diff --git a/notebook_必备数学基础/相关分析/.ipynb_checkpoints/相关分析-checkpoint.ipynb b/notebook_必备数学基础/相关分析/.ipynb_checkpoints/相关分析-checkpoint.ipynb
index 0f44ecc..f49ec71 100644
--- a/notebook_必备数学基础/相关分析/.ipynb_checkpoints/相关分析-checkpoint.ipynb
+++ b/notebook_必备数学基础/相关分析/.ipynb_checkpoints/相关分析-checkpoint.ipynb
@@ -223,7 +223,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 10,
    "metadata": {},
    "outputs": [
     {
@@ -233,24 +233,220 @@
       "correlation: 0.9891763198690562\n",
       "pvalue: 5.926875946481138e-08\n"
      ]
+    },
+    {
+     "data": {
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXQAAAD4CAYAAAD8Zh1EAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjAsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy+17YcXAAATHUlEQVR4nO3df4zcd33n8efrbF9ZKO1eyZbG64BbFbmnJk2crNJwkSJIuDpAlLgBdKlKSxA931XpEarK6FydOIFO4pBPLdwhEaXkrqZQfjQYnxsBJi2kV/5I0Dp2MMGxmuulxOtcvSQ4kGObs837/thxzlnP7MzYs56Zr58PabXf+X4/nnnFSV47+/l+vt9JVSFJGn//aNgBJEmDYaFLUkNY6JLUEBa6JDWEhS5JDbF6WC980UUX1fr164f18pI0lvbu3fvdqppqd2xohb5+/XpmZ2eH9fKSNJaS/F2nY065SFJDWOiS1BAWuiQ1hIUuSQ1hoUtSQ1joktQQQ1u2KEkXil375ti+5xBHji2wdnKCrZs2sHnj9MBfx3fokrSCdu2bY9vOA8wdW6CAuWML/O5n9/Pvdh0Y+GtZ6JK0grbvOcTC8ZMv2lfApx78Drv2zQ30tSx0SVpBR44ttN1fLJb9IFnokrSC1k5OdDzWqezPloUuSSto66YNpMOx5cr+bFjokrSCNm+c5tevedUZpT6xZhVbN20Y6Gv1tGwxyRPAD4CTwImqmllyPMBHgDcBPwRur6qHB5pU0gXlfC31Ox/+w+bLmHn1T634P08/69BfX1Xf7XDsjcBrWl+/DHys9V2S+nZqqd+p1SFzxxbYtnNxmd+4lvrmjdMrnn1QUy63AJ+oRQ8Ck0kuHtBzS7rAtFvqt3D85MBXhTRNr4VewFeS7E2ypc3xaeDJ0x4fbu17kSRbkswmmZ2fn+8/raQLQqfVH4NeFdI0vRb6tVV1JYtTK3ckuW7J8XYnceuMHVV3V9VMVc1MTbX9BCVJ6rj6Y9CrQpqmp0KvqiOt70eBLwBXLxlyGLjktMfrgCODCCjpwrN10wYm1qx60b6VWBXSNF0LPcnLkrz81DbwK8C3lgzbDfxmFl0DPFtVTw08raQLwuaN03zw1suYnpwgwPTkBB+89bKxPSF6vvSyyuWVwBcWVyayGvjTqvpykn8NUFV3AV9kccni4ywuW3znysSVNMoGudTwfKwKaZquhV5Vfwtc3mb/XadtF3DHYKNJGidNXGo4brxSVNJAuNRw+Cx0SQPhUsPhs9AlDYRLDYfPQpc0EC41HD4/U1TSQJw68dmUG2qNIwtd0sC41HC4nHKRpIaw0CWpISx0SWoIC12SGsJCl6SGsNAlqSEsdElqCAtdkhrCQpekhrDQJakhLHRJaggLXZIawkKXpIboudCTrEqyL8l9bY7dnmQ+yf7W128NNqYkqZt+bp97J3AQ+IkOxz9bVb9z7pEkSWejp3foSdYBbwY+vrJxJElnq9cplw8D7wV+tMyYtyT5ZpJ7k1zSbkCSLUlmk8zOz8/3m1WStIyuhZ7kJuBoVe1dZtifA+ur6peAvwB2tBtUVXdX1UxVzUxNTZ1VYElSe728Q78WuDnJE8BngOuTfPL0AVX1dFU933r4R8BVA00pSeqqa6FX1baqWldV64HbgK9W1dtPH5Pk4tMe3sziyVNJ0nl01h8SneQDwGxV7QbeneRm4ATwDHD7YOJJknqVqhrKC8/MzNTs7OxQXluSxlWSvVU10+6YV4pKUkOc9ZSLpAvLrn1zbN9ziCPHFlg7OcHWTRvYvHF62LF0GgtdUle79s2xbecBFo6fBGDu2ALbdh4AsNRHiFMukrravufQC2V+ysLxk2zfc2hIidSOhS6pqyPHFvrar+Gw0CV1tXZyoq/9Gg4LXVJXWzdtYGLNqhftm1iziq2bNgwpkdrxpKikrk6d+HSVy2iz0CX1ZPPGaQt8xDnlIkkNYaFLUkNY6JLUEBa6JDWEJ0WlEec9VNQrC10aYd5DRf1wykUaYd5DRf2w0KUR5j1U1A8LXRph3kNF/bDQpRHmPVTUD0+KSiPMe6ioHz0XepJVwCwwV1U3LTn2Y8AngKuAp4F/UVVPDDCndMHyHirqVT9TLncCBzscexfwvar6eeAPgQ+dazBJUn96KvQk64A3Ax/vMOQWYEdr+17ghiQ593iSpF71+g79w8B7gR91OD4NPAlQVSeAZ4FXLB2UZEuS2SSz8/PzZxFXktRJ10JPchNwtKr2Ljeszb46Y0fV3VU1U1UzU1NTfcSUJHXTyzv0a4GbkzwBfAa4Psknl4w5DFwCkGQ18JPAMwPMKUnqomuhV9W2qlpXVeuB24CvVtXblwzbDbyjtf3W1pgz3qFLklbOWa9DT/IBYLaqdgP3AH+S5HEW35nfNqB8kqQe9VXoVfUA8EBr+32n7f8H4G2DDCZJ6o+X/ktSQ1joktQQFrokNYSFLkkNYaFLUkNY6JLUEBa6JDWEhS5JDWGhS1JDWOiS1BAWuiQ1hIUuSQ1hoUtSQ1joktQQFrokNYSFLkkNYaFLUkNY6JLUEBa6JDVE10JP8pIk30jySJJHk7y/zZjbk8wn2d/6+q2ViStJ6qSXD4l+Hri+qp5Lsgb4epIvVdWDS8Z9tqp+Z/ARJUm96FroVVXAc62Ha1pftZKhJEn962kOPcmqJPuBo8D9VfVQm2FvSfLNJPcmuaTD82xJMptkdn5+/hxiS5KW6qnQq+pkVV0BrAOuTnLpkiF/Dqyvql8C/gLY0eF57q6qmaqamZqaOpfckqQl+lrlUlXHgAeAG5fsf7qqnm89/CPgqoGkkyT1rJdVLlNJJlvbE8AbgMeWjLn4tIc3AwcHGVKS1F0vq1wuBnYkWcXiD4DPVdV9ST4AzFbVbuDdSW4GTgDPALevVGBJUntZXMRy/s3MzNTs7OxQXluSxlWSvVU10+6YV4pKUkNY6JLUEBa6JDWEhS5JDWGhS1JDWOiS1BAWuiQ1hIUuSQ1hoUtSQ1joktQQFrokNUQvN+eSRsqufXNs33OII8cWWDs5wdZNG9i8cXrYsaShs9A1Vnbtm2PbzgMsHD8JwNyxBbbtPABgqeuC55SLxsr2PYdeKPNTFo6fZPueQ0NKJI0OC11j5cixhb72SxcSC11jZe3kRF/7pQuJha6xsnXTBibWrHrRvok1q9i6acOQEkmjw5OiGiunTny6ykU6k4WusbN547QFLrXRdcolyUuSfCPJI0keTfL+NmN+LMlnkzye5KEk61cirCSps17m0J8Hrq+qy4ErgBuTXLNkzLuA71XVzwN/CHxosDElSd10LfRa9Fzr4ZrWVy0Zdguwo7V9L3BDkgwspSSpq55WuSRZlWQ/cBS4v6oeWjJkGngSoKpOAM8CrxhkUEnS8noq9Ko6WVVXAOuAq5NcumRIu3fjS9/Fk2RLktkks/Pz8/2nlSR11Nc69Ko6BjwA3Ljk0GHgEoAkq4GfBJ5p8+fvrqqZqpqZmpo6q8CSpPZ6WeUylWSytT0BvAF4bMmw3cA7WttvBb5aVWe8Q5ckrZxe1qFfDOxIsorFHwCfq6r7knwAmK2q3cA9wJ8keZzFd+a3rVhiSVJbXQu9qr4JbGyz/32nbf8D8LbBRpMk9cN7uUhSQ1joktQQFrokNYSFLkkNYaFLUkNY6JLUEBa6JDWEhS5JDWGhS1JDWOiS1BAWuiQ1hIUuSQ1hoUtSQ1joktQQFrokNYSFLkkNYaFLUkNY6JLUEBa6JDWEhS5JDdG10JNckuRrSQ4meTTJnW3GvC7Js0n2t77e1+65JEkrZ3UPY04Av1dVDyd5ObA3yf1V9e0l4/66qm4afERJUi+6vkOvqqeq6uHW9g+Ag8D0