Data-Analysis-Examples

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
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   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Data Analysis Examples\n",
    "\n",
    "## Introduction\n",
    "In this notebook, I'm going to demonstrate the process of data analysis in the real world using two relatively simple datasets. I will also try to explain how to apply the scientific method.\n",
    "\n",
    "It is meant to be a supplement to the lecture \"Python Crash Course\" from the course \"Data Science\" in SoftUni.\n",
    "\n",
    "The datasets used are properties of their respective owners."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 1. Gas Consumption\n",
    "### Background\n",
    "The dataset represents data from 15 houses collected during a single day. The measured variables are:\n",
    "* Temperature difference (inside - outside), averaged over the entire day, $ \\Delta T\\ [^\\circ \\text{C}] $\n",
    "* Gas consumption, $ C\\ [\\text{kWh}] $\n",
    "\n",
    "### Question\n",
    "How are these quantities related?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Data Exploration\n",
    "The dataset is pretty small, so we can inspect it directly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from scipy import stats\n",
    "from scipy.stats import ttest_ind"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>temp_diff</th>\n",
       "      <th>power</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>10.3</td>\n",
       "      <td>69.81</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>11.4</td>\n",
       "      <td>82.75</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>11.5</td>\n",
       "      <td>81.75</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>12.5</td>\n",
       "      <td>80.38</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>13.1</td>\n",
       "      <td>85.89</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>13.4</td>\n",
       "      <td>75.32</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>13.6</td>\n",
       "      <td>69.81</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>15.0</td>\n",
       "      <td>78.54</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>15.2</td>\n",
       "      <td>81.29</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>15.3</td>\n",
       "      <td>99.20</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>15.6</td>\n",
       "      <td>86.35</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>16.4</td>\n",
       "      <td>110.23</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>16.5</td>\n",
       "      <td>106.55</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>17.0</td>\n",
       "      <td>85.50</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>17.1</td>\n",
       "      <td>90.02</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    temp_diff   power\n",
       "0        10.3   69.81\n",
       "1        11.4   82.75\n",
       "2        11.5   81.75\n",
       "3        12.5   80.38\n",
       "4        13.1   85.89\n",
       "5        13.4   75.32\n",
       "6        13.6   69.81\n",
       "7        15.0   78.54\n",
       "8        15.2   81.29\n",
       "9        15.3   99.20\n",
       "10       15.6   86.35\n",
       "11       16.4  110.23\n",
       "12       16.5  106.55\n",
       "13       17.0   85.50\n",
       "14       17.1   90.02"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gas_data = pd.read_table(\"temp_gas.csv\")\n",
    "gas_data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The data appears to be clean - there are no missing values. We cannot be sure if there are any erroneous values (such as \"human errors\"). We can check for outliers by plotting histograms or boxplots."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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2CHtK0kZErG5X1/Y9ST9tm/eapCt0LHsyIv5Fly//qRYXF5tuCgAMpaYvz5qzfUrSqopz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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x21535ff3cc0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "matplotlib.rcParams[\"text.usetex\"] = True\n",
    "\n",
    "def plot_histogram(data, axis, xlabel, ylabel):\n",
    "    axis.hist(data, bins = 5, normed = 1)\n",
    "    axis.set_xlabel(xlabel)\n",
    "    axis.set_ylabel(ylabel)\n",
    "\n",
    "f, (ax1, ax2) = plt.subplots(2, figsize = (5, 6))\n",
    "plot_histogram(gas_data.temp_diff, ax1, r\"$\\Delta T\\ [^{\\circ} \\mathrm{C}]$\", r\"$\\mathrm{Frequency}$\")\n",
    "plot_histogram(gas_data.power, ax2, r\"$C\\ [\\mathrm{kWh}]$\", r\"$\\mathrm{Frequency}$\")\n",
    "\n",
    "f.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Indeed, there are no outliers.\n",
    "\n",
    "### Correlation\n",
    "To see if there is any correlation between the two variables, we'll plot $C = f(T)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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cVi5Pq1qdTbGq/tt12HD334czROYlvePuX/f61FN3/9MWfd9LWltE7GYvfy4AAL0wPDys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\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x21535ff3390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(gas_data.temp_diff, gas_data.power)\n",
    "plt.xlabel(r\"$\\Delta T\\ [^{\\circ}\\mathrm{C}]$\")\n",
    "plt.ylabel(r\"$C\\ [\\mathrm{kWh}]$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There appears to be an upward trend. To see how strong it is, we will check the Pearson correlation coefficient."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Correlation: 0.626803947457\n"
