CLT Experiments.ipynb

{
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"Callysto_Notebook-Banner_Top_06.06.18.jpg\"/>"
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  {
   "cell_type": "code",
   "execution_count": null,
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   "source": [
    "#Running this cell displays a button to toggle hidden code\n",
    "#From: http://chris-said.io/2016/02/13/how-to-make-polished-jupyter-presentations-with-optional-code-visibility/\n",
    "\n",
    "from IPython.display import HTML\n",
    "\n",
    "HTML('''<script>\n",
    "  function code_toggle() {\n",
    "    if (code_shown){\n",
    "      $('div.input').hide('500');\n",
    "      $('#toggleButton').val('Show Code')\n",
    "    } else {\n",
    "      $('div.input').show('500');\n",
    "      $('#toggleButton').val('Hide Code')\n",
    "    }\n",
    "    code_shown = !code_shown\n",
    "  }\n",
    "  \n",
    "  $( document ).ready(function(){\n",
    "    code_shown=false;\n",
    "    $('div.input').hide()\n",
    "  });\n",
    "</script>\n",
    "<form action=\"javascript:code_toggle()\"><input type=\"submit\" id=\"toggleButton\" value=\"Show Code\"></form>''')"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Central Limit Theorem Experiments\n",
    "\n",
    "**Please note: If you have hidden the code using the toggle in the cell above, the cells will appear as an empty line. However, they can still be run by pressing shift-enter with the empty cell selected.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Experiment 1(a): \n",
    "\n",
    "Toss a fair coin.  If it lands heads, you win 100 dollars, and if it lands tails, you win 0 dollars.  Run this experiment X times, where X is chosen by the user.  We get a graph of the results, and discover that we get roughly a mean of 50 dollars and a standard deviation of 50 dollars.\n",
    "\n",
    "Note: In this plot and all others, the mean will be displayed as a vertical red line, and one standard deviation away from the mean on both sides will be displayed as a horizontal red line."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#EXPERIMENT 1A\n",
    "\n",
    "#SETUP\n",
    "\n",
    "import numpy as np\n",
    "import plotly\n",
    "import plotly.offline as py\n",
    "import plotly.graph_objs as go\n",
    "py.init_notebook_mode(connected=False)\n",
    "%matplotlib inline\n",
    "\n",
    "#Takes input from the user for the value of X:\n",
    "print(\"How many times would you like to run the experiment? \")\n",
    "X = int(input())\n",
    "X1 = X\n",
    "\n",
    "#Stores the data on wins and losses:\n",
    "win_data = []\n",
    "\n",
    "#Performs the experiment X times:\n",
    "for n in range(0, X):\n",
    "    \n",
    "    #Flips a coin. 100 represents heads, 0 represents tails:\n",
    "    flip = np.random.choice((100, 0))\n",
    "    \n",
    "    #Updates win_data with a win or a loss:\n",
    "    win_data.append(flip)\n",
    "\n",
    "#Calculates the mean:\n",
    "mean = np.mean(win_data)\n",
    "\n",
    "#Calculates the standard deviation:\n",
    "sd = np.std(win_data)\n",
    "\n",
    "#Graphs a histogram of the data:\n",
    "data = [go.Histogram(x = win_data, xbins=dict(\n",
    "        start=-10,\n",
    "        end=110,\n",
    "        size=10\n",
    "    ))]\n",
    "\n",
    "title = \"Experiment 1(a) Winnings (Mean: {0:.2f}, Standard deviation: {1:.2f})\".format(mean, sd)\n",
    "\n",
    "layout = go.Layout(\n",
    "title=title,\n",
    "xaxis=dict(\n",
    "    title='Winnings',\n",
    "    range = [0,110],\n",
    "    fixedrange = True,\n",
    "),\n",
    "yaxis=dict(\n",
    "    title='Count',\n",
    "    fixedrange = False\n",
    "),\n",
    "shapes=[dict({\n",
    "        'type': 'line',\n",
    "        'x0': mean,\n",
    "        'y0': 0,\n",
    "        'x1': mean,\n",
    "        'y1': 1,\n",
    "        'xref': 'x',\n",
    "        'yref': 'paper',\n",
    "        'line': {\n",
