diff --git a/1. NumPy.ipynb b/1. NumPy.ipynb
index d157876..a2ca585 100644
--- a/1. NumPy.ipynb	
+++ b/1. NumPy.ipynb	
@@ -4,7 +4,10 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Initialization"
+    "\n",
+    "### Initialization\n",
+    "\n",
+    "Numpy Array Creation Routines"
    ]
   },
   {
@@ -18,7 +21,45 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "x = np.empty([3,3])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([[0.00000000e+000, 0.00000000e+000, 6.94582274e-310],\n",
+       "       [6.94582971e-310, 6.94582969e-310, 6.94582963e-310],\n",
+       "       [6.94582965e-310, 6.94582971e-310, 3.95252517e-322]])"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "x"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "An empty array initializes a numpy array with random values"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -27,7 +68,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 5,
    "metadata": {},
    "outputs": [
     {
@@ -46,7 +87,7 @@
        "        [0., 0., 0.]]])"
       ]
      },
-     "execution_count": 6,
+     "execution_count": 5,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -57,7 +98,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 10,
+   "execution_count": 6,
    "metadata": {},
    "outputs": [
     {
@@ -70,7 +111,7 @@
        "        [1, 1, 1, 1]]])"
       ]
      },
-     "execution_count": 10,
+     "execution_count": 6,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -79,9 +120,16 @@
     "np.ones((2,2,4), dtype=int)"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "By using dttype, we can set the type of values that the numpy array contains."
+   ]
+  },
   {
    "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": 7,
    "metadata": {},
    "outputs": [
     {
@@ -91,7 +139,7 @@
        "       ['hellott', 'hellott']], dtype='<U7')"
       ]
      },
-     "execution_count": 12,
+     "execution_count": 7,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -100,19 +148,26 @@
     "np.full((2,2),\"hellottwyyw\", dtype='<U7')"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Note : Only the first 7 characters of the input string were taken into the array elements when we set dtype = '<U7'"
+   ]
+  },
   {
    "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 8,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([0.12520064, 0.07562049, 0.12900815, 0.55747886, 0.11532656,\n",
-       "       0.53774122, 0.06926235, 0.29796341, 0.09515193, 0.05531125])"
+       "array([0.66726332, 0.43897927, 0.04688608, 0.93075331, 0.31688651,\n",
+       "       0.53722039, 0.33152139, 0.53403819, 0.21897812, 0.02441027])"
       ]
      },
-     "execution_count": 13,
+     "execution_count": 8,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -123,7 +178,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 15,
+   "execution_count": 9,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -134,24 +189,24 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Access of NumPy "
+    "### Accessing Numpy Arrays"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 20,
+   "execution_count": 10,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([[6, 6, 7, 7, 5, 6, 9, 9, 8, 7],\n",
-       "       [8, 5, 9, 5, 7, 8, 5, 9, 6, 8],\n",
-       "       [7, 7, 6, 6, 5, 9, 6, 6, 5, 9],\n",
-       "       [5, 6, 9, 6, 8, 9, 9, 6, 9, 8]])"
+       "array([[7, 5, 6, 5, 7, 6, 7, 5, 5, 8],\n",
+       "       [5, 6, 7, 5, 8, 5, 8, 9, 8, 7],\n",
+       "       [8, 5, 9, 9, 9, 6, 5, 8, 8, 6],\n",
+       "       [7, 8, 7, 6, 8, 8, 6, 7, 5, 5]])"
       ]
      },
-     "execution_count": 20,
+     "execution_count": 10,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -162,19 +217,19 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 22,
+   "execution_count": 11,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([[5, 6],\n",
-       "       [7, 8],\n",
-       "       [5, 9],\n",
-       "       [8, 9]])"
+       "array([[7, 6],\n",
+       "       [8, 5],\n",
+       "       [9, 6],\n",
+       "       [8, 8]])"
       ]
      },
-     "execution_count": 22,
+     "execution_count": 11,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -185,17 +240,17 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 28,
+   "execution_count": 12,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([[5, 6],\n",
-       "       [7, 8]])"
+       "array([[7, 6],\n",
+       "       [8, 5]])"
       ]
      },
-     "execution_count": 28,
+     "execution_count": 12,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -208,12 +263,61 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Combine Data"
+    "## Array Manipulation\n",
+    "\n",
+    "A list of functions is listed below for reference\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "##### CHANGING SHAPE\n",
+    "\n",
+    "reshape - Gives new shape to an array without changing its data \n",
+    "#flat - A 1-D iterator over the array\n",
+    "#flatten - returns the copy of the array collapsed in one dimension\n",
+    "#ravel - returns contiguous flattened array\n",
+    "\n",
+    "##### TRANSPOSE OPERATIONS: \n",
+    "\n",
+    "#transpose - permutes the dimnsions of the array\n",
+    "#ndarray.T - same as self.transpose()\n",
+    "#rollaxis - rolls the specified axis backwards\n",
+    "#swapaxes - interchanges the two axes of an array\n",
+    "\n",
+    "##### CHANGING DIMENSIONS:\n",
+    "\n",
+    "#broadcast - Produces an object that mimics broadcasting\n",
+    "#broadcast_to - broadcasts array to a new shape\n",
+    "#expand_dims - expands the shape of an array\n",
+    "#squeeze - Removes single dimensional entries from the shape of an array\n",
+    "\n",
+    "##### JOINING ARRAYS:\n",
+    "\n",
+    "#concatenate - Joins a sequence of arrays along an existing axis\n",
+    "#stack - Joins a sequence of arrays along a new axis \n",
+    "#hstack - Stacks arrays in sequence horizontally\n",
+    "#vstack - stacks arrays in sequence vertically\n",
+    "\n",
+    "##### SPLITTING ARRAYS: \n",
+    "\n",
+    "#split - splits an array into multiple sub-arrays\n",
