diff --git a/SRECon16 - Theory.ipynb b/SRECon16 - Theory.ipynb
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+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "metadata": {
+    "collapsed": true,
+    "slideshow": {
+     "slide_type": "skip"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "import math\n",
+    "import numpy as np\n",
+    "from matplotlib import pyplot as plt\n",
+    "import seaborn as sns\n",
+    "%matplotlib inline"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "nbpresent": {
+     "id": "769dee7d-5d46-414a-8713-fae761d2cd10"
+    },
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# #StatsForEngineers - Theory\n",
+    "\n",
+    "1. Summary Statistics\n",
+    "\n",
+    "2. Deviation Measures\n",
+    "\n",
+    "3. Measuring Distributions\n",
+    "\n",
+    "4. Mergability and Robustness"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "nbpresent": {
+     "id": "6d170ee1-0905-48d6-9874-c80e01fe9254"
+    },
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# (1) Summary Statistics\n",
+    "\n",
+    "Problem statment:\n",
+    "\n",
+    "Given a dataset X:\n",
+    "\n",
+    "* Characterize the distribution in one or two values\n",
+    "* Characterization should be robust, i.e. not severly affected by single outliers\n",
+    "\n",
+    "Equivalent of an elevator pitch for a data sets.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "nbpresent": {
+     "id": "e45021d3-f422-4831-9e82-69a390cdfe53"
+    },
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## The Mean Value\n",
+    "\n",
+    "The _mean value_ of $X=[x_1, \\dots, x_n]$ is defined as\n",
+    "\n",
+    "$$ \\mu = mean(X) = \\frac{1}{n} \\sum_{i=1}^n x_i. $$\n",
+    "\n",
+    "- Represnets center of mass\n",
+    "- If the values are close together this is a good representative\n",
+    "- Can be easily affected by outliers"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Computation of the mean value\n",
+    "\n",
+    "def mean(X):\n",
+    "    return float(sum(X)) / len(X)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Medians\n",
+    "\n",
+    "A _median value_ for $x_1, \\dots, x_n$ is number $m$ such that\n",
+    "  \n",
+    " $$ \\# \\{ i \\,|\\, x_i \\lt m \\} = \\# \\{ i \\,|\\, x_i \\gt m \\}. $$\n",
+    "\n",
+    "So the number of samples smaller than $m$ is equal to the number of samples larger than $m$.\n",
+    "(Both should be roughly $n/2$).\n",
+    "  \n",
+    "Remark:\n",
+    "\n",
+    "* A Median always exists\n",
+    "* Median is not always unique\n",
+    "* Very robust to outliers\n",
+    "* Less \"precise\" then the mean for well behaved datasets"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# The central median\n",
+    "\n",
+    "Medians can be computed by sorting the dataset.\n",
+    "\n",
+    "`S = sorted(X)`\n",
+    "\n",
+    "Consider the following cases:\n",
+    "\n",
+    "- if `len(X)` is odd, then `S[(len(X)-1)/2]` is the (unique and) central median.\n",
+    "- if `len(X)` is even, then all values between `S[len(X)/2-1]` and `S[len(X)/2]` are median values.  \n",
+    "  The central median is defined as the mean of those two values.\n",
+    "\n",
+    "We write $median(X)$ for the central median of $X$.\n",
+    "\n",
+    "Example:\n",
+    "* `median([1,2,3]) = 2`\n",
+    "* `median([1,2]) = (1 + 2) / 2`"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "metadata": {
+    "collapsed": true,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Computation of the median\n",
+    "def median_range(X):\n",
+    "    S = sorted(X)\n",
+    "    n = len(X)\n",
+    "    lower_mid = int(math.floor((n-1)/2.))\n",
+    "    upper_mid = int(math.ceil((n-1)/2.))\n",
+    "    return (S[lower_mid], S[upper_mid])\n",
+    "\n",
+    "def median(X):\n",
+    "    a,b = median_range(X)\n",
+    "    return (a+b)/2"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Challenge\n",
+    "\n",
+    "Just like the mean value, medians can be computed in linear time.\n",
+    "\n",
+    "Can you find the algorithm?\n",
+    "\n",
+    "Hint: Start with a quicksort."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
+   "metadata": {
+    "collapsed": true,
+    "slideshow": {
+     "slide_type": "skip"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "def plot_median_range(X):\n",
+    "    plt.figure(figsize=(14,1))\n",
+    "    a, b = median_range(X)\n",
+    "    c = (a+b)/2.\n",
+    "    sns.rugplot(X, color=\"green\", height=0.5)\n",
+    "    plt.plot([a,a],[0.5,1],'r-')\n",
+    "    plt.plot([b,b],[0.5,1],'r-')\n",
+    "    plt.plot([c,c],[0.5,1],'r-')\n",
+    "    plt.plot([a,b],[0.5,0.5],'r--')\n",
+    "    plt.title(\"Median Range: {:.3f} .. {:,.3f}\".format(a,b))\n",
+    "    plt.ylim([0,1])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x1066f3850>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "plot_median_range([1,2,3])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 23,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x10d8a5390>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "plot_median_range([1,2,3,4])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Truncated Means\n",
+    "\n",
+    "* Truncated means \"interpolate\" between means and medianse.\n",
+    "* In many cases they provid the increased \"precision\" of means, while keeping the robustnes against outliers.\n",
+    "\n",
+    "To arrive at the k-truncated mean of a dataset X, perform the following steps:\n",
+    "* Remove a minimum and a maximum value from X, a total of k times.\n",
+    "* The remaining samples are called the k-truncation of X and denoted by $T_k X$.\n",
+    "* Caculate the mean value of the remaining samples.\n",
+    "\n",
+    "We deonte the k-truncated mean by `tmean(X,k)`."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 24,
+   "metadata": {
+    "collapsed": true,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "def tmean(X,k):\n",
+    "    X = list(X)\n",
+    "    k = int(k)\n",
+    "    if len(X) <= 2*k:\n",
+    "        return None\n",
+    "    for i in range(k):\n",
+    "        X.remove(max(X))\n",
+    "        X.remove(min(X))\n",
+    "    return mean(X)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 25,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "22.2\n",
+      "3.33333333333\n",
+      "3.0\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Truncated mean examples\n",
+    "print tmean([1,2,3,5,100],0)\n",
+    "print tmean([1,2,3,5,100],1)\n",
+    "print tmean([1,2,3,5,100],2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 83,
+   "metadata": {
+    "collapsed": false,
+    "nbpresent": {
+     "id": "150f1522-96cb-4ea6-ba08-e9a4acb9554c"
+    },
+    "slideshow": {
+     "slide_type": "skip"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Plot\n",
+    "def mark(m,height=1,style='r'):\n",
+    "    plt.plot([m,m],[0,height],style)\n",
+    "\n",
+    "def plot_mean_median(X):\n",
+    "    plt.figure(figsize=(14,1))\n",
+    "    sns.rugplot(X, color='grey', height=0.5)\n",
+    "    mark(mean(X));                     print \"mean:        {:4.2f} (red)\".format(mean(X))\n",