    "            'color': '#f44242',\n",
    "            'width': 3\n",
    "}}),\n",
    "       dict({\n",
    "        'type': 'line',\n",
    "        'x0': mean - sd,\n",
    "        'y0': 0,\n",
    "        'x1': mean + sd,\n",
    "        'y1': 0,\n",
    "        'line': {\n",
    "            'color': '#f44242',\n",
    "            'width': 10\n",
    "}})]\n",
    ")\n",
    "\n",
    "fig = go.Figure(data=data, layout=layout)\n",
    "py.iplot(fig, filename='Experiment 1 Data')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Experiment 1(b)\n",
    "\n",
    "Toss a fair coin one hundred times, getting 1 dollar for each head and 0 dollars for each tail.  Run this 100-trial experiment X times, where X is chosen by the user.  We get a graph of the results, and discover that we get roughly a mean of 50 dollars and a standard deviation of 5 dollars."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#EXPERIMENT 1B\n",
    "\n",
    "#Takes input from the user for the value of X:\n",
    "print(\"How many times would you like to run the experiment? \")\n",
    "X = int(input())\n",
    "X2 = X\n",
    "\n",
    "#Stores the data on wins and losses:\n",
    "win_data = []\n",
    "\n",
    "#Performs the experiment X times:\n",
    "for n in range(0, X1):\n",
    "    \n",
    "    #Stores the amount won:\n",
    "    winnings = 0\n",
    "    \n",
    "    #Flips 100 coins:\n",
    "    for m in range(0, 100):\n",
    "        \n",
    "        #Flips a coin. 1 represents heads, 0 represents tails:\n",
    "        flip = np.random.choice((1, 0))\n",
    "        \n",
    "        #Updates winnings:\n",
    "        winnings += flip\n",
    "    \n",
    "    #Updates win_data with the final total for this run of the experiment:\n",
    "    win_data.append(winnings)\n",
    "\n",
    "#Calculates the mean:\n",
    "mean = np.mean(win_data)\n",
    "\n",
    "#Calculates the standard deviation:\n",
    "sd = np.std(win_data)\n",
    "\n",
    "#Graphs a histogram of the data:\n",
    "data = [go.Histogram(x = win_data, xbins=dict(\n",
    "        start=-10,\n",
    "        end=110,\n",
    "        size=5\n",
    "    ))]\n",
    "\n",
    "title = \"Experiment 1(b) Winnings (Mean: {0:.2f}, Standard deviation: {1:.2f})\".format(mean, sd)\n",
    "\n",
    "layout = go.Layout(\n",
    "title=title,\n",
    "xaxis=dict(\n",
    "    title='Winnings',\n",
    "    range = [0,110],\n",
    "    fixedrange = True,\n",
    "),\n",
    "yaxis=dict(\n",
    "    title='Count',\n",
    "    fixedrange = False\n",
    "),\n",
    "shapes=[dict({\n",
    "        'type': 'line',\n",
    "        'x0': mean,\n",
    "        'y0': 0,\n",
    "        'x1': mean,\n",
    "        'y1': 1,\n",
    "        'xref': 'x',\n",
    "        'yref': 'paper',\n",
    "        'line': {\n",
    "            'color': '#f44242',\n",
    "            'width': 3\n",
    "}}),\n",
    "       dict({\n",
    "        'type': 'line',\n",
    "        'x0': mean - sd,\n",
    "        'y0': 0,\n",
    "        'x1': mean + sd,\n",
    "        'y1': 0,\n",
    "        'line': {\n",
    "            'color': '#f44242',\n",
    "            'width': 10\n",
    "}})]\n",
    ")\n",
    "\n",
    "fig = go.Figure(data=data, layout=layout)\n",
    "py.iplot(fig, filename='Experiment 1 Data')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Experiment 2(a)\n",
    "\n",
    "For experiments 2(a) and 2(b), we'll re-run the above experiments, and this time you can specify the value of N.\n",
    "\n",
    "Flip X coins (with the same X from experiment 1(a)), and on each coin flip, you win N dollars or win 0 dollars."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#EXPERIMENT 2A\n",
    "\n",
    "#Takes input from the user for the value of N:\n",
    "print(\"How much money will a winning flip be worth? \")\n",
    "N = int(input())\n",
    "\n",
    "print(f\"The experiment will be run {X1} times, as in experiment 1(a).\")\n",
    "\n",
