central_limit_theorem_and_standard_error.ipynb

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  "metadata": {
    "colab": {
      "provenance": [],
      "authorship_tag": "ABX9TyMp13shpIoQ//quPlzVLrV2"
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "This notebook investigates the Central Limit Theorem (CLT) for both observational data and discrete random variables. Also, the standard error is briefly discussed at the end."
      ],
      "metadata": {
        "id": "D68mbOCL53Fh"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# The Central Limit Theorem (CLT)\n",
        "\n",
        "When drawing a sample of size n from a population with mean $\\mu$ and variance $\\sigma^2$, the distribution of the sample means approaches a normal distribution $N(\\mu, \\sigma^2/n)$ as n increases.\n",
        "\n",
        "Example:\n",
        "- For a sample of size 5, the variance of the sample means is $\\frac{\\sigma^2}{5}$\n",
        "- For a sample of size 100, the variance of the sample means is $\\frac{\\sigma^2}{100}$\n",
        "\n",
        "In the following cells, we will investigate whether the CLT holds true for observational data and discrete random variables. We will observe how the variance of the sample means decreases as the sample size n increases, verifying the behavior predicted by the CLT."
      ],
      "metadata": {
        "id": "B9v7czQM87Kq"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Notation for Population and Sample\n",
        "\n",
        "Before we proceed to explore the CLT through examples, let's quickly review the notation and formulas used to describe population and sample statistics.\n",
        "\n",
        "Population:\n",
        "\n",
        "- Size: $N$\n",
        "- Mean: $$\\mu = \\frac{1}{N} \\sum_{i=1}^{N} x_i$$\n",
        "- Variance: $$\\sigma^2 = \\frac{1}{N} \\sum_{i=1}^{N} (x_i - \\mu)^2$$\n",
        "- Standard deviation: $$\\sigma = \\sqrt{\\sigma^2}$$\n",
        "\n",
        "\n",
        "Sample:\n",
        "\n",
        "- Size: $n$\n",
        "- Mean: $$\\bar{X} = \\frac{1}{n} \\sum_{i=1}^{n} x_i$$\n",
        "- Variance (biased): $$s_n^2 = \\frac{1}{n} \\sum_{i=1}^{n} (x_i - \\bar{X})^2$$\n",
        "- Variance (unbiased): $$s_{n-1}^2 = \\frac{1}{n-1} \\sum_{i=1}^{n} (x_i - \\bar{X})^2$$\n",
        "- Standard deviation: $$s = \\sqrt{s_{n-1}^2}$$\n",
        "\n"
      ],
      "metadata": {
        "id": "ayMSSwp4-b5P"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Import libs\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "plt.rcParams[\"figure.figsize\"] = (5, 3)"
      ],
      "metadata": {
        "id": "B81eXysmIRC6"
      },
      "execution_count": 16,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# 1. Verification of the Central Limit Theorem with Observational Data\n",
        "\n",
        "def variance_observational_data(n):\n",
        "  mu = 100 # populcation mean\n",
        "  sigma = 5 # population standard deviation\n",
        "  pop_var = sigma ** 2 # population var\n",
        "  attempt = 1_000 # number of attempts\n",
        "\n",
        "  # to store sample means\n",
        "  sample_means = []\n",
        "\n",
