Day 9.ipynb

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      "source": [
        "# Demonstration of Central Limit Theorem"
      ]
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      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "import seaborn as sns\n",
        "from scipy.stats import truncnorm\n",
        "from tqdm.auto import tqdm\n",
        "\n",
        "def get_data(low, high, mean, std, size):\n",
        "    data = np.round(truncnorm((low-mean)/std, (high-mean)/std, mean, std).rvs(size))\n",
        "    return data\n",
        "\n",
        "np.random.seed(seed=0) # for reproducibility\n",
        "lazy = get_data(0, 100, 30, 15, 400000)\n",
        "hardworking = get_data(0, 100, 75, 10, 600000)\n",
        "all_data = np.concatenate([lazy, hardworking])"
      ]
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      "source": [
        "## Part 1: Description of the data\n",
        "\n",
        "- Note: this dataset is randomly generated, does not reflect reality.\n",
        "\n",
        "- Assume that we can obtain the math exam scores from all junior high school students studying in a country.\n",
        "\n",
        "- This is a called a **population.**\n",
        "\n",
        "- Let's look at the distribution of math grades for this population."
      ]
    },
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          "height": 314
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      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 1500x300 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ],
      "source": [
        "plt.figure(figsize=(15,3))\n",
        "plt.hist(all_data, bins=50)\n",
        "plt.xticks(np.arange(0,101,5))\n",
        "plt.yticks([])\n",
        "plt.grid()\n",
        "plt.title('Distribution of grades')\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "2c708b94",
      "metadata": {
        "papermill": {
          "duration": 0.015284,
          "end_time": "2022-02-27T23:26:58.209200",
          "exception": false,
          "start_time": "2022-02-27T23:26:58.193916",
          "status": "completed"
        },
        "tags": [],
        "id": "2c708b94"
      },
      "source": [
        "- This distribution actually makes a lot sense because\n",
        "\n",
        "    1. The population contains some lazy students with a mean of 30\n",
        "    2. The population contains many hard-working students with a mean of 75\n",
        "    \n",
        "- We can also compute the mean and standard deviation of our population:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "id": "b6cdd8d1",
      "metadata": {
        "execution": {
          "iopub.execute_input": "2022-02-27T23:26:58.243478Z",
          "iopub.status.busy": "2022-02-27T23:26:58.242410Z",
          "iopub.status.idle": "2022-02-27T23:26:58.259034Z",
          "shell.execute_reply": "2022-02-27T23:26:58.259673Z",
          "shell.execute_reply.started": "2021-12-18T19:58:57.767454Z"
        },
        "papermill": {
          "duration": 0.035768,
          "end_time": "2022-02-27T23:26:58.259880",
          "exception": false,
          "start_time": "2022-02-27T23:26:58.224112",
          "status": "completed"
        },
        "tags": [],
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "b6cdd8d1",
        "outputId": "b4bdff9e-c387-4516-fa59-8f2e6ca332f3"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The mean of the data is 57.24\n",
            "The standard deviation of the data is 24.55\n"
          ]
        }
      ],
      "source": [
        "print('The mean of the data is', np.round(np.mean(all_data),2))\n",
        "print('The standard deviation of the data is', np.round(np.std(all_data),2))"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "eea75a11",
      "metadata": {
        "papermill": {
          "duration": 0.015117,
          "end_time": "2022-02-27T23:26:58.290365",
          "exception": false,
          "start_time": "2022-02-27T23:26:58.275248",
          "status": "completed"
        },
        "tags": [],
        "id": "eea75a11"
      },
      "source": [
        "Let's call these two numbers $\\mu$ and $\\sigma$. ($\\mu = 57.24, \\ \\sigma=24.55$)"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "7e05d74b",
      "metadata": {
        "papermill": {
          "duration": 0.015084,
          "end_time": "2022-02-27T23:26:58.321108",
          "exception": false,
          "start_time": "2022-02-27T23:26:58.306024",
          "status": "completed"
        },
        "tags": [],
        "id": "7e05d74b"
      },
      "source": [
        "## Part 2: Select a sample\n",
        "\n",
        "Now let's select a sample of size 100 students and find the sample mean."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "id": "7e83556c",