+    "#hsplit - splits an arra into multiple sub arrays horizontally(column-wise)\n",
+    "#vsplit - splits an arra into multiple sub arrays vertically(row-wise)\n",
+    "\n",
+    "##### ADDING/REMOVING ELEMENTS: \n",
+    "\n",
+    "#resize - Returns a new array with the specified shape\n",
+    "#append - appends the values to the end of the array\n",
+    "#insert - Inserts the values to the end of an array\n",
+    "#delete - returns a new array with sub arrays along an axis deleted\n",
+    "\n"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 29,
+   "execution_count": 13,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -222,7 +326,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 31,
+   "execution_count": 14,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -231,20 +335,20 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 32,
+   "execution_count": 15,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([[ 8, 10, 14,  6, 13,  6],\n",
-       "       [ 6, 10, 14, 10, 14, 13],\n",
-       "       [12,  5, 13,  5,  8, 13],\n",
-       "       [ 5, 14, 10, 12,  9,  6],\n",
-       "       [ 9,  7, 14, 13,  9,  8]])"
+       "array([[11, 12,  7, 10,  9,  6],\n",
+       "       [11,  9,  9, 11, 13, 14],\n",
+       "       [11,  8,  5, 11,  9, 13],\n",
+       "       [ 7,  8,  8,  5, 11, 12],\n",
+       "       [11, 14, 13,  7,  5,  7]])"
       ]
      },
-     "execution_count": 32,
+     "execution_count": 15,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -253,27 +357,34 @@
     "np.hstack([data1,data2])"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "np.hstack() takes input as two numpy arrays and stacks them together into a single array 'horizontally'"
+   ]
+  },
   {
    "cell_type": "code",
-   "execution_count": 33,
+   "execution_count": 16,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([[ 8, 10, 14],\n",
-       "       [ 6, 10, 14],\n",
-       "       [12,  5, 13],\n",
-       "       [ 5, 14, 10],\n",
-       "       [ 9,  7, 14],\n",
-       "       [ 6, 13,  6],\n",
-       "       [10, 14, 13],\n",
-       "       [ 5,  8, 13],\n",
-       "       [12,  9,  6],\n",
-       "       [13,  9,  8]])"
+       "array([[11, 12,  7],\n",
+       "       [11,  9,  9],\n",
+       "       [11,  8,  5],\n",
+       "       [ 7,  8,  8],\n",
+       "       [11, 14, 13],\n",
+       "       [10,  9,  6],\n",
+       "       [11, 13, 14],\n",
+       "       [11,  9, 13],\n",
+       "       [ 5, 11, 12],\n",
+       "       [ 7,  5,  7]])"
       ]
      },
-     "execution_count": 33,
+     "execution_count": 16,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -282,22 +393,29 @@
     "np.vstack([data1, data2])"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "np.vstack() takes input as two numpy arrays and stacks them together into a single array 'vertically'"
+   ]
+  },
   {
    "cell_type": "code",
-   "execution_count": 34,
+   "execution_count": 17,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([[ 8, 10, 14,  6, 13,  6],\n",
-       "       [ 6, 10, 14, 10, 14, 13],\n",
-       "       [12,  5, 13,  5,  8, 13],\n",
-       "       [ 5, 14, 10, 12,  9,  6],\n",
-       "       [ 9,  7, 14, 13,  9,  8]])"
+       "array([[11, 12,  7, 10,  9,  6],\n",
+       "       [11,  9,  9, 11, 13, 14],\n",
+       "       [11,  8,  5, 11,  9, 13],\n",
+       "       [ 7,  8,  8,  5, 11, 12],\n",
+       "       [11, 14, 13,  7,  5,  7]])"
       ]
      },
-     "execution_count": 34,
+     "execution_count": 17,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -306,9 +424,16 @@
     "np.concatenate([data1, data2], axis=1)"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "np.concatenate() appends the second input array at the end pf the first one along the specified axis parameter."
+   ]
+  },
   {
    "cell_type": "code",
-   "execution_count": 35,
+   "execution_count": 18,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -317,7 +442,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 38,
+   "execution_count": 19,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -326,34 +451,34 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 41,
+   "execution_count": 20,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([[[ 5, 14, 14, 14],\n",
-       "        [ 8,  9, 10,  8],\n",
-       "        [12,  9, 14,  5]],\n",
+       "array([[[ 9, 11, 10,  9],\n",
+       "        [ 6,  5, 12, 12],\n",
+       "        [13, 10, 13, 14]],\n",
        "\n",
-       "       [[ 6, 11,  6, 12],\n",
-       "        [ 9,  6,  5, 10],\n",
-       "        [10,  6,  9,  8]],\n",
+       "       [[14,  9, 12, 12],\n",
+       "        [ 8,  5, 11, 12],\n",
+       "        [ 8,  6,  6,  6]],\n",
        "\n",
-       "       [[11, 13,  8,  5],\n",
-       "        [13, 14, 12,  6],\n",
-       "        [ 7,  9, 14,  6]],\n",
+       "       [[11, 12,  9,  9],\n",
+       "        [ 5, 12, 14, 12],\n",
+       "        [ 9, 12,  5,  6]],\n",
        "\n",
-       "       [[13,  9,  5, 11],\n",
-       "        [ 5, 10,  5,  7],\n",
-       "        [ 6,  5, 14,  6]],\n",
+       "       [[10, 13, 13, 14],\n",
+       "        [11, 13,  5,  9],\n",
+       "        [ 8,  7, 10,  7]],\n",
        "\n",
-       "       [[ 5, 14, 11,  8],\n",
-       "        [ 5,  5,  6,  9],\n",
-       "        [ 7, 11, 11, 11]]])"
+       "       [[13, 14,  6,  6],\n",
+       "        [ 8,  6,  8,  7],\n",
+       "        [11, 12, 12, 10]]])"
       ]
      },
-     "execution_count": 41,
+     "execution_count": 20,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -371,7 +496,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 44,
+   "execution_count": 21,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -380,25 +505,25 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 45,
+   "execution_count": 22,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([[9, 6, 5, 7, 9],\n",
-       "       [8, 9, 9, 6, 9],\n",
-       "       [9, 9, 8, 6, 7],\n",
-       "       [5, 5, 5, 6, 5],\n",
-       "       [6, 6, 7, 7, 5],\n",
-       "       [8, 5, 9, 5, 7],\n",