        "  for _ in range(attempt):\n",
        "    sample = np.random.normal(mu, sigma, n)\n",
        "    sample_mean = np.mean(sample)\n",
        "    sample_means.append(sample_mean)\n",
        "\n",
        "  plt.hist(sample_means)\n",
        "  plt.axvline(mu, color='red')  # draw population mean\n",
        "\n",
        "  print(f\"Expected variance (σ^2/n): {pop_var/n:.3f}\")\n",
        "  print(f\"Actual variance of sample means: {np.var(sample_means):.3f}\")\n",
        "\n",
        "n = 5 # sample size\n",
        "variance_observational_data(n)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 327
        },
        "id": "HzeGp-YiGI4n",
        "outputId": "11fc45ed-e493-4748-957f-d0d4365eecae"
      },
      "execution_count": 19,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Expected variance (σ^2/n): 5.000\n",
            "Actual variance of sample means: 4.909\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 500x300 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "n = 100 # sample size\n",
        "variance_observational_data(n)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 327
        },
        "id": "93cZA8DqoTcw",
        "outputId": "b46f6506-7dbc-483c-f606-5f18544567e7"
      },
      "execution_count": 23,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Expected variance (σ^2/n): 0.250\n",
            "Actual variance of sample means: 0.244\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 500x300 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# 2. Verification of the Central Limit Theorem with discrete random variables\n",
        "\n",
        "def variance_discrete_random_variable(n):\n",
        "  attempt = 1_000 # number of attempts\n",
        "\n",
        "  # store possible values of a die in an array\n",
        "  dice_data = np.arange(1, 7)\n",
        "\n",
        "  # to store sample means\n",
        "  sample_means = np.array([])\n",
        "\n",
        "  for _ in range(attempt):\n",
        "    sample_means = np.append(sample_means, np.random.choice(dice_data, n).mean())\n",
        "\n",
        "  # expected value and variance of a die roll\n",
        "  E = dice_data.sum() / len(dice_data)\n",
        "  pop_var = np.square(dice_data - E).sum() / len(dice_data)\n",
        "\n",
        "  plt.hist(sample_means)\n",
        "  plt.axvline(E, color='red') # draw expected value\n",
        "\n",
        "  print(f\"Expected variance (σ^2/n): {pop_var/n:.3f}\")\n",
        "  print(f\"Actual variance of sample means: {sample_means.var():.3f}\")\n",
        "\n",
        "n = 5 # sample size\n",
        "variance_discrete_random_variable(n)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 327
        },
        "id": "nA3v8KGc8dtT",
        "outputId": "c4df05a4-0738-4c24-e669-ea7fd31b1f21"
      },
      "execution_count": 24,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Expected variance (σ^2/n): 0.583\n",
            "Actual variance of sample means: 0.574\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 500x300 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "n = 100 # sample size\n",
