More explicit wording to identify curves.

I was saying 'blue curve' 'green curve', but those have no meaning,
especially if printed in black and white. Switched to explicit
$\sigma=0.2$.

Also added second std in the chart showing the 68-95-99.7 rule.

03-Gaussians.ipynb

@@ -28,6 +28,7 @@
        "@import url('http://fonts.googleapis.com/css?family=Source+Code+Pro');\n",
        "@import url('http://fonts.googleapis.com/css?family=Vollkorn');\n",
        "@import url('http://fonts.googleapis.com/css?family=Arimo');\n",
+       "@import url('http://fonts.googleapis.com/css?family=Fira_sans');\n",
        "\n",
        "    div.cell{\n",
        "        width: 900px;\n",
@@ -118,9 +119,9 @@
        "        white-space: nowrap;\n",
        "    }\n",
        "    div.text_cell_render{\n",
-       "        font-family: 'Arimo',verdana,arial,sans-serif;\n",
+       "        font-family: 'Fira sans', verdana,arial,sans-serif;\n",
        "        line-height: 125%;\n",
-       "        font-size: 120%;\n",
+       "        font-size: 115%;\n",
        "        text-align:justify;\n",
        "        text-justify:inter-word;\n",
        "    }\n",
@@ -455,7 +456,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x440c9b0>"
+       "<matplotlib.figure.Figure at 0x4f969b0>"
       ]
      },
      "metadata": {},
@@ -491,7 +492,7 @@
      "data": {
       "image/png": "iVBORw0KGgoAAAANSUhEUgAAAvIAAADxCAYAAACpkFPLAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7ed1208>"
+       "<matplotlib.figure.Figure at 0x772d898>"
       ]
      },
      "metadata": {},
@@ -675,7 +676,7 @@
      "data": {
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       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7ef21d0>"
+       "<matplotlib.figure.Figure at 0x7e9f748>"
       ]
      },
      "metadata": {},
@@ -769,7 +770,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7f01828>"
+       "<matplotlib.figure.Figure at 0x7ee9c50>"
       ]
      },
      "metadata": {},
@@ -807,7 +808,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8011ef0>"
+       "<matplotlib.figure.Figure at 0x7f97748>"
       ]
      },
      "metadata": {},
@@ -846,7 +847,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8085a90>"
+       "<matplotlib.figure.Figure at 0x7ed8f60>"
       ]
      },
      "metadata": {},
@@ -904,7 +905,9 @@
     "    \"\"\"returns normal distribution for x given a \n",
     "    gaussian with the specified mean and variance. \n",
     "    \"\"\"\n",
-    "    return np.exp((-0.5*(x-mean)**2)/var) / \\\n",
+    "    return (np.exp((-0.5*(np.asarray(x)-mean)**2)/var) /\n",
+    "            math.sqrt(2*math.pi*var))\n",
+    "\n",
     "```"
    ]
   },
@@ -927,7 +930,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8003ef0>"
+       "<matplotlib.figure.Figure at 0x7fb7208>"
       ]
      },
      "metadata": {},
@@ -965,7 +968,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 31,
+   "execution_count": 16,
    "metadata": {
     "collapsed": false
    },
@@ -1053,9 +1056,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7945e80>"
+       "<matplotlib.figure.Figure at 0x7f2b860>"
       ]
      },
      "metadata": {},
@@ -1067,9 +1070,9 @@
     "import matplotlib.pyplot as plt\n",
     "    \n",
     "xs = np.arange(15, 30, 0.05)\n",
-    "plt.plot(xs, gaussian(xs, 23, 0.2), label='var=0.2')\n",
-    "plt.plot(xs, gaussian(xs, 23, 1), label='var=1')\n",
-    "plt.plot(xs, gaussian(xs, 23, 5), label='var=5')\n",
+    "plt.plot(xs, gaussian(xs, 23, 0.2), label='$\\sigma^2$=0.2')\n",
+    "plt.plot(xs, gaussian(xs, 23, 1), label='$\\sigma^2$=1', ls=':')\n",
+    "plt.plot(xs, gaussian(xs, 23, 5), label='$\\sigma^2$=5', ls='--')\n",
     "plt.legend();"
    ]
   },
@@ -1077,11 +1080,13 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "So what is this telling us? The blue gaussian is very narrow. It is saying that we believe $x=23$, and that we are very sure about that. In contrast, the red gaussian also believes that $x=23$, but we are much less sure about that. Our believe that $x=23$ is lower, and so our belief about the likely possible values for $x$ is spread out - we think it is quite likely that $x=20$ or $x=26$, for example. The blue gaussian has almost completely eliminated $22$ or $24$ as possible value - their probability is almost $0\\%$, whereas the red curve considers them nearly as likely as $23$.\n",
+    "So what is this telling us? The Gaussian with $\\sigma^2=0.2$ is very narrow. It is saying that we believe $x=23$, and that we are very sure about that. In contrast, the Gaussian with $\\sigma^2=5$ also believes that $x=23$, but we are much less sure about that. Our believe that $x=23$ is lower, and so our belief about the likely possible values for $x$ is spread out - we think it is quite likely that $x=20$ or $x=26$, for example. $\\sigma^2=0.2$ has almost completely eliminated $22$ or $24$ as possible value - their probability is almost $0\\%$, whereas $\\sigma^2=5$ considers them nearly as likely as $23$.\n",
     "\n",
-    "If we think back to the thermometer, we can consider these three curves as representing the readings from three different thermometers. The blue curve represents a very accurate thermometer, and the black one represents a fairly inaccurate one. Green of course represents one in between the two others. Note the very powerful property the Gaussian distribution affords us - we can entirely represent both the reading and the error of a thermometer with only two numbers - the mean and the variance.\n",
+    "If we think back to the thermometer, we can consider these three curves as representing the readings from three different thermometers. The curve for $\\sigma^2=0.2$ represents a very accurate thermometer, and curve for $\\sigma^2=5$ represents a fairly inaccurate one. Note the very powerful property the Gaussian distribution affords us - we can entirely represent both the reading and the error of a thermometer with only two numbers - the mean and the variance.\n",
     "\n",
-    "It is worth spending a few words on standard deviation now. The standard deviation is a measure of how much variation from the mean exists. For Gaussian distributions, 68% of all the data falls within one standard deviation ($1\\sigma$) of the mean, 95% falls within two standard deviations ($2\\sigma$), and 99.7% within three ($3\\sigma$). This is often called the 68-95-99.7 rule. So if you were told that the average test score in a class was 71 with a standard deviation of 9.4, you could conclude that 95% of the students received a score between 52.2 and 89.8 if the distribution is normal (that is calculated with $71 \\pm (2 * 9.4)$). "
+    "It is worth spending a few words on standard deviation now. The standard deviation is a measure of how much variation from the mean exists. For Gaussian distributions, 68% of all the data falls within one standard deviation ($1\\sigma$) of the mean, 95% falls within two standard deviations ($2\\sigma$), and 99.7% within three ($3\\sigma$). This is often called the 68-95-99.7 rule. So if you were told that the average test score in a class was 71 with a standard deviation of 9.4, you could conclude that 95% of the students received a score between 52.2 and 89.8 if the distribution is normal (that is calculated with $71 \\pm (2 * 9.4)$). \n",
+    "\n",
+    "Finally, these are not arbitrary numbers. If the Gaussian for our position is $\\mu=22$ meters, then the standard deviation also has units meters. Thus $\\sigma=0.2$ implies that 68% of the measurements range from 21.8 to 22.2 meters. Variance is the standard deviation squared, so $\\sigma^2 = .04$ meters$^2$."
    ]
   },
   {
@@ -1100,9 +1105,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7feb0b8>"
+       "<matplotlib.figure.Figure at 0x90ab2e8>"
       ]
      },
      "metadata": {},
@@ -1147,7 +1152,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9142748>"
+       "<matplotlib.figure.Figure at 0x80c4908>"
       ]
      },
      "metadata": {},
@@ -1288,9 +1293,9 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[ 1.665  5.067 -3.73   1.994 -2.977 -3.43   3.368\n",
-      " -0.468  4.66  -1.389  0.625  2.565  4.079  1.163\n",
-      " -0.795]\n"
+      "[ -1.67    1.966   2.794   2.159   2.462  -0.012\n",
+      "  12.025   6.336   3.566  -1.321  -1.545   2.25\n",
+      "   4.888   2.674   1.885]\n"
      ]
     }
    ],
@@ -1394,7 +1399,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x79563c8>"
+       "<matplotlib.figure.Figure at 0x803e5c0>"
       ]
      },
      "metadata": {},
@@ -1449,9 +1454,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9200780>"
+       "<matplotlib.figure.Figure at 0x7f4d908>"
       ]
      },
      "metadata": {},
@@ -1502,9 +1507,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x923c550>"
+       "<matplotlib.figure.Figure at 0x7efcb38>"
       ]
      },
      "metadata": {},
@@ -1594,7 +1599,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.4.1"
+   "version": "3.4.3"
   }
  },
  "nbformat": 4,