-       "       [7, 7, 6, 6, 5],\n",
-       "       [5, 6, 9, 6, 8],\n",
-       "       [6, 9, 6, 6, 7],\n",
-       "       [5, 7, 5, 5, 6]])"
+       "array([[7, 7, 7, 6, 7],\n",
+       "       [9, 5, 5, 9, 5],\n",
+       "       [9, 5, 5, 5, 7],\n",
+       "       [9, 5, 7, 7, 9],\n",
+       "       [7, 5, 6, 5, 7],\n",
+       "       [5, 6, 7, 5, 8],\n",
+       "       [8, 5, 9, 9, 9],\n",
+       "       [7, 8, 7, 6, 8],\n",
+       "       [5, 6, 8, 9, 5],\n",
+       "       [8, 9, 8, 9, 7]])"
       ]
      },
-     "execution_count": 45,
+     "execution_count": 22,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -409,7 +534,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 47,
+   "execution_count": 23,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -418,20 +543,20 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 48,
+   "execution_count": 24,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([[9, 6, 5, 7, 9, 6, 5, 5, 5, 6],\n",
-       "       [8, 9, 9, 6, 9, 7, 5, 6, 7, 8],\n",
-       "       [9, 9, 8, 6, 7, 7, 6, 6, 9, 5],\n",
-       "       [5, 5, 5, 6, 5, 9, 9, 9, 8, 9],\n",
-       "       [6, 6, 7, 7, 5, 6, 9, 9, 8, 7]])"
+       "array([[7, 7, 7, 6, 7, 9, 6, 6, 5, 6],\n",
+       "       [9, 5, 5, 9, 5, 8, 6, 5, 6, 9],\n",
+       "       [9, 5, 5, 5, 7, 7, 7, 8, 6, 6],\n",
+       "       [9, 5, 7, 7, 9, 7, 7, 9, 6, 6],\n",
+       "       [7, 5, 6, 5, 7, 6, 7, 5, 5, 8]])"
       ]
      },
-     "execution_count": 48,
+     "execution_count": 24,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -442,17 +567,17 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 49,
+   "execution_count": 25,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([[8, 5, 9, 5, 7, 8, 5, 9, 6, 8],\n",
-       "       [7, 7, 6, 6, 5, 9, 6, 6, 5, 9]])"
+       "array([[5, 6, 7, 5, 8, 5, 8, 9, 8, 7],\n",
+       "       [8, 5, 9, 9, 9, 6, 5, 8, 8, 6]])"
       ]
      },
-     "execution_count": 49,
+     "execution_count": 25,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -463,18 +588,18 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 50,
+   "execution_count": 26,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([[5, 6, 9, 6, 8, 9, 9, 6, 9, 8],\n",
-       "       [6, 9, 6, 6, 7, 7, 5, 8, 5, 8],\n",
-       "       [5, 7, 5, 5, 6, 5, 6, 6, 5, 6]])"
+       "array([[7, 8, 7, 6, 8, 8, 6, 7, 5, 5],\n",
+       "       [5, 6, 8, 9, 5, 5, 8, 8, 5, 8],\n",
+       "       [8, 9, 8, 9, 7, 6, 5, 7, 9, 5]])"
       ]
      },
-     "execution_count": 50,
+     "execution_count": 26,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -492,7 +617,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 52,
+   "execution_count": 27,
    "metadata": {},
    "outputs": [
     {
@@ -501,7 +626,7 @@
        "(10, 10)"
       ]
      },
-     "execution_count": 52,
+     "execution_count": 27,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -512,7 +637,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 53,
+   "execution_count": 28,
    "metadata": {},
    "outputs": [
     {
@@ -521,7 +646,7 @@
        "2"
       ]
      },
-     "execution_count": 53,
+     "execution_count": 28,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -532,7 +657,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 54,
+   "execution_count": 29,
    "metadata": {},
    "outputs": [
     {
@@ -541,7 +666,7 @@
        "100"
       ]
      },
-     "execution_count": 54,
+     "execution_count": 29,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -552,21 +677,21 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 56,
+   "execution_count": 30,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([[9, 6, 5, 7, 9, 6, 5, 5, 5, 6, 8, 9, 9, 6, 9, 7, 5, 6, 7, 8, 9, 9,\n",
-       "        8, 6, 7, 7, 6, 6, 9, 5, 5, 5, 5, 6, 5, 9, 9, 9, 8, 9, 6, 6, 7, 7,\n",
-       "        5, 6, 9, 9, 8, 7],\n",
-       "       [8, 5, 9, 5, 7, 8, 5, 9, 6, 8, 7, 7, 6, 6, 5, 9, 6, 6, 5, 9, 5, 6,\n",
-       "        9, 6, 8, 9, 9, 6, 9, 8, 6, 9, 6, 6, 7, 7, 5, 8, 5, 8, 5, 7, 5, 5,\n",
-       "        6, 5, 6, 6, 5, 6]])"
+       "array([[7, 7, 7, 6, 7, 9, 6, 6, 5, 6, 9, 5, 5, 9, 5, 8, 6, 5, 6, 9, 9, 5,\n",
+       "        5, 5, 7, 7, 7, 8, 6, 6, 9, 5, 7, 7, 9, 7, 7, 9, 6, 6, 7, 5, 6, 5,\n",
+       "        7, 6, 7, 5, 5, 8],\n",
+       "       [5, 6, 7, 5, 8, 5, 8, 9, 8, 7, 8, 5, 9, 9, 9, 6, 5, 8, 8, 6, 7, 8,\n",
+       "        7, 6, 8, 8, 6, 7, 5, 5, 5, 6, 8, 9, 5, 5, 8, 8, 5, 8, 8, 9, 8, 9,\n",
+       "        7, 6, 5, 7, 9, 5]])"
       ]
      },
-     "execution_count": 56,
+     "execution_count": 30,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -577,7 +702,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 57,
+   "execution_count": 31,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -586,7 +711,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 59,
+   "execution_count": 32,
    "metadata": {},
    "outputs": [
     {
@@ -595,7 +720,7 @@
        "1"
       ]
      },
-     "execution_count": 59,
+     "execution_count": 32,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -606,7 +731,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 60,
+   "execution_count": 33,
    "metadata": {},
    "outputs": [
     {
@@ -615,7 +740,7 @@
        "(8,)"
       ]
      },
-     "execution_count": 60,
+     "execution_count": 33,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -626,7 +751,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 61,
+   "execution_count": 34,
    "metadata": {},
    "outputs": [
     {
@@ -635,7 +760,7 @@
        "8"
       ]
      },
-     "execution_count": 61,
+     "execution_count": 34,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -646,7 +771,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 62,
+   "execution_count": 35,
    "metadata": {},
    "outputs": [
     {
@@ -662,7 +787,7 @@
        "       [9]])"
       ]
      },
-     "execution_count": 62,
+     "execution_count": 35,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -673,20 +798,20 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 65,