        "variance_discrete_random_variable(n)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 327
        },
        "id": "F07Ns7txou_p",
        "outputId": "17ebb46a-4023-4a74-c1bd-9ddfffb3ce66"
      },
      "execution_count": 25,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Expected variance (σ^2/n): 0.029\n",
            "Actual variance of sample means: 0.028\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 500x300 with 1 Axes>"
            ],
            "image/png": 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ChAmaNWuWVqxYodbWVuXn52vq1KlciQkACImAw+7jjz/W7bff7lsuKCiQJM2YMUMrV67UY489puPHj2v27Nk6cuSIxowZo4qKCvXt29d3n1WrVik/P18ZGRnq1auXcnJy9NxzzwVhdwAA+DGbZVlWpIsIlMfjkcPhUEtLC+fvQmTw/HWRLqFH63fqpPYt+4Ukaejcf+h/o/ue4x74vq+WZEe6BITJ+eZBt7gaEwCAC0HYAQCMR9gBAIxH2AEAjEfYAQCMR9gBAIxH2AEAjEfYAQCMR9gBAIxH2AEAjEfYAQCMR9gBAIxH2AEAjEfYAQCMR9gBAIxH2AEAjEfYAQCMR9gBAIxH2AEAjEfYAQCMR9gBAIxH2AEAjNc70gUAQLANnr8u0iX4+WpJdqRL6PE4sgMAGI+wAwAYj7ADABiPsAMAGI+wAwAYj7ADABiPsAMAGI+wAwAYL+hht2jRItlsNr+Wmprq6z958qTy8vI0YMAAXXrppcrJyVFzc3OwywAAwCckR3Y//elPdfDgQV/btm2br2/u3Ll677339NZbb6mqqkoHDhzQ3XffHYoyAACQFKKvC+vdu7ecTueP1re0tOiVV15RWVmZ7rjjDknSq6++qqFDh2r79u265ZZbQlEOAKCHC8mR3RdffKHk5GRdffXVys3NVWNjoySppqZGra2tyszM9I1NTU3VVVddperq6jNuz+v1yuPx+DUAAM5X0MMuPT1dK1euVEVFhZYvX66GhgbddtttOnr0qNxut6KjoxUXF+d3n8TERLnd7jNus6SkRA6Hw9dSUlKCXTYAwGBBfxtz4sSJvttpaWlKT0/XoEGD9Pe//139+vXr0jYLCwtVUFDgW/Z4PAQeAOC8hfyjB3FxcbruuutUX18vp9OpU6dO6ciRI35jmpubOz3Hd5rdbldsbKxfAwDgfIU87I4dO6b9+/crKSlJI0eOVJ8+fbRp0yZff11dnRobG+VyuUJdCgCghwr625iPPPKIJk+erEGDBunAgQMqLi5WVFSU7r33XjkcDs2cOVMFBQWKj49XbGysHnroIblcLq7EBACETNDD7uuvv9a9996rb7/9VpdffrnGjBmj7du36/LLL5ckLVu2TL169VJOTo68Xq+ysrL0wgsvBLsMAAB8gh52q1evPmt/3759VVpaqtLS0mA/NAAAneK7MQEAxiPsAADGI+wAAMYj7AAAxiPsAADGI+wAAMYj7AAAxiPsAADGI+wAAMYj7AAAxiPsAADGI+wAAMYL+hdBo+sGz18X6RIAwEgc2QEAjEfYAQCMR9gBAIxH2AEAjEfYAQCMx9WYABBiF9OV1l8tyY50CRHBkR0AwHiEHQDAeIQdAMB4hB0AwHiEHQDAeIQdAMB4hB0AwHg9/nN2F9PnXwAAocGRHQDAeIQdAMB4hB0AwHiEHQDAeIQdAMB4EQu70tJSDR48WH379lV6ero++uijSJUCADBcRMLuzTffVEFBgYqLi/XJJ5/ohhtuUFZWlg4dOhSJcgAAhovI5+z+/Oc/a9asWfr1r38tSVqxYoXWrVunv/71r5o/f/6Pxnu9Xnm9Xt9yS0uLJMnj8VxwLR3eExe8DSDY2k+d1Olnd7v3hDqsjojWA3NcNfetSJfg89njWRe8jdM5YFnW2QdaYeb1eq2oqChrzZo1fuunT59u3XnnnZ3ep7i42JJEo9FoNFqnramp6azZE/Yju2+++Ubt7e1KTEz0W5+YmKj//Oc/nd6nsLBQBQUFvuWOjg4dPnxYAwYMkM1mC0pdHo9HKSkpampqUmxsbFC22R0xD99hHr7DPDAHp12s82BZlo4ePark5OSzjusWXxdmt9tlt9v91sXFxYXksWJjYy+qf8hIYR6+wzx8h3lgDk67GOfB4XCcc0zYL1AZOHCgoqKi1Nzc7Le+ublZTqcz3OUAAHqAsIdddHS0Ro4cqU2bNvnWdXR0aNOmTXK5XOEuBwDQA0TkbcyCggLNmDFDo0aN0s9+9jM9++yzOn78uO/qzEiw2+0qLi7+0dulPQ3z8B3m4TvMA3NwWnefB5tlnet6zdD4y1/+oqefflput1s33nijnnvuOaWnp0eiFACA4SIWdgAAhAvfjQkAMB5hBwAwHmEHADAeYQcAMF6PCLvly5crLS3N98l/l8ulf/7zn2e9z1tvvaXU1FT17dtXI0aM0Pr168NUbegEOg8vvfSSbrvtNl122WW67LLLlJmZacRPMXXl+XDa6tWrZbPZNGXKlNAWGQZdmYcjR44oLy9PSUlJstvtuu6667r130ZX5uDZZ5/V9ddfr379+iklJUVz587VyZMnw1RxeCxZskQ2m01z5sw567hu9ToZhO92vui9++671rp166zPP//cqqursxYsWGD16dPH+uyzzzod/8EHH1hRUVHW0qVLrb1791pFRUVWnz59rN27d4e58uAKdB7uu+8+q7S01Pr000+tffv2Wffff7/lcDisr7/+OsyVB1eg83BaQ0ODdcUVV1i33Xabddddd4Wn2BAKdB68Xq81atQoa9KkSda2bdushoYGa8uWLVZtbW2YKw+eQOdg1apVlt1ut1atWmU1NDRYGzZssJKSkqy5c+eGufLQ+eijj6zBgwdbaWlp1sMPP3zGcd3tdbJHhF1nLrvsMuvll1/utO+ee+6xsrOz/dalp6dbv/nNb8JRWlidbR5+qK2tzYqJibFee+21EFcVfueah7a2NuvWW2+1Xn75ZWvGjBlGhF1nzjYPy5cvt66++mrr1KlTYa4qvM42B3l5edYdd9zht66goMAaPXp0OEoLuaNHj1rXXnutVVlZaf3P//zPWcOuu71O9oi3Mb+vvb1dq1ev1vHjx8/49WTV1dXKzMz0W5eVlaXq6upwlBgW5zMPP3TixAm1trYqPj4+xNWFz/nOwxNPPKGEhATNnDkzjNWFz/nMw7vvviuXy6W8vDwlJiZq+PDheuqpp9Te3h7makPjfObg1ltvVU1Nje/t/C+//FLr16/XpEmTwllqyOTl5Sk7O/tHr3+d6W6vk93iVw+CYffu3XK5XDp58qQuvfRSrVmzRsOGDet0rNvt7vQniNxudzhKDalA5uGH5s2bp+Tk5PP6Q7jYBTIP27Zt0yuvvKLa2trwFhkGgczDl19+qc2bNys3N1fr169XfX29fvvb36q1tVXFxcVhrjx4ApmD++67T998843GjBkjy7LU1tamBx54QAsWLAhz1cG3evVqffLJJ9q5c+d5je9ur5M95sju+uuvV21trXbs2KEHH3xQM2bM0N69eyNdVth1dR6WLFmi1atXa82aNerbt28YKg2t852Ho0ePatq0aXrppZc0cODACFQaWoE8Hzo6OpSQkKAXX3xRI0eO1C9/+Uv9/ve/14oVK8JcdXAFMgdbtmzRU089pRdeeEGffPKJ3n77ba1bt06LFy8Oc9XB1dTUpIcfflirVq0y4u+7U5F+HzVSMjIyrNmzZ3fal5KSYi1btsxv3cKFC620tLQwVBZeZ5uH055++mnL4XBYO3fuDFNV4Xemefj0008tSVZUVJSv2Ww2y2azWVFRUVZ9fX0Eqg2dsz0fxo4da2VkZPitW79+vSXJ8nq94SgvLM42B2PGjLEeeeQRv3V/+9vfrH79+lnt7e3hKC8k1qxZ86PnuSTf87ytre1H9+lur5M95sjuhzo6OuT1ejvtc7lcfj9BJEmVlZVG/gTR2eZBkpYuXarFixeroqJCo0aNCmNl4XWmeUhNTdXu3btVW1vra3feeaduv/121dbWKiUlJQLVhs7Zng+jR49WfX29Ojo6fOs+//xzJSUlKTo6OlwlhtzZ5uDEiRPq1cv/ZTMqKkrSd7+Y3V1lZGT86Hk+atQo5ebmqra21reP39ftXicjnbbhMH/+fKuqqspqaGiwdu3aZc2fP9+y2WzWxo0bLcuyrGnTplnz58/3jf/ggw+s3r17W88884y1b98+q7i4+KK+pPZ8BToPS5YssaKjo61//OMf1sGDB33t6NGjkdqFoAh0Hn7IlKsxA52HxsZGKyYmxsrPz7fq6uqs8vJyKyEhwXryyScjtQsXLNA5KC4utmJiYqw33njD+vLLL62NGzda11xzjXXPPfdEahdC5odXY3b318kecYHKoUOHNH36dB08eFAOh0NpaWnasGGDfv7zn0uSGhsb/f63duutt6qsrExFRUVasGCBrr32Wq1du1bDhw+P1C4ERaDzsHz5cp06dUq/+MUv/LZTXFysRYsWhbP0oAp0HkwV6DykpKRow4YNmjt3rtLS0nTFFVfo4Ycf1rx58yK1Cxcs0DkoKiqSzWZTUVGR/vvf/+ryyy/X5MmT9cc//jFSuxA23f11kp/4AQAYz/z/vgIAejzCDgBgPMIOAGA8wg4AYDzCDgBgPMIOAGA8wg4AYDzCDgBgPMIOAGA8wg4AYDzCDgBgvP8DF4bOdzGUHEMAAAAASUVORK5CYII=\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Standard Error (SE)\n",
        "\n",
        "The above experiments confirm that the CLT holds for observational data and discrete random variables.\n",
        "\n",
        "As the sample size increases, the variance of the sample means approaches $\\frac{\\sigma^2}{n}$, so the standard deviation of the sample means approaches $\\frac{\\sigma}{\\sqrt{n}}$.\n",
        "\n",
        "We take n samples from a population and examine the sample mean. How different is the obtained sample mean from the population mean?\n",
        "The difference between the mean of a sample of size n and the population mean can be estimated by the standard deviation of the sample means, known as the standard error (SE).\n",
        "\n",
        "$$\\text{Standard deviation of the sample means} = \\text{Standard Error (SE)}$$\n",
        "\n",
        "However, since the population variance is usually unknown, we use the unbiased sample variance $s_{n-1}^2$ to calculate the SE. It has been shown by the CLT that the standard deviation of the sample means approaches $\\frac{\\sigma}{\\sqrt{n}}$, so:\n",
        "\n",
        "$$SE = \\sqrt{\\frac{s_{n-1}^2}{n}} = \\frac{s_{n-1}}{\\sqrt{n}}$$\n",
        "\n",
        "The standard error is widely used in statistical inference such as t-tests."
      ],
      "metadata": {
        "id": "-wU_Z8LURIGR"
      }
    }
  ]
}