SgeTJPWnrzn0JOuBjcBDbQ6/NskjSb6U5Bc7/PktSWaTzM7Pz/cdVpLUWc+FnuTHgc8D76mq7y85/DDw6qq6HPgvwK52z1FVd1fVTFXNTE1NnW1mSVIbPRV6kjUslvmnqmrn0uNV9f2qeq61/UVgTZKLBppUkrSsXla5BLgHOFhVf9BhzM+0xpHk6tbzPj3IoJKk5fWyyuVa4DeAA0n2t/b9PvAqgKq6C3gr8NtJTgALwG1VVSuQV5LUQddCr6qvA+ky5qPARwcVSpLUP68UlaSGsNAlqSEsdElqCAtdkhrCQpekhrDQJakhLHRJaggLXZIawkKXpIaw0CWpISx0SWoIC12SGsJCl6SG6OX2uRqgXfvm2L7nEEeOLbB2coKtmzaweaMf0Srp3Fno59GufXNs23mAheMnAZg7tsC2nQcALHVJ58wpl/No+55DL5T5KQvHT7J9z6EhJZLUJBb6eXTk2EJf+yWpHxb6ebR2cqKv/ZLUDwv9PNq6aQMTa1a9aN/EmlVs3bRhSIkkNUnXQk9ySZKvJTmY5NEkd7YZkyT/OcnjSb6Z5MqViTveNm+c5oO3Xsb05AQBpicn+OCtl3lCVNJA9LLK5QTwe1X1cJKXA3uT3F9V3z5tzBuB17S+fhn4WOu7lti8cdoCl7Qiur5Dr6qnqurh1vYPgIPA0ka6BfhELXoQmExy8cDTSpI66msOPcl6YCPw0JJD08CTpz0+zJmlT5ItSWaTzM7Pz/eXVJK0rJ4LPcmPA58H3lNV3196uM0fqTN2VN1dVTNVNTM1NdVfUknSsnoq9CRrWCzzT1XVzjZDDgOXnPZ4HXDk3ONJknrVyyqXAPcAB6vqDzoM2w38Zmu1yzXAs1X11ABzSpK66GWVy7XAbwAHkuxv7ft94FUAVXUX8EXgTcDjwA+Bdw4+qiRpOV0Lvaq+Tvs58tPHFHDHoEJJkvrnlaKS1BAWuiQ1hIUuSQ1hoUtSQ1joktQQY/kRdH4upySdaewK3c/llKT2xm7Kxc/llKT2xq7Q/VxOSWpv7Ardz+WUpPbGrtBf/wtTZ9yHwM/llKQxK/Rd++b4/N65F91oPcBbrvJj3SRprAq93QnRAr72mJ9+JEljVeieEJWkzsaq0D0hKkmdjVWhb920gYk1q160zxOikrRorK4UPXXi08v+JelMY1XosFjqFrgknWmsplwkSZ1Z6JLUEF0LPcl/TXI0ybc6HH9dkmeT7G99vW/wMSVJ3fQyh/7HwEeBTywz5q+r6qaBJJIknZWu79Cr6n8Az5yHLJKkczCoOfTXJnkkyZeS/GKnQUm2JJlNMjs/7+X6kjRIqarug5L1wH1VdWmbYz8B/KiqnkvyJuAjVfWaHp5zHvi7ZYZcBHy3a7jRMC5ZzTl445LVnIM3rKyvrqqpdgfOudDbjH0CmKmqc/oHTTJbVTPn8hzny7hkNefgjUtWcw7eKGY95ymXJD+TJK3tq1vP+fS5Pq8kqT9dV7kk+TTwOuCiJIeBfw+sAaiqu4C3Ar+d5ASwANxWvbztlyQNVNdCr6pf63L8oywuaxy0u1fgOVfKuGQ15+CNS1ZzDt7IZe1pDl2SNPq89F+SGsJCl6SGGLlCT/KSJN9oXaj0aJL3DzvTcpKsSrIvyX3DztJJkieSHGjda2d22HmWk2Qyyb1JHktyMMlrh51pqSQbTrt30f4k30/ynmHnaifJ77b+P/pWkk8necmwM3WS5M5WzkdH6e+z3f2skvxUkvuT/E3r+z8ZZsZTRq7QgeeB66vqcuAK4MYk1ww503LuBA4OO0QPXl9VV4zautk2PgJ8uap+AbicEfy7rapDrb/LK4CrgB8CXxhyrDMkmQbezeJ1IZcCq4DbhpuqvSSXAv8SuJrFf+83Jel6geJ58sfAjUv2/VvgL1sXUf5l6/HQjVyh16LnWg/XtL5G8sxtknXAm4GPDztLE7SuOr4OuAegqv5vVR0bbqqubgD+Z1Utd9XzMK0GJpKsBl4KHBlynk7+KfBgVf2wqk4AfwX86pAzAR3vZ3ULsKO1vQPYfF5DdTByhQ4vTGPsB44C91fVQ8PO1MGHgfcCPxp2kC4K+EqSvUm2DDvMMn4OmAf+W2sa6+NJXjbsUF3cBnx62CHaqao54D8B3wGeAp6tqq8MN1VH3wKuS/KKJC8F3gRcMuRMy3llVT0F0Pr+00POA4xooVfVydavs+uAq1u/jo2UJDcBR6tq77Cz9ODaqroSeCNwR5Lrhh2og9XAlcDHqmoj8H8YkV9l20nyj4GbgT8bdpZ2WvO6twA/C6wFXpbk7cNN1V5VHQQ+BNwPfBl4BDgx1FBjaCQL/ZTWr9sPcOb81Si4Fri5de+azwDXJ/nkcCO1V1VHWt+PsjjXe/VwE3V0GDh82m9k97JY8KPqjcDDVfX3ww7SwRuA/1VV81V1HNgJ/LMhZ+qoqu6pqiur6joWpzj+ZtiZlvH3SS4GaH0/OuQ8wAgWepKpJJOt7QkW/6N8bLipzlRV26pqXVWtZ/HX7q9W1ci9+0nysiQvP7UN/AqLv96OnKr638CTSTa0dt0AfHuIkbr5NUZ0uqXlO8A1SV7aut/SDYzgSeZTkvx06/urgFsZ7b/b3cA7WtvvAP77ELO8oJdPLDrfLgZ2JFnF4g+cz1XVyC4JHAOvBL7Qun/aauBPq+rLw420rH8DfKo1nfG3wDuHnKet1jzvPwf+1bCzdFJVDyW5F3iYxemLfYzg5eqn+XySVwDHgTuq6nvDDgQd72f1H4HPJXkXiz843za8hP+fl/5LUkOM3JSLJOnsWOiS1BAWuiQ1hIUuSQ1hoUtSQ1joktQQFrokNcT/A25n/syy3KbXAAAAAElFTkSuQmCC\n",
+      "text/plain": [
+       "<Figure size 432x288 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
     }
    ],
    "source": [
     "from scipy import stats\n",
+    "import matplotlib.pyplot as plt\n",
     "x = [10.35, 6.24,3.18,8.46,3.21,7.65,4.32,8.66,9.12,10.31]\n",
     "y = [5.1, 3.15,1.67,4.33,1.76,4.11,2.11,4.88,4.99,5.12]\n",
     "correlation, pvalue = stats.stats.pearsonr(x,y)\n",
-    "print('correlation:', correlation)\n",
-    "print('pvalue:', pvalue)"
+    "print('correlation:', correlation)  # 相关系数高\n",
+    "print('pvalue:', pvalue)\n",
+    "plt.scatter(x,y)\n",
+    "plt.show()  # 类斜线"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 等级变量的相关分析\n",
+    "### 斯皮尔曼等级变量的相关分析\n",
     "当测量得到的数据不是等距或等比数据，而是具有等级顺序的数据；或者得到的数据是等距或等比数据，但其所来自的总体分布不是正态的，不满足求皮尔森相关系数(积差相关)的要求。这时就要运用等级相关系数。"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "小实验，两个基因A、B，他们的表达量关系是B=2A，在8个样本中的表达了如下：\n",
+    "<img src=\"assets/20201120223504.png\" width=\"70%\">"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "计算得出，他们的皮尔森相关系数=1，P-vlaue=0，从以上可以直观看出，如果两个基因的表达量呈线性关系，则具有显著的皮尔森相关性\n",
+    "<br><br>\n",
+    "以上是两个基因呈线性关系的结果。如果两者呈非线性关系，例如幂函数关系(曲线关系)，那又如何呢？\n",
+    "<br><br>两个基因A、D，他们的关系是D=A^10，在8个样本中的表达量值如下：\n",
+    "<img src=\"assets/20201120223836.png\" width=\"50%\">\n",
+    "<img src=\"assets/20201120223916.png\" width=\"70%\">"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "correlation: 0.7659287963138055\n",
+      "pvalue: 0.026696497208768055\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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\n",