    "#Stores the data on wins and losses:\n",
    "win_data = []\n",
    "\n",
    "#Performs the experiment X times:\n",
    "for n in range(0, X):\n",
    "    \n",
    "    #Flips a coin. N represents heads, 0 represents tails:\n",
    "    flip = np.random.choice((N, 0))\n",
    "    \n",
    "    #Updates win_data with a win or a loss:\n",
    "    win_data.append(flip)\n",
    "\n",
    "#Calculates the mean:\n",
    "mean = np.mean(win_data)\n",
    "\n",
    "#Calculates the standard deviation:\n",
    "sd = np.std(win_data)\n",
    "\n",
    "#Graphs a histogram of the data:\n",
    "data = [go.Histogram(x = win_data, xbins=dict(\n",
    "        start=min(win_data),\n",
    "        end=max(win_data) + (max(win_data) - min(win_data))/15,\n",
    "        size=(max(win_data) - min(win_data))/15\n",
    "    ))]\n",
    "\n",
    "title = \"Experiment 2(a) Winnings (Mean: {0:.2f}, Standard deviation: {1:.2f})\".format(mean, sd)\n",
    "\n",
    "layout = go.Layout(\n",
    "title=title,\n",
    "xaxis=dict(\n",
    "    title='Winnings',\n",
    "    #range = [0,110],\n",
    "    #fixedrange = True,\n",
    "),\n",
    "yaxis=dict(\n",
    "    title='Count',\n",
    "    fixedrange = False\n",
    "),\n",
    "shapes=[dict({\n",
    "        'type': 'line',\n",
    "        'x0': mean,\n",
    "        'y0': 0,\n",
    "        'x1': mean,\n",
    "        'y1': 1,\n",
    "        'xref': 'x',\n",
    "        'yref': 'paper',\n",
    "        'line': {\n",
    "            'color': '#f44242',\n",
    "            'width': 3\n",
    "}}),\n",
    "       dict({\n",
    "        'type': 'line',\n",
    "        'x0': mean - sd,\n",
    "        'y0': 0,\n",
    "        'x1': mean + sd,\n",
    "        'y1': 0,\n",
    "        'line': {\n",
    "            'color': '#f44242',\n",
    "            'width': 10\n",
    "}})]\n",
    ")\n",
    "\n",
    "fig = go.Figure(data=data, layout=layout)\n",
    "py.iplot(fig, filename='Experiment 1 Data')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Experiment 2(b)\n",
    "\n",
    "Toss a fair coin N times, getting 1 dollar for each head and 0 dollars for each tail.  Run this N-trial experiment X times, where X is the same as specified in experiment 1(b)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#EXPERIMENT 2B\n",
    "\n",
    "print(f\"The experiment will be run {X2} times, as in experiment 1(b).\")\n",
    "\n",
    "#Stores the data on wins and losses:\n",
    "win_data = []\n",
    "\n",
    "#Performs the experiment X times:\n",
    "X = X2\n",
    "for n in range(0, X):\n",
    "    \n",
    "    #Stores the amount won:\n",
    "    winnings = 0\n",
    "    \n",
    "    #Flips 100 coins:\n",
    "    for m in range(0, N):\n",
    "        \n",
    "        #Flips a coin. 1 represents heads, 0 represents tails:\n",
    "        flip = np.random.choice((1, 0))\n",
    "        \n",
    "        #Updates winnings:\n",
    "        winnings += flip\n",
    "    \n",
    "    #Updates win_data with the final total for this run of the experiment:\n",
    "    win_data.append(winnings)\n",
    "\n",
    "#Calculates the mean:\n",
    "mean = np.mean(win_data)\n",
    "\n",
    "#Calculates the standard deviation:\n",
    "sd = np.std(win_data)\n",
    "\n",
    "#Graphs a histogram of the data:\n",
    "data = [go.Histogram(x = win_data, xbins=dict(\n",
    "        start=min(win_data),\n",
    "        end=max(win_data) + (max(win_data) - min(win_data))/15,\n",
    "        size=(max(win_data) - min(win_data))/15\n",
    "    ))]\n",
    "\n",
    "title = \"Experiment 2(b) Winnings (Mean: {0:.2f}, Standard deviation: {1:.2f})\".format(mean, sd)\n",
    "\n",
    "layout = go.Layout(\n",
    "title=title,\n",
    "xaxis=dict(\n",
    "    title='Winnings',\n",
    "    #range = [0,110],\n",
    "    #fixedrange = True,\n",
    "),\n",
    "yaxis=dict(\n",
    "    title='Count',\n",
    "    fixedrange = False\n",
    "),\n",
    "shapes=[dict({\n",
    "        'type': 'line',\n",
