", "sage: Class Marks$=\\sage{latex(class_marks)}$\n", "sage: temp=copy(class_marks)\n", "sage: temp[0]=list_of_class_boundaries[0]\n", "sage: temp[len(class_marks)-1]=list_of_class_boundaries[len(list_of_class_boundaries)-1]\n", "sage: histogram(temp,bins=len(temp), weights=class_frequency)\n", "sage: mean_of_frequency_distribution=vector(class_frequency).dot_product(vector(class_marks))/sum(class_frequency)\n", "sage: %latex\n", "sage: $m=\\displaystyle\\sage{latex(mean_of_frequency_distribution)}$\n", "sage: squar_of_class_marks=[]\n", "sage: for i in range(len(class_marks)):\n", "... squar_of_class_marks.insert(i,class_marks[i]^2)\n", "sage: %latex\n", "sage: squar of class marks : $\\displaystyle=\\sage{latex(squar_of_class_marks)}$\n", "sage: variance_of_frequency_distribution=vector(squar_of_class_marks).dot_product(vector(class_frequency))/sum(class_frequency)-mean_of_frequency_distribution^2\n", "sage: %latex\n", "sage: $\\sigma^2\\displaystyle=\\sage{latex(variance_of_frequency_distribution)}$\n", "sage: standard_deviation_of_frequency_distribution=sqrt(variance_of_frequency_distribution)\n", "sage: %latex\n", "sage: $\\sigma\\displaystyle=\\sage{latex(standard_deviation_of_frequency_distribution.simplify())}$\n", "~~~" ], "metadata": { "id": "axKWI8U7L-X7" } } ] }