     ]
    }
   ],
   "source": [
    "corr = np.corrcoef(gas_data.temp_diff, gas_data.power)[1][0]\n",
    "print(\"Correlation: \" + str(corr))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This indicates a positive correlation, that is, the gas consumption tends to increase when the temperature difference increases.\n",
    "\n",
    "We can also add a trendline to the plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x21536614a58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "a, b, r_value, p_value, std_err  = stats.linregress(gas_data.temp_diff, gas_data.power)\n",
    "plt.scatter(gas_data.temp_diff, gas_data.power, label = None)\n",
    "regr_label = r\"$y = \" + str(round(a, 2)) + \"x + \" + str(round(b,2)) + \"$\"\n",
    "plt.plot(gas_data.temp_diff, gas_data.temp_diff * a + b, color = \"red\", linestyle = \"--\", label = regr_label)\n",
    "plt.xlabel(r\"$\\Delta T\\ [^{\\circ}\\mathrm{C}]$\")\n",
    "plt.ylabel(r\"$C\\ [\\mathrm{kWh}]$\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 2. Cloud Seeding\n",
    "### Background\n",
    "The dataset represents data from seeded and unseeded clouds. Each cloud was randomly chosen to be seeded or not. The values represent rainfall from the clouds $R [\\text{mm}]$.\n",
    "\n",
    "### Question\n",
    "Does seeding increase rainfall?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Data Exploration\n",
    "Once again, the dataset is small enough, so let's just review it. We'll also rename the column names to make them \"Pythonic\"."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>unseeded</th>\n",
       "      <th>seeded</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1202.6</td>\n",
       "      <td>2745.6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>830.1</td>\n",
       "      <td>1697.8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>372.4</td>\n",
       "      <td>1656.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>345.5</td>\n",
       "      <td>978.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>321.2</td>\n",
       "      <td>703.4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>244.3</td>\n",
       "      <td>489.1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>163.0</td>\n",
       "      <td>430.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>147.8</td>\n",
       "      <td>334.1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>95.0</td>\n",
       "      <td>302.8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>87.0</td>\n",
       "      <td>274.7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>81.2</td>\n",
       "      <td>274.7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>68.5</td>\n",
       "      <td>255.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>47.3</td>\n",
       "      <td>242.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>41.1</td>\n",
       "      <td>200.7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>36.6</td>\n",
       "      <td>198.6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>29.0</td>\n",
       "      <td>129.6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>28.6</td>\n",
       "      <td>119.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>26.3</td>\n",
       "      <td>118.3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>26.1</td>\n",
       "      <td>115.3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>24.4</td>\n",
       "      <td>92.4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>21.7</td>\n",
       "      <td>40.6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>17.3</td>\n",
       "      <td>32.7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>11.5</td>\n",
       "      <td>31.4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>4.9</td>\n",
       "      <td>17.5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>4.9</td>\n",
       "      <td>7.7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>1.0</td>\n",
       "      <td>4.1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    unseeded  seeded\n",
       "0     1202.6  2745.6\n",
       "1      830.1  1697.8\n",
       "2      372.4  1656.0\n",
       "3      345.5   978.0\n",
       "4      321.2   703.4\n",
       "5      244.3   489.1\n",
       "6      163.0   430.0\n",
       "7      147.8   334.1\n",
       "8       95.0   302.8\n",
       "9       87.0   274.7\n",
       "10      81.2   274.7\n",
       "11      68.5   255.0\n",
       "12      47.3   242.5\n",
       "13      41.1   200.7\n",
       "14      36.6   198.6\n",
       "15      29.0   129.6\n",
       "16      28.6   119.0\n",
       "17      26.3   118.3\n",
       "18      26.1   115.3\n",
       "19      24.4    92.4\n",
       "20      21.7    40.6\n",
       "21      17.3    32.7\n",
       "22      11.5    31.4\n",
       "23       4.9    17.5\n",
       "24       4.9     7.7\n",
       "25       1.0     4.1"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cloud_data = pd.read_table(\"clouds.csv\", header = 0)\n",
    "cloud_data.columns = [\"unseeded\", \"seeded\"]\n",
    "cloud_data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see how the values are distributed, again using histograms."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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EPMpXq1atc1BmnouIw8DdiOgAH+/019+k0hjU0jbVp2cPZebF/vb61O6x/k5m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\nkqSCGdSSJBXMoJYkqWAGtSRJBfvvfUnH6lZWZMwAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2153656fb70>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_histogram_clouds(data, axis, xlabel, ylabel, title):\n",
