SOR
138 lines
4.7 KiB
Plaintext
138 lines
4.7 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 角度直接测量与不确定度评定\n",
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"\n",
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"本 Notebook 用于对分光计等仪器测得的角度多次测量数据（支持度分格式 `度.分`）进行不确定度合成与保留一位有效数字的规范修约输出。\n",
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"\n",
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"---\n",
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"### 实验原理与数学公式\n",
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"\n",
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"#### 1. 角度单位换算与平均值计算\n",
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"角度采用“分（arcminute）”为内部基准单位换算：\n",
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"$$D^\\circ M' = D \\times 60' + M'$$\n",
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"对 $n$ 次测量值 $\\theta_1, \\theta_2, \\dots, \\theta_n$（以分为单位）：\n",
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"$$\\bar{\\theta} = \\frac{1}{n} \\sum_{i=1}^n \\theta_i$$\n",
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"$$s = \\sqrt{\\frac{1}{n-1} \\sum_{i=1}^n (\\theta_i - \\bar{\\theta})^2}$$\n",
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"\n",
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"#### 2. 不确定度评定公式\n",
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"* **A 类不确定度**：\n",
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"$$u_A = \\frac{s}{\\sqrt{n}}$$\n",
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"* **B 类不确定度**（分光计游标盘分度值误差 $\\Delta_{\\mathrm{inst}}$ 通常为 $1'$）：\n",
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"$$u_B = \\frac{\\Delta_{\\mathrm{inst}}}{\\sqrt{3}}$$\n",
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"* **合成不确定度**：\n",
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"$$u = \\sqrt{u_A^2 + u_B^2}$$\n",
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"\n",
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"#### 3. 结果修约规则\n",
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"不确定度 $u$ 按照“四舍六入五凑偶”保留一位有效数字，角度平均值末位与不确定度对齐，并还原输出为度分格式 $D^\\circ M'$。"
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],
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"id": "b15a8bae68dd9273"
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"import math\n",
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"from decimal import Decimal\n",
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"from python.utils import parse_angle_to_minutes, format_minutes_as_angle, scientific_round\n",
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"\n",
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"print(\"角度换算与修约模块加载完成。\")"
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],
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"id": "afc36384df4dec97"
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### 1. 角度测量数据与分度值输入\n",
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"> **输入格式**：字符串 `'度.分'`（例如 `'120.15'` 代表 $120^\\circ 15'$）。"
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],
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"id": "2d31b33f8ef516db"
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"# 测量数据列表 (度.分 格式)\n",
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"angle_data = [\"120.15\", \"120.16\", \"120.14\", \"120.15\", \"120.17\", \"120.15\"]\n",
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"\n",
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"# 仪器分度值 (单位: 分 ')\n",
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"res_str = \"1\"\n",
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"\n",
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"print(f\"输入角度数据: {angle_data}\")\n",
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"print(f\"仪器分度值: {res_str} '\")"
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],
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"id": "50959966653d1094"
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### 2. 统计计算与修约输出"
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],
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"id": "d91d9838fb92eb89"
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"# 转换为内部单位“分”\n",
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"data_minutes = [parse_angle_to_minutes(s) for s in angle_data]\n",
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"res = Decimal(res_str)\n",
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"n = len(data_minutes)\n",
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"\n",
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"# 1. 平均值与标准差\n",
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"mean_minutes = sum(data_minutes) / n\n",
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"variance = sum((x - mean_minutes) ** 2 for x in data_minutes) / (n - 1) if n > 1 else Decimal(\"0\")\n",
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"s = Decimal(str(math.sqrt(float(variance))))\n",
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"\n",
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"# 2. A类、B类及合成不确定度\n",
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"u_A = s / Decimal(str(math.sqrt(n))) if n > 1 else Decimal(\"0\")\n",
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"u_B = res / Decimal(str(math.sqrt(3)))\n",
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"u = Decimal(str(math.sqrt(float(u_A**2 + u_B**2))))\n",
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"\n",
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"# 3. 科学修约与格式还原\n",
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"u_float = float(u)\n",
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"if u_float == 0:\n",
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" final_result_str = f\"θ = {format_minutes_as_angle(mean_minutes)} ± 0'\"\n",
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"else:\n",
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" first_digit_pos = math.floor(math.log10(u_float))\n",
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" prec = Decimal('1e' + str(first_digit_pos))\n",
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" mean_final_val, u_final = scientific_round(mean_minutes, u)\n",
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" mean_str = format_minutes_as_angle(mean_final_val, prec)\n",
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" final_result_str = f\"θ = {mean_str} ± {u_final}'\"\n",
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"\n",
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"print(\"=\" * 45)\n",
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"print(\" 角 度 测 量 不 确 定 度 分 析 \")\n",
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"print(\"=\" * 45)\n",
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"print(f\"测量次数 n : {n}\")\n",
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"print(f\"角度平均值 : {format_minutes_as_angle(mean_minutes)} ({mean_minutes:.4f}')\")\n",
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"print(f\"样本标准差 s : {s:.6f}'\")\n",
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"print(f\"A 类不确定度 u_A : {u_A:.6f}'\")\n",
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"print(f\"B 类不确定度 u_B : {u_B:.6f}'\")\n",
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"print(f\"合成不确定度 u : {u:.6f}'\")\n",
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"print(\"-\" * 45)\n",
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"print(f\"★ 最终修约结果 : {final_result_str}\")\n",
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"print(\"=\" * 45)"
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],
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"id": "ef15abbb8e0d991c"
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}
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],
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"metadata": {
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"language_info": {
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"name": "python"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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