code/gaussian_internal.py

@@ -48,23 +48,42 @@ def plot_gaussian (mu, variance,
 
 def display_stddev_plot():
     xs = np.arange(10,30,0.1)
-    var = 8; stddev = math.sqrt(var)
+    var = 8;
+    stddev = math.sqrt(var)
     p2, = plt.plot (xs,[stats.gaussian(x, 20, var) for x in xs])
     x = 20+stddev
+    # 1std vertical lines
     y = stats.gaussian(x, 20, var)
     plt.plot ([x,x], [0,y],'g')
     plt.plot ([20-stddev, 20-stddev], [0,y], 'g')
+
+    #2std vertical lines
+    x = 20+2*stddev
+    y = stats.gaussian(x, 20, var)
+    plt.plot ([x,x], [0,y],'g')
+    plt.plot ([20-2*stddev, 20-2*stddev], [0,y], 'g')
+
     y = stats.gaussian(20,20,var)
     plt.plot ([20,20],[0,y],'b')
+
+    x = 20+stddev
     ax = plt.axes()
     ax.annotate('68%', xy=(20.3, 0.045))
     ax.annotate('', xy=(20-stddev,0.04), xytext=(x,0.04),
                 arrowprops=dict(arrowstyle="<->",
                                 ec="r",
                                 shrinkA=2, shrinkB=2))
-    ax.xaxis.set_ticks ([20-stddev, 20, 20+stddev])
-    ax.xaxis.set_ticklabels(['$-\sigma$','$\mu$','$\sigma$'])
+    ax.annotate('95%', xy=(20.3, 0.02))
+    ax.annotate('', xy=(20-2*stddev,0.015), xytext=(20+2*stddev,0.015),
+                arrowprops=dict(arrowstyle="<->",
+                                ec="r",
+                                shrinkA=2, shrinkB=2))
+
+
+    ax.xaxis.set_ticks ([20-2*stddev, 20-stddev, 20, 20+stddev, 20+2*stddev])
+    ax.xaxis.set_ticklabels(['$-2\sigma$', '$-1\sigma$','$\mu$','$1\sigma$', '$2\sigma$'])
     ax.yaxis.set_ticks([])
+    ax.grid(None, 'both', lw=0)
     plt.show()
 
 if __name__ == '__main__':