+    "    k = int(math.ceil(10*len(X)/100.))\n",
+    "    mark(tmean(X,k), style='b');       print \"tmean(X,{:2}): {:4.2f} (blue)\".format(k,tmean(X,k))\n",
+    "    mark(median(X), style='g');        print \"median:      {:4.2f} (green)\".format(median(X))\n",
+    "    plt.show()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 84,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "mean:        0.10 (red)\n",
+      "tmean(X,50): 0.09 (blue)\n",
+      "median:      0.05 (green)\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x120f2d4d0>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "mean:        1316.31 (red)\n",
+      "tmean(X, 8): 1315.61 (blue)\n",
+      "median:      1399.66 (green)\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x11f141e90>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "mean:        45.77 (red)\n",
+      "tmean(X,10): 30.52 (blue)\n",
+      "median:      29.10 (green)\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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SpMXgF/8lSZIkDRWLHEmSJElDxSJHkiRJ0lCxyJEkSZI0VCxyJEmSJA0VixxJkiRJQ8Ui\nR5IkSdJQ+X8fOM3ZIfpXqgAAAABJRU5ErkJggg==\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x11f1523d0>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "# Normally distributed data\n",
+    "Z = [ np.random.normal() for x in range(500) ]\n",
+    "plot_mean_median(Z)\n",
+    "\n",
+    "# Mean value of some request rates\n",
+    "X = np.loadtxt(\"DataSets/RequestRates.csv\", delimiter=\",\")[:,1]\n",
+    "plot_mean_median(X)\n",
+    "\n",
+    "# Request Latency Dataset\n",
+    "Y = np.genfromtxt(\"DataSets/WebLatency.csv\", delimiter=\",\")[:,2][0:500] - 85\n",
+    "plot_mean_median(Y[0:100])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 30,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "mean:        41.60 (red)\n",
+      "tmean(X, 1): 35.00 (blue)\n",
+      "median:      3.00 (green)\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAzkAAABaCAYAAACbmphwAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n1hBOR/MAAAAASUVORK5CYII=\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x10d3fc610>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "# Mean values can be atypical\n",
+    "plot_mean_median([1,2,3,100,102])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "\n",
+    "# Mean and Median Values in Monitoring\n",
+    "\n",
+    "* Spike Erosion in Graphs [demo](https://demo-hh2.circonus.com/trending/graphs/view/6944e456-b711-e532-def1-b92c8a118898#set-OqHjffwi)\n",
+    "* Request latency mean values can be missleading [demo](https://demo-hh2.circonus.com/trending/graphs/view/359c41b6-025e-656f-d490-e5548191033b)\n",
+    "* Rolling and windowed means as smoothing techniques [demo](https://demo-hh2.circonus.com/trending/graphs/add)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# (2) Deviation Measures\n",
+    "\n",
+    "Goal measure \"typical\" deviation form a central value."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "1. The _maximal deviation_ is defined as\n",
+    "\n",
+    "   $$ maxdev(X) = max \\{ |X[i] - \\mu| \\,|\\, i=1,\\dots,n\\}.$$\n",
+    "\n",
+    "2. The _mean absolute deviation_ is defined as\n",
+    "   \n",
+    "   $$ mad(X) = \\frac{1}{n} \\sum_{i=1}^n |X[i] - \\mu|.$$\n",
+    "\n",
+    "3. The _standard deviation_ is defined as\n",
+    "\n",
+    "   $$ \\sigma(X) = stddev(X) =  \\sqrt{\\frac{1}{n} \\sum_{i=1}^n (X[i] - \\mu)^2}.$$\n",
+    "\n",
+    "4. The _inter quartile range_ is defined as difference of the 75- and 25-Percentile:\n",
+    "\n",
+    "   $$ IQR(X) = P(75,X) - P(25,X) $$\n",
+    "\n",
+    "Properties:\n",
+    "* Measure the 'typical' displacement from the mean value.\n",
+    "* Standard deviation is popular because it has extremely nice mathematical properties.\n",
+    "* The IQR is very robust to outliers."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 31,
+   "metadata": {
+    "collapsed": true,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "def max_dev(X):\n",
+    "    m = mean(X)\n",
+    "    return max(abs(x - m) for x in X)\n",
+    "\n",
+    "def mad(X):\n",
+    "    m = mean(X)\n",
+    "    return sum(abs(x - m) for x in X) / float(len(X))\n",
+    "\n",
+    "def stddev(X):\n",
+    "    m = mean(X)\n",
+    "    return math.pow(sum((x - m)**2 for x in X) / len(X), 0.5)\n",
+    "\n",
+    "def IQR(X): return percentile(75,X) - percentile(0.25,X)\n",
+    "\n",
+    "# computation of percentiles is given below"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 32,
+   "metadata": {
+    "collapsed": true,
+    "slideshow": {
+     "slide_type": "skip"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Plotting helper function\n",
+    "def plot_mean_dev(X, m, s, new_canvas=True):\n",
+    "    print \"center    = \", m\n",
+    "    print \"deviation = \", s\n",
+    "    if new_canvas: plt.figure(figsize=(14,1))\n",
+    "    sns.rugplot(X, color='grey', height=0.5)\n",
+    "    plt.plot([m,m],[0,1],'r-' )\n",
+    "    plt.plot([m-s,m-s],[0,1],'b-')\n",
+    "    plt.plot([m+s,m+s],[0,1],'b-')\n",
+    "    plt.plot([m-s,m+s],[0.5,0.5],'b--')\n",
+    "    if new_canvas: plt.show()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 85,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Maximal deviation\n",
+      "center    =  0.0994650556128\n",
+      "deviation =  2.63265757082\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x120f2d450>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Standard Deviation\n",
+      "center    =  0.0994650556128\n",
+      "deviation =  0.954918342285\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x1224dce50>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Mean Absolute Deviation\n",
+      "center    =  0.0994650556128\n",
+      "deviation =  0.779523036406\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x122bf9e50>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Median IQR\n",
+      "center    =  0.0541148890544\n",
+      "deviation =  1.61999449286\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x12283fe50>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "# Standard deviation is a good deviation for normal distributed data\n",
+    "\n",
+    "print \"Maximal deviation\"\n",
+    "plot_mean_dev(Z,mean(Z),max_dev(Z))\n",
+    "\n",
+    "print \"Standard Deviation\"\n",
+    "plot_mean_dev(Z,mean(Z),stddev(Z))\n",
+    "\n",
+    "print \"Mean Absolute Deviation\"\n",
+    "plot_mean_dev(Z,mean(Z),mad(Z))\n",
+    "\n",
+    "print \"Median IQR\"\n",
+    "plot_mean_dev(Z,median(Z),IQR(Z)/2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 86,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Maximal deviation\n",
+      "center    =  1316.30863471\n",
+      "deviation =  738.072957089\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x121671590>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Standard Deviation\n",
+      "center    =  1316.30863471\n",
+      "deviation =  374.572363954\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x12178dfd0>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Mean Absolute Deviation\n",