+      "text/plain": [
+       "<Figure size 432x288 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "x = [0.6,0.7,1,2.1,2.9,3.2,5.5,6.7]\n",
+    "y = np.power(x, 10)\n",
+    "correlation, pvalue = stats.stats.pearsonr(x, y)\n",
+    "print('correlation:', correlation)  # 相关系数高\n",
+    "print('pvalue:', pvalue)\n",
+    "plt.scatter(x,y)\n",
+    "plt.show()  # 类斜线"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "可以看到,基因A、D相关系数，无论数值还是显著性都下降了。皮尔森相关系数是一种线性相关系数，因此如果两个变量呈线性关系的时候，具有最大的显著性。对于非线性关系(例如A、D的幂函数关系)，则其对相关性的检测功效会下降\n",
+    "\n",
+    "<br>这时我们可以考虑另外一个相关系数计算方法:斯皮尔曼等级相关。"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**斯皮尔曼等级相关**\n",
+    "当两个变量值以等级次序排列或以等级次序表示时,两个相应总体并不一定呈正态分布,样本容量也不一定大于30,表示这两变量之间的相关,称为 Spearman等级相关。\n",
+    "\n",
+    "<br>简单点说,就是无论两个变量的数据如何变化，符合什么样的分布，我们只关心每个数值在变量内的排列顺序。如果两个变量的对应值，在各组内的排序顺位是相同或类似的，则具有显著的相关性。\n",
+    "$$\n",
+    "p = 1-\\frac{6\\sum{d^2_i}}{n^2-n}\n",
+    "$$\n",
+    "<ul>\n",
+    "    <li>n 为等级个数\n",
+    "    <li>d 为二列成对变量的等级差数"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "将上表转换成排序等级\n",
+    "<img src=\"assets/20201120224505.png\" width=\"50%\">\n",
+    "\n",
+    "利用斯皮尔曼等级相关计算A、D基因表达量的相关性，结果是:r=1, p-value=4.96e-05\n",
+    "\n",
+    "这里斯皮尔曼等级相关的显著性显然高于皮尔森相关。这是因为虽然两个基因的表达量是非线性关系，但两个基因表达量在所有样本中的排列顺序是完全相同的，因为具有极显著的斯皮尔曼等级相关性。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "correlation: 0.9999999999999999\n",
+      "pvalue: 6.646897422032013e-64\n"
+     ]
+    }
+   ],
+   "source": [
+    "x = [10.35,6.24,3.18,8.46,3.21,7.65,4.32,8.66,9.12,10.31]\n",
+    "y = [5.13,3.15,1.67,4.33,1.76,4.11,2.11,4.88,4.99,5.12]\n",
+    "correlation, pvalue = stats.stats.spearmanr(x, y)\n",
+    "print('correlation:', correlation)  # 相关系数高\n",
+    "print('pvalue:', pvalue)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[10.  4.  1.  6.  2.  5.  3.  7.  8.  9.] [10.  4.  1.  6.  2.  5.  3.  7.  8.  9.]\n",
+      "correlation: 0.9999999999999999\n",
+      "pvalue: 6.646897422032013e-64\n"
+     ]
+    }
+   ],
+   "source": [
+    "import scipy\n",
+    "x = [10.35,6.24,3.18,8.46,3.21,7.65,4.32,8.66,9.12,10.31]\n",
+    "y = [5.13,3.15,1.67,4.33,1.76,4.11,2.11,4.88,4.99,5.12]\n",
+    "x = scipy.stats.stats.rankdata(x)  # 原本的数据转换成等级数据\n",