+   "execution_count": 36,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([[9, 6, 5, 7, 9, 6, 5, 5, 5, 6, 8, 9, 9, 6, 9, 7, 5, 6, 7, 8],\n",
-       "       [9, 9, 8, 6, 7, 7, 6, 6, 9, 5, 5, 5, 5, 6, 5, 9, 9, 9, 8, 9],\n",
-       "       [6, 6, 7, 7, 5, 6, 9, 9, 8, 7, 8, 5, 9, 5, 7, 8, 5, 9, 6, 8],\n",
-       "       [7, 7, 6, 6, 5, 9, 6, 6, 5, 9, 5, 6, 9, 6, 8, 9, 9, 6, 9, 8],\n",
-       "       [6, 9, 6, 6, 7, 7, 5, 8, 5, 8, 5, 7, 5, 5, 6, 5, 6, 6, 5, 6]])"
+       "array([[7, 7, 7, 6, 7, 9, 6, 6, 5, 6, 9, 5, 5, 9, 5, 8, 6, 5, 6, 9],\n",
+       "       [9, 5, 5, 5, 7, 7, 7, 8, 6, 6, 9, 5, 7, 7, 9, 7, 7, 9, 6, 6],\n",
+       "       [7, 5, 6, 5, 7, 6, 7, 5, 5, 8, 5, 6, 7, 5, 8, 5, 8, 9, 8, 7],\n",
+       "       [8, 5, 9, 9, 9, 6, 5, 8, 8, 6, 7, 8, 7, 6, 8, 8, 6, 7, 5, 5],\n",
+       "       [5, 6, 8, 9, 5, 5, 8, 8, 5, 8, 8, 9, 8, 9, 7, 6, 5, 7, 9, 5]])"
       ]
      },
-     "execution_count": 65,
+     "execution_count": 36,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -704,16 +829,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 68,
+   "execution_count": 37,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "683"
+       "679"
       ]
      },
-     "execution_count": 68,
+     "execution_count": 37,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -724,16 +849,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 72,
+   "execution_count": 38,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([63, 74, 72, 70, 70, 70, 66, 75, 67, 56])"
+       "array([66, 67, 65, 72, 61, 68, 73, 67, 67, 73])"
       ]
      },
-     "execution_count": 72,
+     "execution_count": 38,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -744,25 +869,25 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 71,
+   "execution_count": 39,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([[9, 6, 5, 7, 9, 6, 5, 5, 5, 6],\n",
-       "       [8, 9, 9, 6, 9, 7, 5, 6, 7, 8],\n",
-       "       [9, 9, 8, 6, 7, 7, 6, 6, 9, 5],\n",
-       "       [5, 5, 5, 6, 5, 9, 9, 9, 8, 9],\n",
-       "       [6, 6, 7, 7, 5, 6, 9, 9, 8, 7],\n",
-       "       [8, 5, 9, 5, 7, 8, 5, 9, 6, 8],\n",
-       "       [7, 7, 6, 6, 5, 9, 6, 6, 5, 9],\n",
-       "       [5, 6, 9, 6, 8, 9, 9, 6, 9, 8],\n",
-       "       [6, 9, 6, 6, 7, 7, 5, 8, 5, 8],\n",
-       "       [5, 7, 5, 5, 6, 5, 6, 6, 5, 6]])"
+       "array([[7, 7, 7, 6, 7, 9, 6, 6, 5, 6],\n",
+       "       [9, 5, 5, 9, 5, 8, 6, 5, 6, 9],\n",
+       "       [9, 5, 5, 5, 7, 7, 7, 8, 6, 6],\n",
+       "       [9, 5, 7, 7, 9, 7, 7, 9, 6, 6],\n",
+       "       [7, 5, 6, 5, 7, 6, 7, 5, 5, 8],\n",
+       "       [5, 6, 7, 5, 8, 5, 8, 9, 8, 7],\n",
+       "       [8, 5, 9, 9, 9, 6, 5, 8, 8, 6],\n",
+       "       [7, 8, 7, 6, 8, 8, 6, 7, 5, 5],\n",
+       "       [5, 6, 8, 9, 5, 5, 8, 8, 5, 8],\n",
+       "       [8, 9, 8, 9, 7, 6, 5, 7, 9, 5]])"
       ]
      },
-     "execution_count": 71,
+     "execution_count": 39,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -773,35 +898,35 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 73,
+   "execution_count": 40,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([[ 0.41211849, -0.2794155 , -0.95892427,  0.6569866 ,  0.41211849,\n",
-       "        -0.2794155 , -0.95892427, -0.95892427, -0.95892427, -0.2794155 ],\n",
-       "       [ 0.98935825,  0.41211849,  0.41211849, -0.2794155 ,  0.41211849,\n",
-       "         0.6569866 , -0.95892427, -0.2794155 ,  0.6569866 ,  0.98935825],\n",
-       "       [ 0.41211849,  0.41211849,  0.98935825, -0.2794155 ,  0.6569866 ,\n",
-       "         0.6569866 , -0.2794155 , -0.2794155 ,  0.41211849, -0.95892427],\n",
-       "       [-0.95892427, -0.95892427, -0.95892427, -0.2794155 , -0.95892427,\n",
-       "         0.41211849,  0.41211849,  0.41211849,  0.98935825,  0.41211849],\n",
-       "       [-0.2794155 , -0.2794155 ,  0.6569866 ,  0.6569866 , -0.95892427,\n",
-       "        -0.2794155 ,  0.41211849,  0.41211849,  0.98935825,  0.6569866 ],\n",
-       "       [ 0.98935825, -0.95892427,  0.41211849, -0.95892427,  0.6569866 ,\n",
-       "         0.98935825, -0.95892427,  0.41211849, -0.2794155 ,  0.98935825],\n",
-       "       [ 0.6569866 ,  0.6569866 , -0.2794155 , -0.2794155 , -0.95892427,\n",
-       "         0.41211849, -0.2794155 , -0.2794155 , -0.95892427,  0.41211849],\n",
-       "       [-0.95892427, -0.2794155 ,  0.41211849, -0.2794155 ,  0.98935825,\n",
-       "         0.41211849,  0.41211849, -0.2794155 ,  0.41211849,  0.98935825],\n",
-       "       [-0.2794155 ,  0.41211849, -0.2794155 , -0.2794155 ,  0.6569866 ,\n",
-       "         0.6569866 , -0.95892427,  0.98935825, -0.95892427,  0.98935825],\n",
-       "       [-0.95892427,  0.6569866 , -0.95892427, -0.95892427, -0.2794155 ,\n",
-       "        -0.95892427, -0.2794155 , -0.2794155 , -0.95892427, -0.2794155 ]])"
+       "array([[ 0.6569866 ,  0.6569866 ,  0.6569866 , -0.2794155 ,  0.6569866 ,\n",
+       "         0.41211849, -0.2794155 , -0.2794155 , -0.95892427, -0.2794155 ],\n",
+       "       [ 0.41211849, -0.95892427, -0.95892427,  0.41211849, -0.95892427,\n",
+       "         0.98935825, -0.2794155 , -0.95892427, -0.2794155 ,  0.41211849],\n",
+       "       [ 0.41211849, -0.95892427, -0.95892427, -0.95892427,  0.6569866 ,\n",
+       "         0.6569866 ,  0.6569866 ,  0.98935825, -0.2794155 , -0.2794155 ],\n",
+       "       [ 0.41211849, -0.95892427,  0.6569866 ,  0.6569866 ,  0.41211849,\n",
+       "         0.6569866 ,  0.6569866 ,  0.41211849, -0.2794155 , -0.2794155 ],\n",
+       "       [ 0.6569866 , -0.95892427, -0.2794155 , -0.95892427,  0.6569866 ,\n",
+       "        -0.2794155 ,  0.6569866 , -0.95892427, -0.95892427,  0.98935825],\n",
+       "       [-0.95892427, -0.2794155 ,  0.6569866 , -0.95892427,  0.98935825,\n",
+       "        -0.95892427,  0.98935825,  0.41211849,  0.98935825,  0.6569866 ],\n",
+       "       [ 0.98935825, -0.95892427,  0.41211849,  0.41211849,  0.41211849,\n",
+       "        -0.2794155 , -0.95892427,  0.98935825,  0.98935825, -0.2794155 ],\n",
+       "       [ 0.6569866 ,  0.98935825,  0.6569866 , -0.2794155 ,  0.98935825,\n",