+      "center    =  1316.30863471\n",
+      "deviation =  327.4656915\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x1214a8b90>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Mean Median IQR\n",
+      "center    =  1399.66320801\n",
+      "deviation =  452.709228516\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x1214a8050>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "X = np.loadtxt(\"DataSets/RequestRates.csv\", delimiter=\",\")[:,1]\n",
+    "    \n",
+    "print \"Maximal deviation\"\n",
+    "plot_mean_dev(X,mean(X),max_dev(X))\n",
+    "\n",
+    "print \"Standard Deviation\"\n",
+    "plot_mean_dev(X,mean(X),stddev(X))\n",
+    "\n",
+    "print \"Mean Absolute Deviation\"\n",
+    "plot_mean_dev(X,mean(X),mad(X))\n",
+    "\n",
+    "print \"Mean Median IQR\"\n",
+    "plot_mean_dev(X,median(X),IQR(X)/2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 87,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Maximal deviation\n",
+      "center    =  47.9512\n",
+      "deviation =  1175.0488\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x11fadb610>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Standard Deviation\n",
+      "center    =  47.9512\n",
+      "deviation =  92.4618512607\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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mfjAi3lhpnn2eT/Hw14XP/+ZsBmYiYiPFr0LvBcYr7eahN+4Gbs/MY8CBiLgP+OFKu3lo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\nf/Si1FPhnTCKAAAAAElFTkSuQmCC\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x11f420a10>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Mean Absolute Deviation\n",
+      "center    =  47.9512\n",
+      "deviation =  34.9059744\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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XzF+j/Esffy8/R/szTfv3g50r08KKiKspjuhckplPtWc/hPkbZCOZ+XBmntO+1uqngS9l\n5kZqzN2C3Y2N4mKxNRGxuz29bgH7VnWbgNOBmyLi14EWxTmUt7YvDHsMuDszWxHxUYoPaCMUF3V+\nbbGC1kn9MnCHuRt87bvMXBgRD1HkZQPwJLDN/DXCR4CPR8RnKe6m9z7gEcxfk1T+exkRHwO2R8SD\nFHf+eseiRa3pU6FuAZ4CdkREC/inzPwN8zfQWidryMyv1JW7kVbrpP1IkiRJUmN5gwBJkiRJQ8li\nR5IkSdJQstiRJEmSNJQsdiRJkiQNJYsdSZIkSUPJYkeSJEnSULLYkSRJkjSU/h+6m08Z5NfWiQAA\nAABJRU5ErkJggg==\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x122b3ad90>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Mean Median IQR\n",
+      "center    =  28.2\n",
+      "deviation =  9.5\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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FnV9brqB1Ur8K3GLuhl/5LTPnRcR9FHnZAjwK7DR/jfAh4KMR8WmKb9N7N/AA5q9Jav+9\njIiPALsi4l6Kb/56y7JFrfapUDcAjwG7I6IF/HNm/pb5G2qtkw1k5pf7lbuxVuuk65EkSZKkxvIL\nAiRJkiSNJIsdSZIkSSPJYkeSJEnSSLLYkSRJkjSSLHYkSZIkjSSLHUmSJEkjyWJHkiRJ0kj6f0fW\nNF3dyy++AAAAAElFTkSuQmCC\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x11ec05e90>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "print \"Maximal deviation\"\n",
+    "plot_mean_dev(Y,mean(Y),max_dev(Y))\n",
+    "\n",
+    "print \"Standard Deviation\"\n",
+    "plot_mean_dev(Y,mean(Y),stddev(Y))\n",
+    "\n",
+    "print \"Mean Absolute Deviation\"\n",
+    "plot_mean_dev(Y,mean(Y),mad(Y))\n",
+    "\n",
+    "print \"Mean Median IQR\"\n",
+    "plot_mean_dev(Y,median(Y),IQR(Y)/2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "collapsed": true,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Caution with Standard Deviation\n",
+    "\n",
+    "- Everybody Learns about standard deviation in school\n",
+    "- Beautiful mathematical properties!\n",
+    "- Everybody knows \n",
+    "  - \"68% of data falls within 1 std-dev of the mean\"\n",
+    "  - \"95% falls within 2 std-dev of the mean\"\n",
+    "  - \"99.7\" falls within 3 std-dev of the mean\"\n",
+    "* \"Problem is: this is utter nonsense\". Only true for normally distributed data.\n",
+    "\n",
+    "* Not good for measuring outliers!\n",
+    "\n",
+    "\n",
+    "_Source:_ Janert - Data Analysis with Open Source Tools"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## War Story:\n",
+    "\n",
+    "- Looking at SLA for DB response times\n",
+    "- Outlier defined as value larger than $\\mu + 3\\sigma$\n",
+    "- Look at code: Takes '0.3' percentile!\n",
+    "- So always have outliers.\n",
+    "- And 0.3-percentile was way too large (hours of latency).\n",
+    "- Programmer changed code for 1%, 5%, 10% quantiles.\n",
+    "- Finally handcoded a threshold\n",
+    "- The SLA was never changed\n",
+    "\n",
+    "Source: Janert - Data Analysis with Open Source Tools"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Tukey's outlier Definition\n",
+    "\n",
+    "A k-outlier is a data point X which is either\n",
+    "\n",
+    "* larger than `P(75) + k * IQR(X)` or\n",
+    "* smaller than `P(25) - k * IQR(X)`.\n",
+    "\n",
+    "An outlier (without k) is an 1.5-outlier."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 34,
+   "metadata": {
+    "collapsed": true,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "def outliers(X, k=1.5):\n",
+    "    iqr = IQR(X)\n",
+    "    lower_bound = percentile(25,X) - k * iqr\n",
+    "    upper_bound = percentile(75,X) + k * iqr\n",
+    "    out = []\n",
+    "    for x in X:\n",
+    "        if x < lower_bound: out.append(x)\n",
+    "        if x > upper_bound: out.append(x)\n",
+    "    return out"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 35,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "skip"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "def outlier_plot(X):\n",
+    "    plt.figure(figsize=(14,2))\n",
+    "    sns.rugplot(X, color=\"green\", height=0.5)\n",
+    "    a,b = percentile(25,X), percentile(75,X)\n",
+    "    O = outliers(X)\n",
+    "    print \"IQR:\", b-a\n",
+    "    print \"Outlier count:\", len(O)\n",
+    "    mark_range(a,b)\n",
+    "    for o in O:\n",
+    "        plt.plot([o,o],[0.7,0.8],'b--')\n",
+    "    plt.ylim([0,1])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 57,
+   "metadata": {
+    "collapsed": false,
+    "scrolled": true,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "IQR: 1.38359374912\n",
+      "Outlier count: 0\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x11fa3c990>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "# Normal Data\n",
+    "outlier_plot(Z)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 58,
+   "metadata": {
+    "collapsed": false,
+    "scrolled": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "IQR: 725.891693115\n",
+      "Outlier count: 0\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x113e86250>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "# Web request Rates\n",
+    "outlier_plot(X)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 59,
+   "metadata": {
+    "collapsed": false,
+    "scrolled": true,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "IQR: 11.2\n",
+      "Outlier count: 22\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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Cfwb8Kq0PzR1HaOVykufm8wlaf+A1TyLiHcDXMvPvOJmz7veYuRtvr6D1DLMfp/XN1Wcw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\nix1JkiRJtfT/J7MLTc+mjJ4AAAAASUVORK5CYII=\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x1202d9210>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "# Request Latencies\n",