+    "y = scipy.stats.stats.rankdata(y)\n",
+    "print(x, y)  # 转换成等级后，可以看到等级是极其相似的。\n",
+    "# x内3.18是最小的所以是1，y内1.67是最小的所以是1，其它值同样\n",
+    "\n",
+    "correlation, pvalue = stats.stats.spearmanr(x, y)\n",
+    "print('correlation:', correlation)  # 相关系数高\n",
+    "print('pvalue:', pvalue)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**斯皮尔曼实例**\n",
+    "10名高三学生学习潜在能力测验与自学能力测验成绩如下表所示，问两者相关情况如何？\n",
+    "<img src=\"assets/20201120225540.png\" width=\"50%\">\n",
+    "计算等级相关系数\n",
+    "$$\n",
+    "r_R = 1-\\frac{2\\sum D^2}{n(n^2-1)}\n",
+    "= 1-\\frac{6*18}{10(10^2-1)}\n",
+    "=0.891\n",
+    "$$\n",
+    "**等级相关系数的显著性检验**\n",
+    "与积差相关系数检验的方法相同\n",
+    "10个学生学习潜在能力与自学能力测验成绩相关系数为0.891，问总体上说，两者是否存在相关？\n",
+    "\n",
+    "计算检验统计量的值：\n",
+    "$$\n",
+    "t = \\frac{r\\sqrt{n-2}}{\\sqrt{1-r^2}} \n",
+    "= \\frac{0.891\\sqrt{10-2}}{\\sqrt{1-0.891^2}}\n",
+    "= 5.551\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "5.551 > t_{(8)0.01} = 3.355\n",
+    "$$\n",
+    "所以学生的学习潜在学习能力与自学能力有较强的正相关。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
   }
  ],
  "metadata": {
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diff --git a/notebook_必备数学基础/相关分析/相关分析.ipynb b/notebook_必备数学基础/相关分析/相关分析.ipynb
index 0f44ecc..f49ec71 100644
--- a/notebook_必备数学基础/相关分析/相关分析.ipynb
+++ b/notebook_必备数学基础/相关分析/相关分析.ipynb
@@ -223,7 +223,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 10,
    "metadata": {},
    "outputs": [
     {
@@ -233,24 +233,220 @@
       "correlation: 0.9891763198690562\n",
       "pvalue: 5.926875946481138e-08\n"
      ]
+    },
+    {
+     "data": {
+      "image/png": "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\n",
+      "text/plain": [
+       "<Figure size 432x288 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
     }
    ],
    "source": [
     "from scipy import stats\n",
+    "import matplotlib.pyplot as plt\n",
     "x = [10.35, 6.24,3.18,8.46,3.21,7.65,4.32,8.66,9.12,10.31]\n",
     "y = [5.1, 3.15,1.67,4.33,1.76,4.11,2.11,4.88,4.99,5.12]\n",
     "correlation, pvalue = stats.stats.pearsonr(x,y)\n",
-    "print('correlation:', correlation)\n",
-    "print('pvalue:', pvalue)"