+       "         0.98935825, -0.2794155 ,  0.6569866 , -0.95892427, -0.95892427],\n",
+       "       [-0.95892427, -0.2794155 ,  0.98935825,  0.41211849, -0.95892427,\n",
+       "        -0.95892427,  0.98935825,  0.98935825, -0.95892427,  0.98935825],\n",
+       "       [ 0.98935825,  0.41211849,  0.98935825,  0.41211849,  0.6569866 ,\n",
+       "        -0.2794155 , -0.95892427,  0.6569866 ,  0.41211849, -0.95892427]])"
       ]
      },
-     "execution_count": 73,
+     "execution_count": 40,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -812,25 +937,25 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 75,
+   "execution_count": 41,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([[9, 8, 9, 5, 6, 8, 7, 5, 6, 5],\n",
-       "       [6, 9, 9, 5, 6, 5, 7, 6, 9, 7],\n",
-       "       [5, 9, 8, 5, 7, 9, 6, 9, 6, 5],\n",
-       "       [7, 6, 6, 6, 7, 5, 6, 6, 6, 5],\n",
-       "       [9, 9, 7, 5, 5, 7, 5, 8, 7, 6],\n",
-       "       [6, 7, 7, 9, 6, 8, 9, 9, 7, 5],\n",
-       "       [5, 5, 6, 9, 9, 5, 6, 9, 5, 6],\n",
-       "       [5, 6, 6, 9, 9, 9, 6, 6, 8, 6],\n",
-       "       [5, 7, 9, 8, 8, 6, 5, 9, 5, 5],\n",
-       "       [6, 8, 5, 9, 7, 8, 9, 8, 8, 6]])"
+       "array([[7, 9, 9, 9, 7, 5, 8, 7, 5, 8],\n",
+       "       [7, 5, 5, 5, 5, 6, 5, 8, 6, 9],\n",
+       "       [7, 5, 5, 7, 6, 7, 9, 7, 8, 8],\n",
+       "       [6, 9, 5, 7, 5, 5, 9, 6, 9, 9],\n",
+       "       [7, 5, 7, 9, 7, 8, 9, 8, 5, 7],\n",
+       "       [9, 8, 7, 7, 6, 5, 6, 8, 5, 6],\n",
+       "       [6, 6, 7, 7, 7, 8, 5, 6, 8, 5],\n",
+       "       [6, 5, 8, 9, 5, 9, 8, 7, 8, 7],\n",
+       "       [5, 6, 6, 6, 5, 8, 8, 5, 5, 9],\n",
+       "       [6, 9, 6, 6, 8, 7, 6, 5, 8, 5]])"
       ]
      },
-     "execution_count": 75,
+     "execution_count": 41,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -841,16 +966,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 76,
+   "execution_count": 42,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([6.3, 7.4, 7.2, 7. , 7. , 7. , 6.6, 7.5, 6.7, 5.6])"
+       "array([6.6, 6.7, 6.5, 7.2, 6.1, 6.8, 7.3, 6.7, 6.7, 7.3])"
       ]
      },
-     "execution_count": 76,
+     "execution_count": 42,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -861,17 +986,17 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 77,
+   "execution_count": 43,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([1.48660687, 1.356466  , 1.4       , 1.84390889, 1.26491106,\n",
-       "       1.54919334, 1.356466  , 1.5       , 1.26885775, 0.66332496])"
+       "array([1.0198039 , 1.73493516, 1.28452326, 1.32664992, 1.04403065,\n",
+       "       1.4       , 1.55241747, 1.1       , 1.55241747, 1.48660687])"
       ]
      },
-     "execution_count": 77,
+     "execution_count": 43,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -880,6 +1005,25 @@
     "data.std(axis=1)"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#### A List of String functions\n",
+    "\n",
+    "#add() - Returns element wise string concatenation for two arrays of str or unicode\n",
+    "#multiply() - returns the string with multiple concatenation, element wise\n",
+    "#center() -  returns the copy of the given string with elements centred ina a string of specified length\n",
+    "#capitalize() - returns a copy of the string with only the first character capitalized\n",
+    "#title() - returns the element wise title cased version of the string or unicode\n",
+    "#lower() - returns an array with elements converted to lowercase\n",
+    "#upper() - returns an array with the elements converted to uppercase\n",
+    "#split() - returns a list of words in the string, uing separator delimiter\n",
+    "#splitlines() - returns a list of lines in the element, breaking at the line boundaries\n",
+    "#strip() - returns a copy with the leading and trailing characters removed\n",
+    "#join() - returns a copy of the string with all occurences of substring replaced by new string"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -889,7 +1033,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 78,
+   "execution_count": 44,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -898,7 +1042,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 79,
+   "execution_count": 45,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -907,7 +1051,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 80,
+   "execution_count": 46,
    "metadata": {},
    "outputs": [
     {
@@ -916,7 +1060,7 @@
        "array([11, 12, 13, 14, 15])"
       ]
      },
-     "execution_count": 80,
+     "execution_count": 46,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -927,7 +1071,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 82,
+   "execution_count": 47,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -936,7 +1080,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 84,
+   "execution_count": 48,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -945,7 +1089,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 85,
+   "execution_count": 49,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -954,7 +1098,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 86,
+   "execution_count": 50,
    "metadata": {},
    "outputs": [
     {
@@ -963,7 +1107,7 @@
        "(5, 1)"
       ]
      },
-     "execution_count": 86,
+     "execution_count": 50,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -974,7 +1118,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 87,
+   "execution_count": 51,
    "metadata": {},
    "outputs": [
     {
@@ -987,7 +1131,7 @@
        "       [ 6,  7,  8,  9, 10]])"
       ]
      },
-     "execution_count": 87,
+     "execution_count": 51,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -998,7 +1142,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 88,
+   "execution_count": 52,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -1007,7 +1151,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 89,
+   "execution_count": 53,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -1016,25 +1160,25 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 94,