+    "outlier_plot(Y)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Tukey's Boxplots\n",
+    "\n",
+    "Show:\n",
+    "\n",
+    "* Median\n",
+    "* Box around 0.25 and 0.75 Quantiles\n",
+    "* \"whiskers\" from min to max\n",
+    "* points for outliers\n",
+    "\n",
+    "Allows visual clues:\n",
+    "\n",
+    "* Where is the data concentracted?\n",
+    "* How far is it spread?\n",
+    "* How skew is the data?"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 63,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA1sAAABWCAYAAAA5SoxGAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nAAAASUVORK5CYII=\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x1216f1610>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "# Normal Data\n",
+    "plt.figure(figsize=(15,1))\n",
+    "sns.boxplot(x=Z, vert=False)\n",
+    "sns.rugplot(Z)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 61,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA2cAAABWCAYAAABLhe7wAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAChtJREFUeJzt3XGMpGddB/Dv7l4P7o4rHHKakGowkjwEEQVEAoWjGokI\nSQ2JAVIbTQw1aSoQCaf1DKREKQSSCsVIDIRYpdEeJiVFI4RgkGtM4NQm2mKf0gTQxATPu4W225a7\n7g1/vHPpdJm7ndmZ2fe5vc/nn9159533/c373fd9n9++78wuDQaDAAAA0K/lvgsAAABAcwYAANAE\nzRkAAEADNGcAAAAN0JwBAAA0QHMGAADQgF2zLuCJJ9YHq6uPzqMWFuDAgb2RT9tk1D4ZtU0+7ZNR\n2+TTPhm17+DB/UvzWM7MV8527VqZRx0siHzaJ6P2yaht8mmfjNomn/bJ6NLhtkYAAIAGaM4AAAAa\noDkDAABowMwfCAJ9ufnmm7K6eqrvMma2srKc9fWzfZfBBcioXWtra1leXsqePXv7LoUL2Oo+dODA\ns3PkyE3zLwigUZozLlqrq6dy8uTJLF22p+9SgJ4MzjyeJHnsTM+FMHeDM4/1XQLAttOccVFbumxP\nnvH8q/suA+jJIw/elSSOAzvQuWwBLiXecwYAANAAzRkAAEADNGcAAAAN0JwBAAA0QHMGAADQAM0Z\nAABAAzRnAAAADdCcAQAANEBzNqOjR2/P0aO3910GAADMzNi2X5qzGR0//tUcP/7VvssAAICZGdv2\nS3MGAADQAM0ZAABAAzRnAAAADdCcAQAANEBzBgAA0ADNGQAAQAM0ZwAAAA3QnAEAADRAcwYAANCA\nXX0XcLFbW1vL6dPfz+HD7+i7lLFWVpazvn627zIWYnX1VAb+vgCwIw3WT2d19fFmz687yU4eK+wU\n25nR6uqp7N79tG1ZFz/MyBYAAKABrpzNaN++fdm3b18+/OFb+y5lrIMH9+fEiYf7LmMhDh9+R049\n9GjfZQCwAEsru3Pg8r3Nnl93kp08VtgptjMjV6v75coZAABAAzRnAAAADdCcAQAANEBzBgAA0ADN\nGQAAQAM0ZwAAAA3QnAEAADRAcwYAANAAzRkAAEADdvVdwMXu5S9/Rd8lAADAXBjb9ktzNqM3v/k3\n+i4BAADmwti2X25rBAAAaIDmDAAAoAGaMwAAgAZozgAAABqgOQMAAGiA5gwAAKABmjMAAIAGaM4A\nAAAa4J9Qc1EbnHksjzx4V99lAD0ZnHksSRwHdqAu2719lwGwrTRnXLQOHHh23yXMxcrKctbXz/Zd\nBhcgo3atrQ2yvLyUPXsM4lu2tX1o7445zgNMSnPGRevIkZv6LmEuDh7cnxMnHu67DC5ARm2TT/tk\nBDAZ7zkDAABogOZsDg4d2vuUr5PMO+2yRx9fccUzxs43btkvetHm80xb27j1T/P8cfNu9bnTbs9Z\n1rUdZl3fdtc7rXH1ne/3aZLnznP+1s379Wx1eed73oWOL4vKYvR4OG4d81rvZsfOcevcbP7zzTfN\n40n2nWmPtdud4bhlT3sunTabSaeNbt9JapzXtp7059tp1vPuVtbTklbrmtU8xoTMn9sa5+D++1ee\n8nWSeadd9mbPP9/0++6bfN2T1nb69NJMzx8371afO+32nGVd22HW9W13vdMaV9/5fp8mee4852/d\nvF/PVpc37TFolnVNU8u4dcxrvVs5Zkx6vN3sWHihx5PsO9Mea7eS76y2cj7Yynlg0t+Rc9NGt+8k\nNc5rW0/68+0063l3K+tpSat1zaqP4zabc+UMAACgAUuDwaDvGgAAAC55rpwBAAA0QHMGAADQAM0Z\nAABAAzRnAAAADdCcAQAANEBzBgAA0ICJ/gl1KeXGJFcnuSzJnyf5SpK/THI2yb211huG812X5HeS\nnEny/lrrPyygZkaUUnYluS3J85I8keS6JOuRTxNKKa9I8sFa6y+WUn4qE+ZSSnl6kk8n+dEkDyX5\nrVrryT5ew062IZ+fS3Jruv3o+0l+s9Z6Qj79Gs1oZNo1SX631vqq4WMZ9WjDfnQwySeSPCvJSrr9\n6Jsy6s+Y49zH0+XwQK31bcN55NOD4RjuU+nGcLuTvD/J12Os0IzzZPTfST6WBY0XNr1yVkp5bZJX\nDk+CVyX5iSS3JDlSa31tkuVSyq+VUn4syduTvDLJ65N8oJRy2XSbgC14Q5KVWuuVSf44yc2RTxNK\nKYfTDVKeNpw0TS7XJ/mPWuuhJH+d5D3b/gJ2uDH5fCTJDbXWX0pyZ5I/kE+/xmSUUspLkvz2yGMZ\n9WhMRh9K8ula61XptvkLZNSfMfm8N8lNw23+9FLKG+XTq2uT/P9wG78+yZ/FWKE14zJa6Hhhktsa\nfyXJvaWUzya5K8nfJ3lprfXY8Of/mOR1SX4hyd211idqrQ8l+UaSF0/2upnBA0l2lVKWkjwzXbcu\nnzY8mORNI49fNmEuP5vk1Uk+PzLvL29PyZeUjfm8pdb6n8PvdyV5PPLp21MyKqX8SJI/SfLOkXlk\n1K+N+9GVSa4opXwxyTVJvhwZ9WljPvckec5wzLA/3ZhBPv05micH6yvprsRMOoaT0fbYmNGZLHi8\nMElz9pwkL0vy6+m6v9s3PO/hJJen28m/NzL9kXTNAov1SJKfTHJ/kr9Id1vW0sjP5dOTWuud6Q60\n50yTy+j0c/MyRxvzqbV+J0lKKa9KckOSP0233eXTk9GMSinLST6Z5F1J1kZmk1GPxhznnpfkVK31\ndUn+J8mNkVFvxuTzjXTjhPvS3Wb15cinN7XWR2uta6WU/Uk+k+SPYqzQlHEZ1Vr/L1nceGGS5uxk\nki8MO8EH0nWHo4P6/Um+m+4+ysvHTGexfi/J52utJV2H/lfp7ok9Rz7tODvy/YVyWR1O379hXhas\nlPKWdO+rfcPwnnD5tOOlSZ6f7v0yf5PkhaWUWyKj1pxM8rnh959L8vPpBiYyasNHk1xZa31hulus\nbol8elVK+fEk/5Tktlrr38ZYoTkbMrpjOG1h44VJmrO70907mVLKc5PsS/Kl4XvRkuRXkxxLcjzJ\nq0spu0spz0zygiT3TrB8ZnMqT3bk3013efUe+TTp30sph4bfb5bLv6R7P2GGX49tXBjzVUq5Nt1f\nwK6qtX57OPlrkU8Llmqt/1pr/ZnhPf5vTfL1Wuu7IqPWHMuT2/1Quiwc59pxMt1f75Pkf9N9cIt8\nejJ8n9IXkvx+rfW24eR7jBXaMS6jRY8XNv20xuEnjbymlPK1dJdar0/yrSSfHL7R7b+S/F2tdVBK\nuTVdM7eU7s2Mpyd76czgI0k+VUr5SrpP07wxyb9FPi16d5JPTJJLKeXjSW4rpRxL90lA1/RW9SVg\neMvcR5N8O8mdpZRBkn+utb5PPk0YnO8HtdbvyKgp7053/rk+3R8Or6m1fk9GzbguyR2llDNJTie5\nzj7Uqz9M1yC/p5Ty3nTHuncm+ZixQjM2ZrSS5KezwPHC0mBw3nMeAAAA28Q/oQYAAGiA5gwAAKAB\nmjMAAIAGaM4AAAAaoDkDAABogOYMAACgAZozAACABmjOAAAAGvADkpO23xBxJW4AAAAASUVORK5C\nYII=\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x11fa0b410>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "# Tukey Boxplots show outliers as 'points' outside the whiskers\n",
+    "\n",
+    "# Web-request dataset\n",
+    "plt.figure(figsize=(15,1))\n",
+    "sns.boxplot(x=X)\n",
+    "sns.rugplot(X)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 62,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA2EAAABWCAYAAABGm563AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x11fafb790>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "# Latency Dataset\n",
+    "plt.figure(figsize=(15,1))\n",
+    "sns.boxplot(x=Y, vert=False)\n",
+    "sns.rugplot(Y)\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "Demo:\n",
+    "- Global duration percentiles with Overlays: [demo](https://demo-hh2.circonus.com/trending/graphs/view/9e85beb8-4658-49f5-95e8-b6cf70ad5ca9#set-YVlpVTrN)\n",
+    "- Latency percentiles in CAQL [demo](https://demo-hh2.circonus.com/trending/graphs/edit/da4ac22a-55b9-6145-abbc-a0371919e876)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Outlier Scores\n",
+    "\n",
+    "The intuitive meaning of outlier is a value in a dataset, that is \"unlikely\" to be generated by the same process as the rest of the distribution.\n",
+    "\n",
+    "A simple outlier score:\n",
+    "\n",
+    "* Pick a summary statistic $M$: E.g. mean\n",
+    "* Pick a deviation measure $D$: E.g. MAD\n",
+    "\n",
+    "The outlier score of a value $x$ in context $X$ is defined as:\n",
+    "\n",
+    "$$OutlierScore(X,y) = \\frac{M(X) - y}{S(X)}$$\n",
+    "\n",
+    "It measures how many deviations a sample is away from the mean.\n",
+    "\n",
+    "* Good for well behaved datasets with a single mode\n",
+    "* Problematic for skew distributions\n",
+    "* Easy to use in practice"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "Demo:\n",
+    "- Outlier Detection in Time Series https://demo-hh2.circonus.com/trending/graphs/view/c7602466-69f4-e6b3-ceae-95049da417c4\n",
+    "- Time range: August 30th - September 7th 2015"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Deviations Measures in Monitoring\n",
+    "\n",
+    "- It's useful to enrich line graphs by deviation measures (e.g. percentile aggregation)\n",
+    "- Jitter (demo?)\n",
+    "- Outlier detection in generated revenue [demo](https://demo-hh2.circonus.com/trending/graphs/edit/c7602466-69f4-e6b3-ceae-95049da417c4)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Takeaway\n",