+    "print('correlation:', correlation)  # 相关系数高\n",
+    "print('pvalue:', pvalue)\n",
+    "plt.scatter(x,y)\n",
+    "plt.show()  # 类斜线"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### 等级变量的相关分析\n",
+    "### 斯皮尔曼等级变量的相关分析\n",
     "当测量得到的数据不是等距或等比数据，而是具有等级顺序的数据；或者得到的数据是等距或等比数据，但其所来自的总体分布不是正态的，不满足求皮尔森相关系数(积差相关)的要求。这时就要运用等级相关系数。"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "小实验，两个基因A、B，他们的表达量关系是B=2A，在8个样本中的表达了如下：\n",
+    "<img src=\"assets/20201120223504.png\" width=\"70%\">"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "计算得出，他们的皮尔森相关系数=1，P-vlaue=0，从以上可以直观看出，如果两个基因的表达量呈线性关系，则具有显著的皮尔森相关性\n",
+    "<br><br>\n",
+    "以上是两个基因呈线性关系的结果。如果两者呈非线性关系，例如幂函数关系(曲线关系)，那又如何呢？\n",
+    "<br><br>两个基因A、D，他们的关系是D=A^10，在8个样本中的表达量值如下：\n",
+    "<img src=\"assets/20201120223836.png\" width=\"50%\">\n",
+    "<img src=\"assets/20201120223916.png\" width=\"70%\">"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "correlation: 0.7659287963138055\n",
+      "pvalue: 0.026696497208768055\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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\n",
+      "text/plain": [
+       "<Figure size 432x288 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "x = [0.6,0.7,1,2.1,2.9,3.2,5.5,6.7]\n",
+    "y = np.power(x, 10)\n",
+    "correlation, pvalue = stats.stats.pearsonr(x, y)\n",
+    "print('correlation:', correlation)  # 相关系数高\n",
+    "print('pvalue:', pvalue)\n",
+    "plt.scatter(x,y)\n",
+    "plt.show()  # 类斜线"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "可以看到,基因A、D相关系数，无论数值还是显著性都下降了。皮尔森相关系数是一种线性相关系数，因此如果两个变量呈线性关系的时候，具有最大的显著性。对于非线性关系(例如A、D的幂函数关系)，则其对相关性的检测功效会下降\n",
+    "\n",
+    "<br>这时我们可以考虑另外一个相关系数计算方法:斯皮尔曼等级相关。"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**斯皮尔曼等级相关**\n",
+    "当两个变量值以等级次序排列或以等级次序表示时,两个相应总体并不一定呈正态分布,样本容量也不一定大于30,表示这两变量之间的相关,称为 Spearman等级相关。\n",
+    "\n",
+    "<br>简单点说,就是无论两个变量的数据如何变化，符合什么样的分布，我们只关心每个数值在变量内的排列顺序。如果两个变量的对应值，在各组内的排序顺位是相同或类似的，则具有显著的相关性。\n",
+    "$$\n",
+    "p = 1-\\frac{6\\sum{d^2_i}}{n^2-n}\n",
+    "$$\n",
+    "<ul>\n",
+    "    <li>n 为等级个数\n",
+    "    <li>d 为二列成对变量的等级差数"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "将上表转换成排序等级\n",
+    "<img src=\"assets/20201120224505.png\" width=\"50%\">\n",