    "        'x0': mean,\n",
    "        'y0': 0,\n",
    "        'x1': mean,\n",
    "        'y1': 1,\n",
    "        'xref': 'x',\n",
    "        'yref': 'paper',\n",
    "        'line': {\n",
    "            'color': '#f44242',\n",
    "            'width': 3\n",
    "}}),\n",
    "       dict({\n",
    "        'type': 'line',\n",
    "        'x0': mean - sd,\n",
    "        'y0': 0,\n",
    "        'x1': mean + sd,\n",
    "        'y1': 0,\n",
    "        'line': {\n",
    "            'color': '#f44242',\n",
    "            'width': 10\n",
    "}})]\n",
    ")\n",
    "\n",
    "fig = go.Figure(data=data, layout=layout)\n",
    "py.iplot(fig, filename='Experiment 1 Data')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Experiment 3(a)\n",
    "Toss a 6-sided die.  In this experiment, we toss X of these dice, where X is specified by the user.  We determine the mean and standard deviation of this experiment."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#EXPERIMENT 3A\n",
    "\n",
    "#Takes input from the user for the value of X:\n",
    "print(\"How many 6-sided dice would you like to roll? \")\n",
    "X = int(input())\n",
    "X3 = X\n",
    "\n",
    "#Stores the data:\n",
    "win_data = []\n",
    "\n",
    "#Performs the experiment X times:\n",
    "for n in range(0, X):\n",
    "    \n",
    "    #Rolls a die:\n",
    "    roll = np.random.choice((1,2,3,4,5,6))\n",
    "    \n",
    "    #Updates win_data with the roll value:\n",
    "    win_data.append(roll)\n",
    "\n",
    "#Calculates the mean:\n",
    "mean = np.mean(win_data)\n",
    "\n",
    "#Calculates the standard deviation:\n",
    "sd = np.std(win_data)\n",
    "\n",
    "#Graphs a histogram of the data:\n",
    "data = [go.Histogram(x = win_data, xbins=dict(\n",
    "        start=min(win_data) - .5,\n",
    "        end=(max(win_data) + (max(win_data) - min(win_data))/15) + .5,\n",
    "        size=1\n",
    "    ))]\n",
    "\n",
    "title = \"Experiment 3(a) Results (Mean: {0:.2f}, Standard deviation: {1:.2f})\".format(mean, sd)\n",
    "\n",
    "layout = go.Layout(\n",
    "title=title,\n",
    "xaxis=dict(\n",
    "    title='Value',\n",
    "    #range = [0,110],\n",
    "    #fixedrange = True,\n",
    "),\n",
    "yaxis=dict(\n",
    "    title='Count',\n",
    "    fixedrange = False\n",
    "),\n",
    "shapes=[dict({\n",
    "        'type': 'line',\n",
    "        'x0': mean,\n",
    "        'y0': 0,\n",
    "        'x1': mean,\n",
    "        'y1': 1,\n",
    "        'xref': 'x',\n",
    "        'yref': 'paper',\n",
    "        'line': {\n",
    "            'color': '#f44242',\n",
    "            'width': 3\n",
    "}}),\n",
    "       dict({\n",
    "        'type': 'line',\n",
    "        'x0': mean - sd,\n",
    "        'y0': 0,\n",
    "        'x1': mean + sd,\n",
    "        'y1': 0,\n",
    "        'line': {\n",
    "            'color': '#f44242',\n",
    "            'width': 10\n",
    "}})]\n",
    ")\n",
    "\n",
    "fig = go.Figure(data=data, layout=layout)\n",
    "py.iplot(fig, filename='Experiment 1 Data')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Experiment 3(b) \n",
    "Toss a 6-sided die.  In this experiment, we toss 4 of these dice, and mark the average.  We do this X times, where X is specified by the user.  We determine the mean and standard deviation of this experiment, showing that the standard deviation reduces by a factor of sqrt(4)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#EXPERIMENT 3B\n",
    "\n",
    "#Takes input from the user for the value of X:\n",
    "print(\"How many experiments would you like to run? \")\n",
    "X = int(input())\n",
    "X4 = X\n",
    "\n",
    "#Stores the data:\n",
    "mean_data = []\n",
    "\n",
    "#Runs X experiments:\n",
    "for m in range(0, X):\n",
    "    \n",
    "    #Stores the results of the 4 rolls in this experiment\n",
    "    temp_data = []\n",
    "\n",
    "    #Rolls 4 dice:\n",
    "    for n in range(0, 4):\n",