+    "\n",
+    "* Don't use the standard-deviation if you can avoid it\n",
+    "* If you need a deviation measure use MAD or IQR:\n",
+    "  - (+) Simple to calculate\n",
+    "  - (+) Simple to interpret\n",
+    "  - (-) Not mergable"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# (3) Measuring Distributions"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "* Classical summary statistics good for describing the body of the distribution\n",
+    "* Need information about the tail of the distributions, e.g. for writing good SLAs\n",
+    "* Determine outliers in a dataset"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Inverse Percentiles\n",
+    "\n",
+    "The inverse percentile y of a dataset $X$, is defined as:\n",
+    "\n",
+    "$$ IP(y, X) = \\text{(the percentage of samples in X that are lower than $y$)} = \\# \\{ i \\, | \\, x_i \\leq y \\} / \\# X \\cdot 100 $$\n",
+    "\n",
+    "**Properties**\n",
+    "\n",
+    "* $0 \\leq IP(y, X) \\leq 100$, dimensionless percentage value\n",
+    "* $IP(y,X)$ is monotonically increasing in $y$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 37,
+   "metadata": {
+    "collapsed": true,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "def IP(y,X):\n",
+    "    lower_count = 0\n",
+    "    for x in X:\n",
+    "        if x <= y: lower_count += 1\n",
+    "    return float(lower_count) / len(X)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 38,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "<matplotlib.axes._subplots.AxesSubplot at 0x11f4ad8d0>"
+      ]
+     },
+     "execution_count": 38,
+     "metadata": {},
+     "output_type": "execute_result"
+    },
+    {
+     "data": {
+      "image/png": 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ccdcmjtS38u5LF7DmfC9ILUmSNJxZFEmSpFOyt6arJKpraOXmyxZy1bmzk44k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\nACeKAAAAAKhEEQAAAAA3j8t8Vz2R9aEAAAAASUVORK5CYII=\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x11f4ad750>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "# Example\n",
+    "import matplotlib as mpl\n",
+    "mpl.rcParams['figure.figsize'] = (15,5)\n",
+    "X = np.loadtxt(\"DataSets/RequestRates.csv\", delimiter=\",\")[:,1]\n",
+    "\n",
+    "grid = np.linspace(0,3000,100)\n",
+    "plt.figure(figsize=(20,5))\n",
+    "plt.plot(grid, [IP(y,X) for y in grid])\n",
+    "plt.title(\"Inverse Percentiles\")\n",
+    "sns.rugplot(X)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Inverse Percentiles in Monitoring\n",
+    "\n",
+    "* Inverse Percentiles to Monitor SLAs (overlay)\n",
+    "* count_lower can be used to compute the number of offended users (slides)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Percentiles\n",
+    "\n",
+    "* Complement or Inverse to Inverse Percentiles:\n",
+    "  - IP: The ratio of samples was below a value $y$ was $?\\%$\n",
+    "  - Percentile: 90% of all queries where faster than $?$\n",
+    "\n",
+    "* The definition is a little involved so cover examples first\n",
+    "\n",
+    "Examples:\n",
+    "\n",
+    "* The minimum is a 0%-percentile\n",
+    "* A median is a 50%-percentile\n",
+    "* The maximum is a 100%-percentile\n",
+    "* Quartiles: $0,25,50,75,100$-percentiles\n",
+    "* Quantiles: The $r$-quantile is the $r*100$-percentile $0<r<1$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# General Definition of Percentiles\n",
+    "\n",
+    "Pick a number $0\\leq p \\leq 100$. A $p$%-percentile for $X$ is a value $y$ such that,\n",
+    "\n",
+    "$$ \\#\\{i \\,|\\, X[i] \\lt y \\} \\leq \\frac{p}{100} \\cdot n $$\n",
+    "\n",
+    "and\n",
+    "\n",
+    "$$ \\#\\{i \\,|\\, X[i] \\gt y \\} \\leq \\frac{100-p}{100} \\cdot n $$\n",
+    "\n",
+    "Roughly speaking, $y$ divides $X$ in $p\\%$ samples that are lower than $y$ and $(100-p)\\%$ samples that are larger than $y$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Example:\n",
+    "<figure>\n",
+    "<img src=\"img/latency_percentile.png\" style=\"width:500px\">\n",
+    "</figure>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "Remarks:\n",
+    "\n",
+    "* Quantiles always exists\n",
+    "* Non unique (like median)\n",
+    "* Lot's of ways to choose a quantile function, i.e. interpolate between $s_a$ and $s_b$ cf.  \n",
+    "  http://en.wikipedia.org/wiki/Quantile#Estimating_the_quantiles_of_a_population"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 88,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Computation of Quantiles\n",
+    "def percentile_range(p,X):\n",
+    "    q = p/100.0\n",
+    "    S = sorted(X)\n",
+    "    n = len(X)\n",
+    "    upper_d = int(math.ceil(q * (n + 1)))\n",
+    "    lower_d = int(math.floor(q * (n + 1)))\n",
+    "    # edge cases\n",
+    "    if upper_d == 0:   upper_d = 1\n",
+    "    if upper_d == n+1: upper_d = n\n",
+    "    if lower_d == 0:   lower_d = 1\n",
+    "    if lower_d == n+1: lower_d = n\n",
+    "    return (S[lower_d-1], S[upper_d-1])\n",
+    "\n",
+    "def percentile(p,X):\n",
+    "    a,b = percentile_range(p,X)\n",
+    "    return (a+b)/2 # mid point percentile. One choice!"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 89,
+   "metadata": {
+    "collapsed": true,
+    "slideshow": {
+     "slide_type": "skip"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "def mark_range(a,b):\n",
+    "    plt.plot([a,a],[0.5,1],'r-')\n",
+    "    plt.plot([b,b],[0.5,1],'r-')\n",
+    "    plt.plot([a,b],[0.75,0.75],'r--')\n",
+    "\n",
+    "def plot_percentile_range(p,X):\n",
+    "    plt.figure(figsize=(14,2))\n",
+    "    sns.rugplot(X, color=\"green\", height=0.5)\n",
+    "    a,b = percentile_range(p,X)\n",
+    "    mark_range(a,b)\n",
+    "    plt.title(\"{:n}%-Percentile Range: {:.3f} .. {:,.3f}\".format(p, a,b))\n",
+    "    plt.ylim([0,1])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 90,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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SpFow3EiSJEmqBcONJEmSpFqY1m6DiOgDTgS2AB4A9snMG4fY7mTgzsz82JhXKUmSJElt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\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x121c84bd0>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "plot_percentile_range(50,[1,2,3,4])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 91,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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njXt1kiRJktSlbsLNDGBZy/KqiFgHICKeCZwAvAPoG//yJEmSJKk7HZ+5AZYD/S3L62Tm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\neIcAAAAASUVORK5CYII=\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x11f120b10>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "plot_percentile_range(40,[1,2,3,4])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 92,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x11ec13610>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "plot_percentile_range(90,X)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# (4) Histograms\n",
+    "\n",
+    "* Basic Idea: Use area to compare sample ratios, instead of point counts\n",