+    "\n",
+    "利用斯皮尔曼等级相关计算A、D基因表达量的相关性，结果是:r=1, p-value=4.96e-05\n",
+    "\n",
+    "这里斯皮尔曼等级相关的显著性显然高于皮尔森相关。这是因为虽然两个基因的表达量是非线性关系，但两个基因表达量在所有样本中的排列顺序是完全相同的，因为具有极显著的斯皮尔曼等级相关性。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "correlation: 0.9999999999999999\n",
+      "pvalue: 6.646897422032013e-64\n"
+     ]
+    }
+   ],
+   "source": [
+    "x = [10.35,6.24,3.18,8.46,3.21,7.65,4.32,8.66,9.12,10.31]\n",
+    "y = [5.13,3.15,1.67,4.33,1.76,4.11,2.11,4.88,4.99,5.12]\n",
+    "correlation, pvalue = stats.stats.spearmanr(x, y)\n",
+    "print('correlation:', correlation)  # 相关系数高\n",
+    "print('pvalue:', pvalue)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[10.  4.  1.  6.  2.  5.  3.  7.  8.  9.] [10.  4.  1.  6.  2.  5.  3.  7.  8.  9.]\n",
+      "correlation: 0.9999999999999999\n",
+      "pvalue: 6.646897422032013e-64\n"
+     ]
+    }
+   ],
+   "source": [
+    "import scipy\n",
+    "x = [10.35,6.24,3.18,8.46,3.21,7.65,4.32,8.66,9.12,10.31]\n",
+    "y = [5.13,3.15,1.67,4.33,1.76,4.11,2.11,4.88,4.99,5.12]\n",
+    "x = scipy.stats.stats.rankdata(x)  # 原本的数据转换成等级数据\n",
+    "y = scipy.stats.stats.rankdata(y)\n",
+    "print(x, y)  # 转换成等级后，可以看到等级是极其相似的。\n",
+    "# x内3.18是最小的所以是1，y内1.67是最小的所以是1，其它值同样\n",
+    "\n",
+    "correlation, pvalue = stats.stats.spearmanr(x, y)\n",
+    "print('correlation:', correlation)  # 相关系数高\n",
+    "print('pvalue:', pvalue)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**斯皮尔曼实例**\n",
+    "10名高三学生学习潜在能力测验与自学能力测验成绩如下表所示，问两者相关情况如何？\n",
+    "<img src=\"assets/20201120225540.png\" width=\"50%\">\n",
+    "计算等级相关系数\n",
+    "$$\n",
+    "r_R = 1-\\frac{2\\sum D^2}{n(n^2-1)}\n",
+    "= 1-\\frac{6*18}{10(10^2-1)}\n",
+    "=0.891\n",
+    "$$\n",
+    "**等级相关系数的显著性检验**\n",
+    "与积差相关系数检验的方法相同\n",
+    "10个学生学习潜在能力与自学能力测验成绩相关系数为0.891，问总体上说，两者是否存在相关？\n",
+    "\n",
+    "计算检验统计量的值：\n",
+    "$$\n",
+    "t = \\frac{r\\sqrt{n-2}}{\\sqrt{1-r^2}} \n",
+    "= \\frac{0.891\\sqrt{10-2}}{\\sqrt{1-0.891^2}}\n",
+    "= 5.551\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "5.551 > t_{(8)0.01} = 3.355\n",
+    "$$\n",
+    "所以学生的学习潜在学习能力与自学能力有较强的正相关。"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
   }
  ],
  "metadata": {