+   "execution_count": 54,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([[[3, 2],\n",
+       "array([[[3, 4],\n",
        "        [4, 2],\n",
-       "        [3, 4],\n",
-       "        [3, 3],\n",
-       "        [3, 1],\n",
-       "        [1, 3],\n",
-       "        [2, 2],\n",
+       "        [1, 2],\n",
        "        [1, 1],\n",
-       "        [4, 3],\n",
-       "        [2, 2]]])"
+       "        [4, 1],\n",
+       "        [4, 4],\n",
+       "        [2, 2],\n",
+       "        [3, 2],\n",
+       "        [2, 4],\n",
+       "        [1, 3]]])"
       ]
      },
-     "execution_count": 94,
+     "execution_count": 54,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1045,69 +1189,69 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 95,
+   "execution_count": 55,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([[[-1,  2],\n",
-       "        [-2,  2],\n",
+       "array([[[-2, -2],\n",
+       "        [-3,  0],\n",
+       "        [ 0,  0],\n",
+       "        [ 0,  1],\n",
+       "        [-3,  1],\n",
+       "        [-3, -2],\n",
        "        [-1,  0],\n",
-       "        [-1,  1],\n",
-       "        [-1,  3],\n",
-       "        [ 1,  1],\n",
-       "        [ 0,  2],\n",
-       "        [ 1,  3],\n",
-       "        [-2,  1],\n",
-       "        [ 0,  2]],\n",
+       "        [-2,  0],\n",
+       "        [-1, -2],\n",
+       "        [ 0, -1]],\n",
        "\n",
-       "       [[-2,  2],\n",
-       "        [-3,  2],\n",
+       "       [[-1, -3],\n",
+       "        [-2, -1],\n",
+       "        [ 1, -1],\n",
+       "        [ 1,  0],\n",
        "        [-2,  0],\n",
-       "        [-2,  1],\n",
-       "        [-2,  3],\n",
-       "        [ 0,  1],\n",
-       "        [-1,  2],\n",
-       "        [ 0,  3],\n",
-       "        [-3,  1],\n",
-       "        [-1,  2]],\n",
+       "        [-2, -3],\n",
+       "        [ 0, -1],\n",
+       "        [-1, -1],\n",
+       "        [ 0, -3],\n",
+       "        [ 1, -2]],\n",
        "\n",
-       "       [[-1,  2],\n",
-       "        [-2,  2],\n",
-       "        [-1,  0],\n",
-       "        [-1,  1],\n",
+       "       [[ 0,  0],\n",
+       "        [-1,  2],\n",
+       "        [ 2,  2],\n",
+       "        [ 2,  3],\n",
        "        [-1,  3],\n",
-       "        [ 1,  1],\n",
+       "        [-1,  0],\n",
+       "        [ 1,  2],\n",
        "        [ 0,  2],\n",
-       "        [ 1,  3],\n",
-       "        [-2,  1],\n",
-       "        [ 0,  2]],\n",
+       "        [ 1,  0],\n",
+       "        [ 2,  1]],\n",
        "\n",
-       "       [[ 1,  0],\n",
+       "       [[-1, -2],\n",
+       "        [-2,  0],\n",
+       "        [ 1,  0],\n",
+       "        [ 1,  1],\n",
+       "        [-2,  1],\n",
+       "        [-2, -2],\n",
        "        [ 0,  0],\n",
-       "        [ 1, -2],\n",
+       "        [-1,  0],\n",
+       "        [ 0, -2],\n",
+       "        [ 1, -1]],\n",
+       "\n",
+       "       [[-1, -3],\n",
+       "        [-2, -1],\n",
        "        [ 1, -1],\n",
-       "        [ 1,  1],\n",
-       "        [ 3, -1],\n",
-       "        [ 2,  0],\n",
-       "        [ 3,  1],\n",
+       "        [ 1,  0],\n",
+       "        [-2,  0],\n",
+       "        [-2, -3],\n",
        "        [ 0, -1],\n",
-       "        [ 2,  0]],\n",
-       "\n",
-       "       [[ 0,  2],\n",
-       "        [-1,  2],\n",
-       "        [ 0,  0],\n",
-       "        [ 0,  1],\n",
-       "        [ 0,  3],\n",
-       "        [ 2,  1],\n",
-       "        [ 1,  2],\n",
-       "        [ 2,  3],\n",
-       "        [-1,  1],\n",
-       "        [ 1,  2]]])"
+       "        [-1, -1],\n",
+       "        [ 0, -3],\n",
+       "        [ 1, -2]]])"
       ]
      },
-     "execution_count": 95,
+     "execution_count": 55,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1118,7 +1262,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 96,
+   "execution_count": 56,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -1128,7 +1272,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 97,
+   "execution_count": 57,
    "metadata": {},
    "outputs": [
     {
@@ -1137,7 +1281,7 @@
        "(10, 2)"
       ]
      },
-     "execution_count": 97,
+     "execution_count": 57,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1148,7 +1292,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 98,
+   "execution_count": 58,
    "metadata": {},
    "outputs": [
     {
@@ -1157,7 +1301,7 @@
        "(5, 3)"
       ]
      },
-     "execution_count": 98,
+     "execution_count": 58,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1168,218 +1312,218 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 99,
+   "execution_count": 59,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "array([[[[-3, -1],\n",
-       "         [ 0,  2],\n",
-       "         [-1,  1]],\n",
+       "array([[[[ 2,  1],\n",
+       "         [ 0, -1],\n",
+       "         [-1, -2]],\n",
        "\n",
-       "        [[-2,  0],\n",
-       "         [ 1,  3],\n",
-       "         [ 0,  2]],\n",
+       "        [[ 3,  3],\n",
+       "         [ 1,  1],\n",
+       "         [ 0,  0]],\n",
        "\n",
-       "        [[ 0, -2],\n",
-       "         [ 3,  1],\n",
-       "         [ 2,  0]],\n",
+       "        [[ 0,  3],\n",
+       "         [-2,  1],\n",
+       "         [-3,  0]],\n",
        "\n",
-       "        [[ 0,  0],\n",
-       "         [ 3,  3],\n",
-       "         [ 2,  2]],\n",
+       "        [[ 2,  0],\n",
+       "         [ 0, -2],\n",
+       "         [-1, -3]],\n",
        "\n",
-       "        [[-1, -3],\n",
-       "         [ 2,  0],\n",
-       "         [ 1, -1]],\n",
+       "        [[ 3,  3],\n",
+       "         [ 1,  1],\n",
+       "         [ 0,  0]],\n",
        "\n",
-       "        [[-2, -2],\n",
+       "        [[ 2,  0],\n",
+       "         [ 0, -2],\n",
+       "         [-1, -3]],\n",
+       "\n",
+       "        [[ 3,  3],\n",
        "         [ 1,  1],\n",
        "         [ 0,  0]],\n",
        "\n",
-       "        [[ 0,  0],\n",