+    "* Easier to answer questios like: \"How many voters are older than 30, when presented with a histogram, than a table or a rugplot.\"\n",
+    "* Both a visualization method and a data-storage format"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Building your own Histogram\n",
+    "\n",
+    "1. Define value range of histogram `bin_min`, `bin_max`\n",
+    "2. Divide range into bins `bin[k], ..., bin[k]`\n",
+    "3. The bins may have variying width \n",
+    "4. Count the number of samples in each bin `sample_count[k]`\n",
+    "5. Draw a rectangle for each bin with height:\n",
+    "   - `h[k] = sample_count[k]` -- non normalized \n",
+    "   - `h[k] = sample_count[k] / bin_width[k]` -- normalized case"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 69,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Bins:  [[500, 700], [700, 800], [800, 900], [900, 1000], [1000, 1500], [1500, 1800], [1800, 2000], [2000, 2200]]\n"
+     ]
+    }
+   ],
+   "source": [
+    "import numpy as np\n",
+    "X = np.loadtxt(\"DataSets/RequestRates.csv\", delimiter=\",\")[:,1]\n",
+    "\n",
+    "# Step 1: Represent bins via list of left-edges\n",
+    "bin_left = [\n",
+    "    500, 700, 800, 900, 1000, 1500, 1800, 2000, 2200\n",
+    "]\n",
+    "\n",
+    "bin_min = bin_left[0]\n",
+    "bin_max = bin_left[-1]\n",
+    "bins = [ [bin_left[i-1],bin_left[i]] for i in xrange(1, len(bin_left)) ]\n",
+    "\n",
+    "print \"Bins: \", bins"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 70,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Counts: [0, 10, 8, 4, 25, 23, 4, 2]\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Step2: Count number of samples in bins\n",
+    "def count(X, bins):\n",
+    "    def findBin(x, bins):\n",
+    "        for i, bin in enumerate(bins):\n",
+    "            left, right = bin\n",
+    "            if left <= x and x < right:\n",
+    "                return i\n",
+    "        return None\n",
+    "    \n",
+    "    count = [0] * len(bins)\n",
+    "    for x in X:\n",
+    "        i = findBin(x, bins)\n",
+    "        if i != None:\n",
+    "            count[i] += 1\n",
+    "\n",
+    "    return count\n",
+    "\n",
+    "sample_count = count(X, bins)\n",
+    "\n",
+    "print \"Counts:\", sample_count"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 71,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x121a9ded0>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "from matplotlib import pyplot as plt\n",
+    "import seaborn as sns\n",
+    "\n",
+    "%matplotlib inline\n",
+    "\n",
+    "# Step3: Plot bar chart\n",
+    "def my_hist(X, bins, normed=True, **kwargs):\n",
+    "    bin_left = [ l for l, r in bins ]\n",
+    "    bin_width = [ r-l  for l,r in bins ]\n",
+    "    bin_height = [ \n",
+    "        float(c) / w if normed else c \n",
+    "        for c,w in zip(count(X, bins), bin_width)\n",
+    "    ]\n",
+    "    plt.bar(bin_left,width=bin_width,height=bin_height, **kwargs)\n",
+    "\n",
+    "plt.figure(figsize=(15,5))\n",
+    "my_hist(X,bins)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 72,
+   "metadata": {
+    "collapsed": true,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "def plot_hist(X,  bin_min, bin_max, bin_width):\n",
+    "    plt.figure(figsize=(15,5))\n",
+    "    bins =[ [i, i+bin_width] for i in np.arange(bin_min, bin_max, bin_width) ]\n",
+    "    my_hist(X,bins)\n",
+    "    print \"Histogram with bin_width \", bin_width\n",
+    "    plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## How bin choices affect the apperance"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 73,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Histogram with bin_width  5\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x11f63d750>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "bin_min = 600\n",
+    "bin_max = 2200\n",
+    "plot_hist(X, bin_min, bin_max, 5)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 74,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Histogram with bin_width  50\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA3kAAAE5CAYAAAAtN+mQAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x113eaa4d0>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "plot_hist(X, bin_min, bin_max, 50)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 75,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Histogram with bin_width  2000\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x121972e90>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "plot_hist(X, bin_min, bin_max, 2000)\n",
+    "# -> Only the total count is displayed"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Common bin-width choices\n",
+    "\n",
+    "Remarks:\n",
+    "* Can convert `bin_count` and `bin_width` using relation $$ bin\\_max - bin\\_min = bin\\_count \\cdot bin\\_width.$$\n",
+    "\n",
+    "* Choices `bin_min=min(X)` and `bin_max=max(X)` are common. But: dependence on input data.\n",
+    "\n",
+    "Choices for bin-width:\n",
+    "\n",
+    "* Excel: Let $n$ be the number of samples, then:\n",
+    "\n",
+    "  $$ bin\\_count = \\sqrt{n} $$\n",
+    "\n",
+    "* Scott's normal reference rule\n",
+    "\n",
+    "  $$ bin\\_width = \\frac{3.5 \\sigma}{n^{1/3}} $$\n",
+    "\n",
+    "  where $\\sigma$ is the standard deviation (cf. later).\n",
+    "  This is optimal for random samples of normally distributed data, in a certain sense.\n",
+    "\n",
+    "* Freedman–Diaconis rule\n",
+    "\n",
+    "  $$ bin\\_width = \\frac{2 IQR}{n^{1/3}} $$\n",
+    "  \n",
+    "  where IQR is the inter-quartile range (cf. later). Like Scott's rule but more robust agains outliers.\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# A pragmatic choice: HDR Histograms\n",
+    "\n",
+    "Example: A decimal HDR Histogram with precision=2 has bin boundaries:\n",
+    "  \n",
+    "    .... 1.0,  1.1,  1.2, ... 1.9,  2.0,  2.1, ...., 9.9; --- bin width = 0.1\n",
+    "         10.0, 11.0, 12.0 ... 19.0, 20.0, 21.0 ...., 99 ; --- bin width = 1\n",
+    "         ...\n",
+    "\n",
+    "** Properties: **\n",
+    "* Captures large part of float range\n",
+    "* Bin boundaries do not depend on data! -> Can aggregate counts!\n",
+    "* Bin width increase with growing values\n",
+    "* Allows compact memory representation\n",
+    "* Implementation available at http://hdrhistogram.org/"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 76,
+   "metadata": {
+    "collapsed": true,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "import math\n",
+    "def HDR_bin_width(x):\n",
+    "    if x == 0: return 0\n",
+    "    sign = x / abs(x)\n",
+    "    return sign * 10 ** ( math.floor(math.log10(abs(x)) - 1 ) )\n",
+    "\n",
+    "def HDR_bin_left(x):\n",
+    "    w = HDR_bin_width(x)\n",
+    "    if w == 0: return 0\n",
+    "    return math.floor(x / w) * w\n",
+    "\n",
+    "def HRD_bin_right(x):\n",
+    "    return bin_left(x) + bin_width(x)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 77,
+   "metadata": {
+    "collapsed": true,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "def HDR_hist(X):\n",
+    "    count = {}\n",
+    "    for x in X:\n",
