011-variance.ipynb

{
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  {
   "metadata": {},
   "cell_type": "markdown",
   "source": "# Descriptive Statistics: Measures of Dispersion (Absolute)\n",
   "id": "a182cc33dc11de41"
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  {
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   "cell_type": "markdown",
   "source": "## Variance",
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   "source": [
    "Variance is the average of the squared differences from the mean.\n",
    "\n",
    "Variance measures how \"spread-out\" the data is."
   ],
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  },
  {
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   "cell_type": "markdown",
   "source": "### Formula (variance)",
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   "source": [
    "#### Population Variance\n",
    "\n",
    "$$\n",
    "\\sigma^2 = \\frac{\\sum_{i=1}^{N} (x_i - \\mu)^2}{N}\n",
    "$$"
   ],
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   "source": [
    "Caclulate the difference between each data point ($x_i$) and the mean ($\\mu$), square the result, and sum them all up.\n",
    "\n",
    "Finally, divide by the number of data points ($N$)."
   ],
   "id": "c17933dc940dd583"
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  {
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   "source": [
    "#### Sample Variance\n",
    "\n",
    "$$\n",
    "s^2 = \\frac{\\sum_{i=1}^{n} (x_i - \\bar{x})^2}{n-1}\n",
    "$$"
   ],
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  {
   "metadata": {},
   "cell_type": "markdown",
   "source": "### Examples (variance)",
   "id": "8e9faf29d5c1b9a"
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  {
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   },
   "cell_type": "code",
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "from scipy import stats\n",
    "import statistics as sts"
   ],
   "id": "35f27210d9eac1dd",
   "outputs": [],
   "execution_count": 23
  },
  {
   "metadata": {},
   "cell_type": "markdown",
   "source": "#### Example 1 (variance)",
   "id": "a403aa77bd04d524"
  },
  {
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    }
   },
   "cell_type": "code",
   "source": "ex_one = [0, 1, 3, 5, 6]",
   "id": "7deca74a4d8c38dc",
   "outputs": [],
   "execution_count": 24
  },
  {
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    }
   },
   "cell_type": "code",
   "source": "sts.pvariance(ex_one) # population variance",
   "id": "59a6c00088a900d4",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.2"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "execution_count": 25
  },
  {
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    }
   },
   "cell_type": "code",
   "source": "sts.variance(ex_one) # sample variance",
   "id": "c40db03d4ef856e1",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.5"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "execution_count": 26
  },
  {
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   },
   "cell_type": "code",
   "source": "np.var(ex_one) # population variance",
   "id": "872fd3399cc08f54",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(5.2)"
      ]
     },
     "execution_count": 35,
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    }
   ],
   "execution_count": 35
  },
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   },
   "cell_type": "code",
   "source": "stats.tvar(ex_one) # sample variance",
   "id": "a1c807b598d46216",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(6.5)"
      ]
     },
     "execution_count": 34,
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   ],
   "execution_count": 34
  },
  {
   "metadata": {},
   "cell_type": "markdown",
   "source": "#### Example 2 (variance)",
   "id": "4488d88aae332814"
  },
  {
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   },
   "cell_type": "code",
   "source": "ex_two = [2, 3, 4, 3, 3]",
   "id": "6793946ae05f5bb4",
   "outputs": [],
   "execution_count": 43
  },
  {
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    }
   },
   "cell_type": "code",
   "source": "sts.pvariance(ex_two)",
   "id": "71ea8327ffb42274",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "execution_count": 44
  },
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   },
   "cell_type": "code",
   "source": "sts.variance(ex_two)",
   "id": "c92d70addb472cec",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "execution_count": 45
  },
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   "cell_type": "code",
   "source": "np.var(ex_two)",
   "id": "41895199e61ce0cc",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(0.4)"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "execution_count": 46
  },
  {
   "metadata": {},
   "cell_type": "markdown",
   "source": "#### Example 3 (variance)",
   "id": "f27f9170a8225dca"
  },
  {
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     "start_time": "2025-01-03T17:00:53.703454Z"
    }
   },
   "cell_type": "code",
   "source": "hurricanes = [18, 21, 13, 19, 24, 17, 14, 12, 15, 14]",
   "id": "1fa1bafafe21a2a0",
   "outputs": [],
   "execution_count": 78
  },
  {
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     "start_time": "2025-01-03T17:04:26.126371Z"
    }
   },
   "cell_type": "code",
   "source": [
    "hurr_mean = np.mean(hurricanes)\n",
    "hurr_sub = np.subtract(hurricanes, hurr_mean)\n",
    "np.sum(np.square(hurr_sub)) / 10"
   ],
   "id": "65ce4f05372e9d2c",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(13.210000000000003)"
      ]
     },
     "execution_count": 86,
     "metadata": {},
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    }
   ],
   "execution_count": 86
  },
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   },
   "cell_type": "code",
   "source": "sts.variance(hurricanes)",
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   "outputs": [
    {
     "data": {
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       "14.677777777777777"
      ]
     },
     "execution_count": 81,
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   ],
   "execution_count": 81
  },
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   },
   "cell_type": "code",
   "source": "sts.pvariance(hurricanes)",
   "id": "73e3c20e435027bc",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.21"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "execution_count": 82
  },
  {
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    }
   },
   "cell_type": "code",
   "source": "np.var(hurricanes)",
   "id": "dbff0532c68c5603",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(13.210000000000003)"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "execution_count": 83
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   },
   "cell_type": "code",
   "source": "stats.tvar(hurricanes)",
   "id": "72ff584b3bcad622",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(14.67777777777778)"
      ]
     },
     "execution_count": 84,
     "metadata": {},
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   ],
   "execution_count": 84
  },
  {
   "metadata": {},
   "cell_type": "markdown",
   "source": "#### Example 4 (variance)",
   "id": "68577d39d423ff63"
  },
  {
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    }
   },
   "cell_type": "code",
   "outputs": [],
   "execution_count": 63,
   "source": "students = [18, 22, 25, 26, 15]",
   "id": "5cddd62c1f4fb7ce"
  },
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     "end_time": "2025-01-03T17:08:19.869058Z",
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   },
   "cell_type": "code",
   "source": [
    "stud_mean = np.mean(students)\n",
    "stud_sub = np.subtract(students, stud_mean)\n",
    "np.sum(np.square(stud_sub)) / 5\n"
   ],
   "id": "27d143aa3611ce66",
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(17.36)"
      ]
     },
     "execution_count": 87,
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    }
   ],
   "execution_count": 87
  }
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