-       "         [ 3,  3],\n",
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+       "         [ 0, -1],\n",
+       "         [-1, -2]],\n",
        "\n",
-       "        [[-3,  0],\n",
-       "         [ 0,  3],\n",
-       "         [-1,  2]],\n",
+       "        [[ 1,  1],\n",
+       "         [-1, -1],\n",
+       "         [-2, -2]],\n",
        "\n",
-       "        [[-1, -3],\n",
-       "         [ 2,  0],\n",
-       "         [ 1, -1]],\n",
+       "        [[ 3,  3],\n",
+       "         [ 1,  1],\n",
+       "         [ 0,  0]]],\n",
        "\n",
-       "        [[-2, -3],\n",
-       "         [ 1,  0],\n",
-       "         [ 0, -1]]],\n",
        "\n",
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+       "         [ 2,  1],\n",
+       "         [ 2,  1]],\n",
        "\n",
-       "       [[[ 0,  2],\n",
-       "         [-2,  0],\n",
-       "         [ 0,  2]],\n",
+       "        [[ 1,  1],\n",
+       "         [ 3,  3],\n",
+       "         [ 3,  3]],\n",
        "\n",
-       "        [[ 1,  3],\n",
-       "         [-1,  1],\n",
-       "         [ 1,  3]],\n",
+       "        [[-2,  1],\n",
+       "         [ 0,  3],\n",
+       "         [ 0,  3]],\n",
        "\n",
-       "        [[ 3,  1],\n",
-       "         [ 1, -1],\n",
-       "         [ 3,  1]],\n",
+       "        [[ 0, -2],\n",
+       "         [ 2,  0],\n",
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        "\n",
-       "        [[ 3,  3],\n",
-       "         [ 1,  1],\n",
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+       "         [ 3,  3],\n",
        "         [ 3,  3]],\n",
        "\n",
-       "        [[ 2,  0],\n",
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        "\n",
        "        [[ 1,  1],\n",
-       "         [-1, -1],\n",
+       "         [ 3,  3],\n",
+       "         [ 3,  3]],\n",
+       "\n",
+       "        [[ 0, -1],\n",
+       "         [ 2,  1],\n",
+       "         [ 2,  1]],\n",
+       "\n",
+       "        [[-1, -1],\n",
+       "         [ 1,  1],\n",
        "         [ 1,  1]],\n",
        "\n",
+       "        [[ 1,  1],\n",
+       "         [ 3,  3],\n",
+       "         [ 3,  3]]],\n",
+       "\n",
+       "\n",
+       "       [[[ 2,  1],\n",
+       "         [ 0, -1],\n",
+       "         [ 0, -1]],\n",
+       "\n",
        "        [[ 3,  3],\n",
        "         [ 1,  1],\n",
-       "         [ 3,  3]],\n",
+       "         [ 1,  1]],\n",
        "\n",
        "        [[ 0,  3],\n",
        "         [-2,  1],\n",
-       "         [ 0,  3]],\n",
+       "         [-2,  1]],\n",
        "\n",
        "        [[ 2,  0],\n",
        "         [ 0, -2],\n",
-       "         [ 2,  0]],\n",
-       "\n",
-       "        [[ 1,  0],\n",
-       "         [-1, -2],\n",
-       "         [ 1,  0]]],\n",
-       "\n",
-       "\n",
-       "       [[[-3, -1],\n",
-       "         [-2,  0],\n",
-       "         [-2,  0]],\n",
-       "\n",
-       "        [[-2,  0],\n",
-       "         [-1,  1],\n",
-       "         [-1,  1]],\n",
-       "\n",
-       "        [[ 0, -2],\n",
-       "         [ 1, -1],\n",
-       "         [ 1, -1]],\n",
+       "         [ 0, -2]],\n",
        "\n",
-       "        [[ 0,  0],\n",
+       "        [[ 3,  3],\n",
        "         [ 1,  1],\n",
        "         [ 1,  1]],\n",
        "\n",
-       "        [[-1, -3],\n",
+       "        [[ 2,  0],\n",
        "         [ 0, -2],\n",
        "         [ 0, -2]],\n",
        "\n",
-       "        [[-2, -2],\n",
+       "        [[ 3,  3],\n",
+       "         [ 1,  1],\n",
+       "         [ 1,  1]],\n",
+       "\n",
+       "        [[ 2,  1],\n",
+       "         [ 0, -1],\n",
+       "         [ 0, -1]],\n",
+       "\n",
+       "        [[ 1,  1],\n",
        "         [-1, -1],\n",
        "         [-1, -1]],\n",
        "\n",
-       "        [[ 0,  0],\n",
+       "        [[ 3,  3],\n",
        "         [ 1,  1],\n",
-       "         [ 1,  1]],\n",
+       "         [ 1,  1]]],\n",
        "\n",
-       "        [[-3,  0],\n",
-       "         [-2,  1],\n",
-       "         [-2,  1]],\n",
        "\n",
-       "        [[-1, -3],\n",
-       "         [ 0, -2],\n",
-       "         [ 0, -2]],\n",
-       "\n",
-       "        [[-2, -3],\n",
+       "       [[[ 0, -1],\n",
        "         [-1, -2],\n",
-       "         [-1, -2]]],\n",
-       "\n",
+       "         [ 0, -1]],\n",
        "\n",
-       "       [[[-1,  1],\n",
-       "         [ 0,  2],\n",
-       "         [ 0,  2]],\n",
+       "        [[ 1,  1],\n",
+       "         [ 0,  0],\n",
+       "         [ 1,  1]],\n",
        "\n",
-       "        [[ 0,  2],\n",
-       "         [ 1,  3],\n",
-       "         [ 1,  3]],\n",
+       "        [[-2,  1],\n",
+       "         [-3,  0],\n",
+       "         [-2,  1]],\n",
        "\n",
-       "        [[ 2,  0],\n",
-       "         [ 3,  1],\n",
-       "         [ 3,  1]],\n",
+       "        [[ 0, -2],\n",
+       "         [-1, -3],\n",
+       "         [ 0, -2]],\n",
        "\n",
-       "        [[ 2,  2],\n",
-       "         [ 3,  3],\n",
-       "         [ 3,  3]],\n",
+       "        [[ 1,  1],\n",
+       "         [ 0,  0],\n",
+       "         [ 1,  1]],\n",
        "\n",
-       "        [[ 1, -1],\n",
-       "         [ 2,  0],\n",
-       "         [ 2,  0]],\n",
+       "        [[ 0, -2],\n",
+       "         [-1, -3],\n",
+       "         [ 0, -2]],\n",
        "\n",
-       "        [[ 0,  0],\n",
-       "         [ 1,  1],\n",
+       "        [[ 1,  1],\n",
+       "         [ 0,  0],\n",
        "         [ 1,  1]],\n",
        "\n",
-       "        [[ 2,  2],\n",
-       "         [ 3,  3],\n",
-       "         [ 3,  3]],\n",
+       "        [[ 0, -1],\n",
+       "         [-1, -2],\n",
+       "         [ 0, -1]],\n",
        "\n",
-       "        [[-1,  2],\n",
-       "         [ 0,  3],\n",
-       "         [ 0,  3]],\n",
+       "        [[-1, -1],\n",
+       "         [-2, -2],\n",
+       "         [-1, -1]],\n",
        "\n",