+    "        l = HDR_bin_left(x)\n",
+    "        count[l] = count.get(l,0) + 1\n",
+    "    bin_left = sorted(count.keys())\n",
+    "    bin_width = [ HDR_bin_width(l) for l in bin_left ]\n",
+    "    bin_height = [ abs(count[l] / HDR_bin_width(l))\n",
+    "        for l in bin_left\n",
+    "    ]\n",
+    "    plt.figure(figsize=(15,5))\n",
+    "    plt.bar(bin_left,width=bin_width,height=bin_height)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 78,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "<matplotlib.axes._subplots.AxesSubplot at 0x120564c90>"
+      ]
+     },
+     "execution_count": 78,
+     "metadata": {},
+     "output_type": "execute_result"
+    },
+    {
+     "data": {
+      "image/png": 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4AACARghwAAAAjRDgAAAAGiHAAQAANEKAAwAAaIQABwAA0AgBDgAAoBECHAAAQCM2DLNR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\nlld+AAAAAElFTkSuQmCC\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x1208205d0>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "Y = np.genfromtxt(\"DataSets/WebLatency.csv\", delimiter=\",\")[:,2][:200] - 85\n",
+    "HDR_hist(Y)\n",
+    "sns.rugplot(Y, color=\"red\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 79,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "<matplotlib.axes._subplots.AxesSubplot at 0x11fdeda50>"
+      ]
+     },
+     "execution_count": 79,
+     "metadata": {},
+     "output_type": "execute_result"
+    },
+    {
+     "data": {
+      "image/png": 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j2bZrz6RLGZmTRx/LbW97Uy666OJJlwLABhPyACDJtl17suP8CyZdBgCsm8/kAQAANETI\nAwAAaIiQBwAA0BAhDwAAoCFCHgAAQEOEPAAAgIa4hAIAazY/P5+DBw9MuoyROXTokUmXMHKLCwtN\nPa6WHgvAuAl5AKxZaxcPP/L5B/OSl18y6TJG6snjh3PL3Y9n265HJ13KSLTYI4BxEfIAOC0tXTz8\n5NEvTrqEsdAjgLOTz+QBAAA0RMgDAABoiJAHAADQECEPAACgIUIeAABAQ4Q8AACAhgh5AAAADRHy\nAAAAGiLkAQAANETIAwAAaIiQBwAA0BAhDwAAoCFCHgAAQEOEPAAAgIYIeQAAAA0R8gAAABoi5AEA\nADREyAMAAGiIkAcAANAQIQ8AAKAhQh4AAEBDhDwAAICGCHkAAAAN2bragFLKpiS3J7k0yVNJrq+1\nHhhY/z1J3p7k6STvr7Xe0V/+mSRH+8MerrX+8IhrBwAAYIlVQ16Sq5OcW2vdV0p5dZJb+8tSStna\nv/2qJE8m2V9K+a9JjiVJrfXKsVQNAADAUF1C3hVJ7k2SWuv9pZTLBtZdkuShWuuxJCmlfCrJa5P8\neZLtpZSPJdmS5GdrrfePtHKgWfPz8zl48MDqA88wu3dfOukSAICzQJeQd16ee9tlkjxTStlca10Y\nsu54kl1JPpvk5lrrnaWUi5N8tJSytz9nWTMzO9dWfQOeeGLH0OW7d+943v7oOm6lsePQtc6N2s6w\n/THMeuav9BhHXedatfIcmp2dzY03fyTbdu2ZdCkjc/LoY/nAO3dk7969ky5lJDbyOAOtGtffgklr\n8TG1RH/ODl1C3rEkg78NmwfC2rH0gt4pO5N8KclDST6XJLXWh0opR5K8NMkXVtrQ4cPHO5bdjrm5\nE8suH9wfXcetNHYcuta5UdsZtj+WG3e681d6jKOucy1mZnY28xyamzuRbbv2ZMf5F0y6lJFrqUfA\n+ozjb8GktfS3qEX6M/1GFcK7fLvm/iRvSJJSyuVJHhhY92CSV5ZSXlxKOSfJa5J8Osl1SW7pz3lZ\neuHv0ZFUDAAAwLK6vJJ3T5KrSin7+7evLaW8Ocn2WusdpZSfTPLbSTYlubPW+mgp5c4k7y+l3Jdk\nIcl1q71VEwAAgPVbNeTVWheT3LBk8ezA+t9K8ltL5jyd5C2jKBAAAIDuXAwdAACgIUIeAABAQ4Q8\nAACAhgh5AAAADeny7ZoAAIzQ4sJCDh16ZNJljNzu3ZdOugQgQh4AwIZ78vjh3HL349m2q53LCJ88\n+lg+8M4dOf/8l066FDjrCXkAABOwbdee7Dj/gkmXATTIZ/IAAAAaIuQBAAA0RMgDAABoiJAHAADQ\nECEPAACgIUIeAABAQ1xCAQCAdVtcWMjDDz+cubkTky5lZC688BXZsmXLpMuANRPyAABYtyePH847\n3vt4tu3aM+lSRuLk0cdy29velIsuunjSpcCaCXkAAIyEC7zDdPCZPAAAgIYIeQAAAA0R8gAAABoi\n5AEAADREyAMAAGiIkAcAANAQl1CABszPz2d2draZC9AeOvTIpEsAgKa0dq5wigvWDyfkQQMOHjyQ\nG2/+SDMXoD3y+QfzkpdfMukyAKAZrZ0rJC5YvxIhDxrR0gVoTx794qRLAIDmtHSuwMp8Jg8AAKAh\nQh4AAEBDhDwAAICGCHlT5N13/cSzP3/LD/z9Vce/756fX9P9Lx3/6f/ycy8Y86GPv2fo/S5d9p9v\n+3vrrutjd924rvnJ8/dZ0m2/LTd/rftz6bZHNXYU1ru9ja53rYbVt9LvZJf5oxw/7c5/7atHdl+n\nu2+Wm/fuu35ixfscRy/efddP5L7f+IVl739U21ztsQ3b5mrjB9cv9/Ny9zO4jbVsZ611rWXdeq2n\nzrX2puuywWPT0nHD/u6s9TGsNmaajl+ns79Hsa1pMcpj7zRZz/GA8fHFK1PkLx859OzP2w58Ll9e\nZfzXf+nRrOXrKb7+S48+7/awhH/RscOd5p4z/3Tn7SznnIVn1jU/ef4+S7rtt+Xmr3V/Lt32qMaO\nwnq3t9H1rtWw+lb6newyf5Tjp93Wzz44svs63X2z3LzV7m8cvRi8z2H3P6ptns4xYy37Y6XHsdLj\n6lJXl/vrun6cz6f11Nm1rq6/I6eWDR6blo4b9vdurY9htTHTdPw6nf09im1Ni1Eee6fJJI7brM4r\neQAAAA0R8gAAABri7ZoAG2BxYSEPP/xw5uZOTLqUzCT53OceWtd9HDr0yGiKAZhSiwsLIz3WjeLY\nux6O22cXIQ9gAzx5/HDe8d7Hs23XnkmXkk8kuem9f7Cu+zjy+QfzkpdfMpqCAKbQk8cP55a7H8+2\nXd2/K2Alozj2rofj9tlFyAPYINt27cmO8y+YdBlJsu46Th5dy9cUAZyZRn3cnuTfAMfts4vP5AEA\nADREyAMAAGiIkAcAANCQVT+TV0rZlOT2JJcmeSrJ9bXWAwPrvyfJ25M8neT9tdY7VpsDAADAeHR5\nJe/qJOfWWvcluSnJradWlFK29m9/R5LXJ/mRUsrMSnMAAAAYny4h74ok9yZJrfX+JJcNrLskyUO1\n1mO11qeT3JfkdavMAQAAYEy6XELhvCRHB24/U0rZXGtdGLLuRJJdSXauMIclTh597NmfTzzxhefd\nXm7cWu4zSZ48Pvfs/Xe5z8Fxa507fP6mTvUMW77a/jg1di37Z9j89c4dV51rremU03lco5y/HsN+\nZ5YaVt9aal7r41vv/ujymDbSKB/P6d7XSvNOd916rHRMGOU2u97XqXGrjR9cv9Kxd7XHtZbtnM74\ntcwdputzaD11rrU3XZcN28ZyfweHLetS1ySeM0ut9dg97rqm8bg97X9bT9d6jgfrMa5zqhZsWlxc\nXHFAKeWWJJ+utX64f/tQrfXr+j9/c5JfqrV+d//2rUk+lWRfkj8YNgcAAIDx6fJ2zf1J3pAkpZTL\nkzwwsO7BJK8spby4lHJOktck+XSS319hDgAAAGPS5ZW8U9+U+S39RdcmeVWS7f1v0vzuJP8svdd/\n76y1/tqwObXW2XE8AAAAAJ6zasgDAADgzOFi6AAAAA0R8gAAABoi5AEAADREyAMAAGhIl4uhj0wp\n5WeSvCnJ16T37Zu/l+TfJFlI8ie11h/rj3trkh9J8nSSX6i1/tZG1nk2KqVsTXJXkguTPJPkrUnm\noz9ToZTy6vSuSfk3SikXpWNfSikvSvLBJHuSHEvyD2utRybxGFq3pEd/Ncm70nsufSXJD9VaD+vR\n5Az2Z2DZNUn+Ua11X/+2/kzQkufQTJJfT/LiJFvSew49rEeTM+QY9570+jBba72+P0Z/JqB/Dve+\n9M7hzknyC0n+NM4VpsYyPTqU5FczpnOFDXslr5TyuiTf1v9j+vokX5fk1iT/tNb6uiSbSyl/u5Ty\nF5P8eJJvS/JdSd5ZSvmajarzLPaGJFtqrd+e5F8m+cXoz1QopbwtvZOdc/uL1tKXG5L8ca31tUk+\nkOTtG/4AzgJDevQrSX6s1nplknuS/LQeTc6Q/qSU8teSXDdwW38maEiPfjnJB2utr09vn3+DHk3O\nkP68I8k/7+/zF5VSvlt/JuotSR7v7+PvSvLuOFeYNsN6NNZzhY18u+bfTPInpZTfSPKRJP8tybfW\nWu/rr/9okquS/PUkn6q1PlNrPZbkoTx3vT3GZzbJ1v41Dnel978H+jMd/izJ9w7cflXHvlya5Iok\n9w6M/Y6NKfmss7RH319rfaD/89YkT0WPJul5/SmlvCTJzye5cWCM/kzW0ufQtyd5eSnld5Jck+ST\n0aNJWtqfP0rytf1zhp3pnTPoz+T8xzx30r8lvVeGup7D6dHGWNqjpzPmc4WNDHlfm95F1P9uemn0\n3y3Z/vEk56V3sDg6sPxEeqGD8TqR5OuTfDbJv07vrWabBtbrz4TUWu9J74B9ylr6Mrj81FhGbGmP\naq1fTJJSyr4kP5bkX6W37/VoAgb7U0rZnOSOJD+Z5MsDw/RngoYc5y5MMldrvSrJnyf5mejRxAzp\nz0PpnSf83/TePvbJ6M/E1FpP1lq/XErZmeQ/JfnZOFeYKsN6VGt9LBnfucJGhrwjST7WT6az6aXV\nwXCwM8mX0nuf6XlDljNe/zjJvbXWkt7/GPzb9N4zfIr+TI+FgZ9X6ssT/eU7l4xlA5RSvj+9zx6/\nof++eT2aDt+a5JXpfZ7o3yf5xlLKrdGfaXMkyW/2f/7NJJeld4KjR9PhtiTfXmv9xvTeOnZr9Gei\nSil/Kcn/SHJXrfU/xLnC1FnSo7v7y8Z2rrCRIe9T6b23NKWUlyXZnuS/9z+rlyR/K8l9Sf5nkitK\nKeeUUnYl+YYkf7KBdZ6t5vLc/xB8Kb2Xjf9If6bS/y6lvLb/82p9+f30Pm+Z/r/3Lb0zRq+U8pb0\n/lfu9bXWR/qL/zB6NGmbaq3/q9b6zf3PQPyDJH9aa/3J6M+0uS/P7ffXptcLx7npcSS9VxOS5P+l\n9wU5+jMh/c9xfSzJP6m13tVf/EfOFabHsB6N+1xhw75ds//NMK8ppfxhei8h35DkYJI7+h8ofDDJ\nh2uti6WUd6UXCjel96HRr25UnWexX0nyvlLK76X37ac/k+Qz0Z9p9FNJfr1LX0op70lyVynlvvS+\nuemaiVV9lui/HfC2JI8kuaeUspjkd2ut/0KPJm5xuRW11i/qz1T5qfT+/tyQ3n9AXlNrPapHU+Ot\nSe4upTyd5KtJ3uo5NFE3pRe0315KeUd6x7obk/yqc4WpsbRHW5L8lYzxXGHT4uKyf/MAAAA4w7gY\nOgAAQEOEPAAAgIYIeQAAAA0R8gAAABoi5AEAADREyAMAAGiIkAcAANCQ/w9A9LRflgYINQAAAABJ\nRU5ErkJggg==\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x11f47de90>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "HDR_hist(X)\n",
+    "sns.rugplot(X, color=\"red\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 80,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(0, 200)"
+      ]
+     },
+     "execution_count": 80,
+     "metadata": {},
+     "output_type": "execute_result"
+    },
+    {
+     "data": {
+      "image/png": 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bue3emzN0/5fyoicfzyNnX5gkuWD/nrzxt+7InR99U/acfWF+41c+ni999E15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\nAAAUTnADAAAonOAGAABQOMENAACgcP8/ebvUJrTcY/IAAAAASUVORK5CYII=\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x113ea6350>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "# Need re-binning near zero:\n",