    "\n",
    "        #Rolls a die:\n",
    "        roll = np.random.choice((1,2,3,4,5,6))\n",
    "\n",
    "        #Updates temp_data with the roll value:\n",
    "        temp_data.append(roll)\n",
    "\n",
    "    #Takes the average of the 4 rolls, appends it to mean_data:\n",
    "    mean = np.mean(temp_data)\n",
    "    mean_data.append(mean)\n",
    "\n",
    "#Calculates the mean:\n",
    "mean = np.mean(mean_data)\n",
    "\n",
    "#Calculates the standard deviation:\n",
    "sd = np.std(mean_data)\n",
    "\n",
    "\n",
    "#Graphs a histogram of the data:\n",
    "data = [go.Histogram(x = mean_data, xbins=dict(\n",
    "        start=min(mean_data),\n",
    "        end=max(mean_data) + (max(mean_data) - min(mean_data))/15,\n",
    "        size=(max(mean_data) - min(mean_data))/15\n",
    "    ))]\n",
    "\n",
    "title = \"Experiment 3(b) Results (Mean: {0:.2f}, Standard deviation: {1:.2f})\".format(mean, sd)\n",
    "\n",
    "layout = go.Layout(\n",
    "title=title,\n",
    "xaxis=dict(\n",
    "    title='Value',\n",
    "    #range = [0,110],\n",
    "    #fixedrange = True,\n",
    "),\n",
    "yaxis=dict(\n",
    "    title='Count',\n",
    "    fixedrange = False\n",
    "),\n",
    "shapes=[dict({\n",
    "        'type': 'line',\n",
    "        'x0': mean,\n",
    "        'y0': 0,\n",
    "        'x1': mean,\n",
    "        'y1': 1,\n",
    "        'xref': 'x',\n",
    "        'yref': 'paper',\n",
    "        'line': {\n",
    "            'color': '#f44242',\n",
    "            'width': 3\n",
    "}}),\n",
    "       dict({\n",
    "        'type': 'line',\n",
    "        'x0': mean - sd,\n",
    "        'y0': 0,\n",
    "        'x1': mean + sd,\n",
    "        'y1': 0,\n",
    "        'line': {\n",
    "            'color': '#f44242',\n",
    "            'width': 10\n",
    "}})]\n",
    ")\n",
    "\n",
    "fig = go.Figure(data=data, layout=layout)\n",
    "py.iplot(fig, filename='Experiment 1 Data')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Experiment 4\n",
    "Re-run the above experiment, except we change 4 with N. What happens to the standard deviations as N gets larger?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#EXPERIMENT 4\n",
    "\n",
    "#Takes input from the user for the value of X:\n",
    "print(\"How many 6-sided dice would you like to roll for each experiment? \")\n",
    "N = int(input())\n",
    "\n",
    "print(f\"The experiment will be run {X4} times, as in experiment 3(b).\")\n",
    "X = X4\n",
    "\n",
    "#Stores the data:\n",
    "mean_data = []\n",
    "\n",
    "#Runs X experiments:\n",
    "for m in range(0, X):\n",
    "    \n",
    "    #Stores the results of the N rolls in this experiment\n",
    "    temp_data = []\n",
    "\n",
    "    #Rolls N dice:\n",
    "    for n in range(0, N):\n",
    "\n",
    "        #Rolls a die:\n",
    "        roll = np.random.choice((1,2,3,4,5,6))\n",
    "\n",
    "        #Updates temp_data with the roll value:\n",
    "        temp_data.append(roll)\n",
    "\n",
    "    #Takes the average of the N rolls, appends it to mean_data:\n",
    "    mean = np.mean(temp_data)\n",
    "    mean_data.append(mean)\n",
    "\n",
    "#Calculates the mean:\n",
    "mean = np.mean(mean_data)\n",
    "\n",
    "#Calculates the standard deviation:\n",
    "sd = np.std(mean_data)\n",
    "\n",
    "print()\n",
    "print(\"The mean of the average rolls was: {:.3f}\".format(mean))\n",
    "print(\"The standard deviation was: {:.3f}\".format(sd))"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "### Experiment 5\n",
    "\n",
    "Toss a 6-sided die, but this time, the six values are chosen by the user.  (For example, it can be 1,2,3,4,5,10000). The user will specify X (the number of dice to be rolled for an individual trial) and N (the number of trials). The average of each of the N trials will be plotted.\n",
    "\n",
    "Experiment with different values to reproduce the Central Limit Theorem!"