    "    axis.hist(data, bins = 10)\n",
    "    axis.set_xlabel(xlabel)\n",
    "    axis.set_ylabel(ylabel)\n",
    "    axis.set_title(title)\n",
    "\n",
    "f, (ax1, ax2) = plt.subplots(2, figsize = (5, 6))\n",
    "plot_histogram_clouds(cloud_data.unseeded, ax1, r\"$R\\ [\\mathrm{mm}]$\", \"$N$\", \"$\\mathrm{Unseeded\\ clouds}$\")\n",
    "plot_histogram_clouds(cloud_data.seeded, ax2, r\"$R\\ [\\mathrm{mm}]$\", \"$N$\", \"$\\mathrm{Seeded\\ clouds}$\")\n",
    "\n",
    "f.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The distributions may look the same but **the units on the x axis** are different. Looking at it, we can see that the seeded clouds have a much greater range.\n",
    "\n",
    "To see the two distributions better, we can plot them together."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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IA0CjRBoAGiXSANAokQaARok0ADRKpAGgUSINAI0SaQBolEgDQKNEGgAaJdIA0CiRBoBG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\nYn4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x215367c8fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f, ax = plt.subplots(figsize = (5, 6))\n",
    "bin_width = 200\n",
    "unseeded_bins = np.arange(min(cloud_data.unseeded), max(cloud_data.unseeded) + bin_width, bin_width)\n",
    "seeded_bins = np.arange(min(cloud_data.seeded), max(cloud_data.seeded) + bin_width, bin_width)\n",
    "ax.hist(cloud_data.unseeded, bins = unseeded_bins, alpha = 0.7, label = \"$\\mathrm{unseeded}$\")\n",
    "ax.hist(cloud_data.seeded, bins = seeded_bins, alpha = 0.7, label = \"$\\mathrm{seeded}$\")\n",
    "ax.set_xlabel(\"$R\\ [\\mathrm{mm}]$\")\n",
    "ax.set_ylabel(\"$N$\")\n",
    "ax.legend()\n",
    "\n",
    "f.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can see that the two distributions have very different ranges. We have to normalize these ranges in order to get meaningful results. The model we're going to run also requires normalized ranges.\n",
    "\n",
    "A data transformation we can try is logarithm. It has the property of \"shrinking\" vast spans of data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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oIg0AgFFEGgAAo4g0AABGEWkAAIwi0gAAGEWkAQAwykyknXPvOediYc8DAAArTETaObdT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\nGgAAo4g0AABGEWkAAIwi0gAAGPX/AWQA6ZS0QO5mAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x215368bcf28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cloud_data[\"log_unseeded\"] = np.log(cloud_data.unseeded)\n",
    "cloud_data[\"log_seeded\"] = np.log(cloud_data.seeded)\n",
    "\n",
    "f, ax = plt.subplots(figsize = (5, 6))\n",
    "bin_width = 0.5\n",
    "log_unseeded_bins = np.arange(min(cloud_data.log_unseeded), max(cloud_data.log_unseeded) + bin_width, bin_width)\n",
    "log_seeded_bins = np.arange(min(cloud_data.log_seeded), max(cloud_data.log_seeded) + bin_width, bin_width)\n",
    "ax.hist(cloud_data.log_unseeded, bins = log_unseeded_bins, alpha = 0.7, label = \"$\\mathrm{unseeded}$\")\n",
    "ax.hist(cloud_data.log_seeded, bins = log_seeded_bins, alpha = 0.7, label = \"$\\mathrm{seeded}$\")\n",
    "ax.set_xlabel(\"$\\ln(R)$\")\n",
    "ax.set_ylabel(\"$N$\")\n",
    "ax.legend()\n",
    "\n",
    "f.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now the two distributions are both comparable (have similar ranges) and similar to normal. We can see that the seeded clouds appear to have more rainfall. We can see this better in a boxplot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x21536799080>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f, ax = plt.subplots(figsize = (5, 6))\n",
    "ax.boxplot([cloud_data.log_unseeded, cloud_data.log_seeded])\n",
    "ax.set_xticklabels([\"$\\mathrm{unseeded}$\", \"$\\mathrm{seeded}$\"])\n",
    "ax.set_ylabel(\"$\\ln(\\mathrm{R})$\")\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can definitely see that the distributions are different and the seeded clouds appear to rain more.\n",
    "\n",
    "### Significance\n",
    "A final question remains - is the result significant? That is, how likely is it that the difference we observe is random? We have to perform a **t-test**. Let's check for $H_0$ = \"Rainfall from seeded clouds is not larger\" and $\\alpha = 0.05$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "p-value: 0.7041%\n"
     ]
    }
   ],
   "source": [
    "p_value = ttest_ind(cloud_data.log_unseeded, cloud_data.log_seeded).pvalue\n",
    "# A detail here is that this p-value is for the \"two-tailed\" t-test but we only want to see\n",
    "# whether one value is larger than the other - a \"one-tailed\" test\n",
    "print(\"p-value: \" + str(round((p_value / 2) * 100, 4)) + \"%\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The p-value is less than $\\alpha$, so we can reject $H_0$. We can even make a stronger claim - that $\\alpha = 0.01$ is also valid.\n",
    "\n",
    "So, in conclusion, we can say that seeded clouds produce significantly more rain.\n",
    "\n",
    "**Note:** These two values of $\\alpha$ are accepted as standards."
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [conda root]",
   "language": "python",
   "name": "conda-root-py"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}