+    "HDR_hist(Z)\n",
+    "sns.rugplot(Z, color=\"red\")\n",
+    "plt.ylim(0,200)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# (5) Mergability\n",
+    "\n",
+    "A summary statistic $S$ assigns (one or many) values to a dataset $X$. (E.g. mean value.)\n",
+    "\n",
+    "Let $X$ and $Y$ be two datasets of size $len(X)=n$ and $len(Y)=m$.\n",
+    "Denote the merged dataset by \n",
+    "\n",
+    "$$X .. Y = [X[1],...,X[n], Y[1],...,Y[m]].$$\n",
+    "\n",
+    "A summary statistic is _mergable_ if there is an algorithm that computes $S(X..Y)$ from $S(X)$ and $S(Y)$.\n",
+    "\n",
+    "Example:\n",
+    "- The sum is mergable: $sum(X..Y) = sum(X)+sum(Y)$.\n",
+    "\n",
+    "References: [Agrawal, et. al. - Mergeable Summaries (pdf)](https://www.cs.utah.edu/~jeffp/papers/merge-summ.pdf)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "Exerciese:\n",
+    "- Show that the maximum is mergable\n",
+    "- Show that the median value is not mergable"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Mergability in Monitoring: Rollups\n",
+    "\n",
+    "It's a common practice in monitoring to rolled-up data to different time periods.\n",
+    "\n",
+    "Example:\n",
+    "- raw data arrives every second at monitoring agent\n",
+    "- the raw data is rolled-up into 1-minute aggregates at the agent and sumbitted downstream \n",
+    "- the 1-minute data is then further rolled-up into 5-minute aggregates\n",
+    "- the 5-minute aggregates are rolled-up into 1h aggregates\n",
+    "\n",
+    "There are two resons for that practice:\n",
+    "- Graphing Performance. Large time range graphs can be constructed from the stored aggregates.\n",
+    "- Storage costs. Many tools delete the one-minute data eventually and only the aggregates for historic data.\n",
+    "\n",
+    "For this proceedure to make sense you need either:\n",
+    "- Store all your raw data\n",
+    "- Use mergable summaries"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Mergability in Monitoring: Service Level Metrics\n",
+    "\n",
+    "When monitoring distributed systems, it's common to have many nodes being dedicated to a single service.\n",
+    "Service level metrics, that measure properties of the whole system are of great interested for operations and accounting.\n",
+    "\n",
+    "Examples of service level metrics include:\n",
+    "- Total requests served\n",
+    "- Latency Percentiles\n",
+    "- Average peer lags\n",
+    "\n",
+    "To collect those service level metrics we need to either:\n",
+    "\n",
+    "- Stream raw data to a central location and compute the metric there (e.g. statsd, logstash) -> Slow, expensive\n",
+    "- Use __mergable summaries__ (e.g. total request counts) that can be computed locally and merged downstream"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Mean values are not mergable (on the nose)\n",
+    "\n",
+    "The summary statistic $S(X)=mean(X)$ is not mergable.\n",
+    "\n",
+    "Proof: Assume contrary that mean is mergable, then there is a computable function $f$ so that:\n",
+    "* `X=[0], Y=[1]; mean(X..Y)=0.5`, hence `f(0,1)=0.5` and\n",
+    "* `X=[0,0,0], Y=[1]; mean(X..Y)=0.2`, hence `f(0,1)=0.2`\n",
+    "\n",
+    "A contradition."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "collapsed": true,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# The mean values can be enriched to a mergable summary\n",
+    "\n",
+    "With a little trick, we can pimp up the mean statistic to something mergable.\n",
+    "\n",
+    "For two datasets $X,Y$ of length $n,m$, we can compute the mean value of the merged dataset as follows:\n",
+    "\n",
+    "$$mean(X..Y) = \\frac{n}{n+m} \\cdot mean(X) + \\frac{m}{n+m} \\cdot mean(Y)$$\n",
+    "\n",
+    "This shows, that the summary statistic $S(X)=(len(X),mean(X))$ is mergable.\n",
+    "\n",
+    "Remarks:\n",
+    "\n",
+    "* Expressions of the form $\\sum w_i X[i]$, with $\\sum w_i = 1$ are called _weighted means_.\n",
+    "\n",
+    "* Exercise: How do you aggregate a greater number of sample sets: $X_1, ..., X_k$ ?"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 81,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "RHS 5.4\n",
+      "LHS 5.4\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Test\n",
+    "print \"RHS\", mean([1,2,3,10,11])\n",
+    "print \"LHS\", (3*mean([1,2,3]) + 2*mean([10,11]))/5"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Median values are not mergable!\n",
+    "\n",
+    "Neither the weighted mean nor the median of medians are the median of the total sample set."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Mergability of Deviation Measures\n",
+    "\n",
+    "* The standard deviation is mergable.  \n",
+    "  More precisely the statistic $S(X)=(len(X),mean(X),stddev(X))$ is mergable.\n",
+    "  \n",
+    "  Hint: Reduce to the case that `T(X)=(len(X),sum(X),sum([x^2 for x in X]))` is mergable.\n",
+    "\n",
+    "* Neither max-deviation nor MAD are mergable on the nose."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
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+   "source": [
+    "# Histograms: The Ulitmate Mergable Summary\n",
+    "\n",
+    "* HDR Histograms as Summary Statistic:  \n",
+    "  The value on a dataset $X$ is the sequence of bin,count pairs:  \n",
+    "  `H(X) = [(bin_1, count_1), ..., (bin_n, count_n)]`\n",
+    "  \n",
+    "  E.g. `H([1,2,101,102]) = [(1,1),(2,1),(100,2)]`\n",
+    "  \n",
+    "* Hisotgrams are mergable: Just add the counts.\n",
+    "\n",
+    "* All summary statistics and deviations measures can be approximated from Histograms:  \n",
+    "  - mean, standard deviation\n",
+    "  - min, max\n",
+    "  - percentiles, median\n",
+    "  \n",
+    "Example: [demo](https://demo-hh2.circonus.com/trending/graphs/edit/b4b3bce1-ce5b-418f-a8eb-adfa369a6bed)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
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+     "slide_type": "slide"
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+   },
+   "source": [
+    "# Overview Monitoring Statistics\n",
+    "\n",
+    "```\n",
+    "| Type         | Statistics     | Robust | Mergable |\n",
+    "|--------------|----------------|--------|----------|\n",
+    "| Central      | mean           | no     | yes      |\n",
+    "|              | median         | yes    | no       |\n",
+    "|              | truncated mean | yes    | no       |\n",
+    "|--------------|----------------|--------|----------|\n",
+    "| Deviation    | stddev         | no     | yes      |\n",
+    "|              | mad            | ~      | no       |\n",
+    "|              | IQR            | yes    | no       |\n",
+    "|--------------|----------------|--------|----------|\n",
+    "| Distribution | Percentile     | yes    | no       |\n",
+    "|              | Histogram      | yes    | yes      |\n",
+    "```"
+   ]
+  }
+ ],
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