   ]
  },
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   "source": [
    "#EXPERIMENT 5\n",
    "\n",
    "#Choosing values for the die:\n",
    "print(\"Choose what you want the values on the dice to be:\")\n",
    "print()\n",
    "\n",
    "print(\"First value: \")\n",
    "V1 = int(input())\n",
    "\n",
    "print(\"Second value: \")\n",
    "V2 = int(input())\n",
    "\n",
    "print(\"Third value: \")\n",
    "V3 = int(input())\n",
    "\n",
    "print(\"Fourth value: \")\n",
    "V4 = int(input())\n",
    "\n",
    "print(\"Fifth value: \")\n",
    "V5 = int(input())\n",
    "\n",
    "print(\"Sixth value: \")\n",
    "V6 = int(input())\n",
    "\n",
    "values = [V1,V2,V3,V4,V5,V6]\n",
    "\n",
    "print()\n",
    "print(\"The values you have chosen are:\")\n",
    "print(values)\n",
    "\n",
    "#Asking for the number of times the dice will be rolled per trial:\n",
    "print()\n",
    "print(\"Please choose a value for N - the number of dice we will be rolling for an individual trial.\")\n",
    "N = int(input())\n",
    "\n",
    "#Asking for the number of trials:\n",
    "print()\n",
    "print(\"Please choose a value for X - the number of trials for which we will roll N dice.\")\n",
    "X = int(input())\n",
    "\n",
    "sample_means = []\n",
    "\n",
    "#Performing all trials:\n",
    "for n in range(0, X):\n",
    "    \n",
    "    sample = []\n",
    "    \n",
    "    #Rolling all dice for a trial:\n",
    "    for m in range(0, N):\n",
    "        num = np.random.choice(values)\n",
    "        sample.append(num)\n",
    "        \n",
    "    #Calculating mean of a trial:\n",
    "    mean = sum(sample)/len(sample)\n",
    "    sample_means.append(mean)\n",
    "    \n",
    "#Plotting the sample means:\n",
    "data = [go.Histogram(x = sample_means)]\n",
    "\n",
    "mean = sum(sample_means)/len(sample_means)\n",
    "sd = np.std(sample_means)\n",
    "\n",
    "title = \"Sample Means (Mean: {0:.2f}, Standard deviation: {1:.2f})\".format(mean, sd)\n",
    "\n",
    "layout = go.Layout(\n",
    "title=title,\n",
    "xaxis=dict(\n",
    "    title='Value',\n",
    "    #range = [min(sample_means),max(sample_means)+1],\n",
    "    fixedrange = True,\n",
    "    #tick0 = 0,\n",
    "    #dtick = 1\n",
    "),\n",
    "yaxis=dict(\n",
    "    title='Count',\n",
    "    fixedrange = False\n",
    "),\n",
    "shapes=[dict({\n",
    "        'type': 'line',\n",
    "        'x0': mean,\n",
    "        'y0': 0,\n",
    "        'x1': mean,\n",
    "        'y1': 1,\n",
    "        'xref': 'x',\n",
    "        'yref': 'paper',\n",
    "        'line': {\n",
    "            'color': '#f44242',\n",
    "            'width': 3\n",
    "}}),\n",
    "       dict({\n",
    "        'type': 'line',\n",
    "        'x0': mean - sd,\n",
    "        'y0': 0,\n",
    "        'x1': mean + sd,\n",
    "        'y1': 0,\n",
    "        'line': {\n",
    "            'color': '#f44242',\n",
    "            'width': 10\n",
    "}})])\n",
    "\n",
    "fig = go.Figure(data=data, layout=layout)\n",
    "py.iplot(fig, filename='Sample')"
   ]
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