-       "        [[ 1, -1],\n",
-       "         [ 2,  0],\n",
-       "         [ 2,  0]],\n",
+       "        [[ 1,  1],\n",
+       "         [ 0,  0],\n",
+       "         [ 1,  1]]],\n",
        "\n",
-       "        [[ 0, -1],\n",
-       "         [ 1,  0],\n",
-       "         [ 1,  0]]],\n",
        "\n",
+       "       [[[ 0, -1],\n",
+       "         [ 0, -1],\n",
+       "         [ 0, -1]],\n",
        "\n",
-       "       [[[-1,  1],\n",
-       "         [ 0,  2],\n",
-       "         [ 0,  2]],\n",
+       "        [[ 1,  1],\n",
+       "         [ 1,  1],\n",
+       "         [ 1,  1]],\n",
        "\n",
-       "        [[ 0,  2],\n",
-       "         [ 1,  3],\n",
-       "         [ 1,  3]],\n",
+       "        [[-2,  1],\n",
+       "         [-2,  1],\n",
+       "         [-2,  1]],\n",
        "\n",
-       "        [[ 2,  0],\n",
-       "         [ 3,  1],\n",
-       "         [ 3,  1]],\n",
+       "        [[ 0, -2],\n",
+       "         [ 0, -2],\n",
+       "         [ 0, -2]],\n",
        "\n",
-       "        [[ 2,  2],\n",
-       "         [ 3,  3],\n",
-       "         [ 3,  3]],\n",
+       "        [[ 1,  1],\n",
+       "         [ 1,  1],\n",
+       "         [ 1,  1]],\n",
        "\n",
-       "        [[ 1, -1],\n",
-       "         [ 2,  0],\n",
-       "         [ 2,  0]],\n",
+       "        [[ 0, -2],\n",
+       "         [ 0, -2],\n",
+       "         [ 0, -2]],\n",
        "\n",
-       "        [[ 0,  0],\n",
+       "        [[ 1,  1],\n",
        "         [ 1,  1],\n",
        "         [ 1,  1]],\n",
        "\n",
-       "        [[ 2,  2],\n",
-       "         [ 3,  3],\n",
-       "         [ 3,  3]],\n",
-       "\n",
-       "        [[-1,  2],\n",
-       "         [ 0,  3],\n",
-       "         [ 0,  3]],\n",
+       "        [[ 0, -1],\n",
+       "         [ 0, -1],\n",
+       "         [ 0, -1]],\n",
        "\n",
-       "        [[ 1, -1],\n",
-       "         [ 2,  0],\n",
-       "         [ 2,  0]],\n",
+       "        [[-1, -1],\n",
+       "         [-1, -1],\n",
+       "         [-1, -1]],\n",
        "\n",
-       "        [[ 0, -1],\n",
-       "         [ 1,  0],\n",
-       "         [ 1,  0]]]])"
+       "        [[ 1,  1],\n",
+       "         [ 1,  1],\n",
+       "         [ 1,  1]]]])"
       ]
      },
-     "execution_count": 99,
+     "execution_count": 59,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1388,6 +1532,256 @@
     "a.reshape(1,10,1,2) - b.reshape(5,1,3,1)"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#### STATISTICAL FUNCTIONS\n",
+    "\n",
+    "\n",
+    "#### numpy.amin() and numpy.amax() \n",
+    "\n",
+    "These functions return the minimum and the maximum from the elements in the given array along the specified axis"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 60,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "a = np.array([[3,7,5],[8,4,3],[2,4,9]])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 61,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[3 3 2]\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(np.amin(a,1))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 62,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[2 4 3]\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(np.amin(a,0))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 63,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[7 8 9]\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(np.amax(a,1))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 64,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[8 7 9]\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(np.amax(a,0))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#### numpy.ptp()\n",
+    "\n",
+    "This function returns the range (maximum-minimum) of the values along an axis.\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 65,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "7\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(np.ptp(a))         #with no argument, it takes the entire array for comparison"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 66,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[4 5 7]\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(np.ptp(a,axis = 1))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 67,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[6 3 6]\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(np.ptp(a,axis = 0))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#### numpy.percentile() \n",
+    "\n",
+    "Percentile (or a centile) is a measure used in statistics indicating the value below which a given percentage of observations in a group of observations fall. The function numpy.percentile() takes the following arguments.\n",
+    "\n",
+    "\n",
+    "#### numpy.percentile(a, q, axis)\n",
+    "\n",
+    "\n",
+    "#a - input array\n",
+    "#q - the percentile to compute (must be between 0 and 100)\n",
+    "#axis - the axis along which percentile is to be caculated"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 68,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[[3 7 5]\n",
+      " [8 4 3]\n",
+      " [2 4 9]]\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(a)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 69,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "4.0\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(np.percentile(a,50)) "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 70,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[5. 4. 4.]\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(np.percentile(a,50, axis = 1)) "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 71,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[3. 4. 5.]\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(np.percentile(a,50, axis = 0))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
   {
    "cell_type": "code",
    "execution_count": null,
@@ -1412,7 +1806,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.6.4"
+   "version": "3.7.3"
   }
  },
  "nbformat": 4,