{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Hughes and Hase, Problem 2.2\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Twelve data points given.  Enter them into a `numpy` array:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = np.array([5.33, 4.95, 4.93, 5.08, 4.95, 4.96, 5.02, 4.99, 5.24, 5.25, 5.23, 5.01])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### i) Calculating the mean: $\\quad \\mu = \\frac{1}{N} \\sum_i x_i$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean =  5.078333333333334\n"
     ]
    }
   ],
   "source": [
    "print('mean = ', np.sum(data)/len(data))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "OR"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean =  5.078333333333334\n"
     ]
    }
   ],
   "source": [
    "print('mean = ', np.mean(data))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### ii) Calculating standard deviation: $\\quad \\sigma_{N-1} = \\sqrt{\\frac{1}{N-1} \\sum_i (x_i - \\mu)^2}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "standard deviation =  0.14357977404617803\n"
     ]
    }
   ],
   "source": [
    "print(\"standard deviation = \",np.sqrt(np.sum((data-np.mean(data))**2)/(len(data)-1)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "OR"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "standard deviation =  0.13746716779734078\n"
     ]
    }
   ],
   "source": [
    "print(\"standard deviation = \",np.std(data))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These results do not agree!!\n",
    "\n",
    "By default the numpy `std` function calculates $\\sigma_N$, which is similar to the $\\sigma_{N−1}$ \n",
    "given in Eq. (2.3) of H&H, except the denominator is $N$ instead of $N−1$. \n",
    "The difference doesn't usually matter, and we won't go into this in any depth now. But if we\n",
    "set the `ddof=1` option, `numpy` will calculate $\\sigma_{N−1}$.\n",
    "\n",
    "Remember: you can see all the details of `np.std` by typing `np.std?`.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "standard deviation =  0.14357977404617803\n"
     ]
    }
   ],
   "source": [
    "print(\"standard deviation = \", np.std(data, ddof=1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### iii) Standard error, or standard deviation of the mean\n",
    "\n",
    "Use Eq. (2.7): \n",
    "$$\\quad \\alpha = \\frac{\\sigma_{N-1}}{\\sqrt{N}}. $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "standard error = 0.04144791059787326\n"
     ]
    }
   ],
   "source": [
    "print(\"standard error =\", np.std(data, ddof=1)/np.sqrt(len(data)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### iv) Formatted result:\n",
    "\n",
    "Sensitivity = $5.08 \\pm 0.04\\, \\mbox{A/W}$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Version information\n",
    "`version_information` is from J.R. Johansson (jrjohansson at gmail.com); see <a href='http://nbviewer.jupyter.org/github/jrjohansson/scientific-python-lectures/blob/master/Lecture-0-Scientific-Computing-with-Python.ipynb'>Introduction to scientific computing with Python</a> for more information and instructions for package installation.\n",
    "\n",
    "`version_information` is installed on the linux network at Bucknell"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "%load_ext version_information"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/json": {
       "Software versions": [
        {
         "module": "Python",
         "version": "3.7.8 64bit [GCC 7.5.0]"
        },
        {
         "module": "IPython",
         "version": "7.17.0"
        },
        {
         "module": "OS",
         "version": "Linux 3.10.0 1127.19.1.el7.x86_64 x86_64 with centos 7.9.2009 Core"
        },
        {
         "module": "numpy",
         "version": "1.19.1"
        }
       ]
      },
      "text/html": [
       "<table><tr><th>Software</th><th>Version</th></tr><tr><td>Python</td><td>3.7.8 64bit [GCC 7.5.0]</td></tr><tr><td>IPython</td><td>7.17.0</td></tr><tr><td>OS</td><td>Linux 3.10.0 1127.19.1.el7.x86_64 x86_64 with centos 7.9.2009 Core</td></tr><tr><td>numpy</td><td>1.19.1</td></tr><tr><td colspan='2'>Sun Jan 16 14:26:17 2022 EST</td></tr></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{|l|l|}\\hline\n",
       "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n",
       "Python & 3.7.8 64bit [GCC 7.5.0] \\\\ \\hline\n",
       "IPython & 7.17.0 \\\\ \\hline\n",
       "OS & Linux 3.10.0 1127.19.1.el7.x86\\_64 x86\\_64 with centos 7.9.2009 Core \\\\ \\hline\n",
       "numpy & 1.19.1 \\\\ \\hline\n",
       "\\hline \\multicolumn{2}{|l|}{Sun Jan 16 14:26:17 2022 EST} \\\\ \\hline\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "Software versions\n",
       "Python 3.7.8 64bit [GCC 7.5.0]\n",
       "IPython 7.17.0\n",
       "OS Linux 3.10.0 1127.19.1.el7.x86_64 x86_64 with centos 7.9.2009 Core\n",
       "numpy 1.19.1\n",
       "Sun Jan 16 14:26:17 2022 EST"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%version_information numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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