      "metadata": {
        "execution": {
          "iopub.execute_input": "2022-02-27T23:26:58.356503Z",
          "iopub.status.busy": "2022-02-27T23:26:58.355730Z",
          "iopub.status.idle": "2022-02-27T23:26:58.363223Z",
          "shell.execute_reply": "2022-02-27T23:26:58.362729Z",
          "shell.execute_reply.started": "2021-12-18T19:58:57.783174Z"
        },
        "papermill": {
          "duration": 0.026726,
          "end_time": "2022-02-27T23:26:58.363370",
          "exception": false,
          "start_time": "2022-02-27T23:26:58.336644",
          "status": "completed"
        },
        "tags": [],
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "7e83556c",
        "outputId": "6da3a04c-82df-489b-88f3-c340415e6889"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The mean of this sample is 58.77\n"
          ]
        }
      ],
      "source": [
        "sample = np.random.choice(all_data, size=100)\n",
        "print('The mean of this sample is', np.round(np.mean(sample),2))"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "5c26357e",
      "metadata": {
        "papermill": {
          "duration": 0.015026,
          "end_time": "2022-02-27T23:26:58.393820",
          "exception": false,
          "start_time": "2022-02-27T23:26:58.378794",
          "status": "completed"
        },
        "tags": [],
        "id": "5c26357e"
      },
      "source": [
        "## Part 3: Select 5 samples\n",
        "\n",
        "Now let's select 5 samples (of again size 100) to see their means:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "id": "b1dbc81d",
      "metadata": {
        "execution": {
          "iopub.execute_input": "2022-02-27T23:26:58.431906Z",
          "iopub.status.busy": "2022-02-27T23:26:58.431165Z",
          "iopub.status.idle": "2022-02-27T23:26:58.437189Z",
          "shell.execute_reply": "2022-02-27T23:26:58.437703Z",
          "shell.execute_reply.started": "2021-12-18T19:58:57.790884Z"
        },
        "papermill": {
          "duration": 0.028749,
          "end_time": "2022-02-27T23:26:58.437928",
          "exception": false,
          "start_time": "2022-02-27T23:26:58.409179",
          "status": "completed"
        },
        "tags": [],
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "b1dbc81d",
        "outputId": "4ec1d4a4-b4dc-4c01-f169-c44f64c34bcc"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The mean of this sample 1 is 59.07\n",
            "The mean of this sample 2 is 61.58\n",
            "The mean of this sample 3 is 61.46\n",
            "The mean of this sample 4 is 56.25\n",
            "The mean of this sample 5 is 60.72\n"
          ]
        }
      ],
      "source": [
        "for i in range(1,6):\n",
        "    sample = np.random.choice(all_data, size=100)\n",
        "    print('The mean of this sample', i, 'is', np.round(np.mean(sample),2))"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "1337e5a4",
      "metadata": {
        "papermill": {
          "duration": 0.015906,
          "end_time": "2022-02-27T23:26:58.470258",
          "exception": false,
          "start_time": "2022-02-27T23:26:58.454352",
          "status": "completed"
        },
        "tags": [],
        "id": "1337e5a4"
      },
      "source": [
        "## Part 4: Understanding Central Limit Theorem\n",
        "\n",
        "- Even if the underlying distribution is non-normal, if the sample size is large enough, the means of samples will approximately be normally distributed.\n",
        "\n",
        "- Furthermore, its mean will be the same as population mean $\\mu$,\n",
        "\n",
        "- And its standard deviation will be the population standard deviation divided by the square root of sample size which is $\\sigma \\big/ \\sqrt{\\text{sample size}}$"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "d77f70c6",
      "metadata": {
        "papermill": {
          "duration": 0.016086,
          "end_time": "2022-02-27T23:26:58.502472",
          "exception": false,
          "start_time": "2022-02-27T23:26:58.486386",
          "status": "completed"
        },
        "tags": [],
        "id": "d77f70c6"
      },
      "source": [
        "## Part 5: Select 10 000 samples\n",
        "\n",
        "We select 10 000 samples of size 100, and write their means to a list."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "id": "652cd17e",
      "metadata": {
        "execution": {
          "iopub.execute_input": "2022-02-27T23:26:58.539116Z",
          "iopub.status.busy": "2022-02-27T23:26:58.538362Z",
          "iopub.status.idle": "2022-02-27T23:26:58.574301Z",
          "shell.execute_reply": "2022-02-27T23:26:58.574884Z",
          "shell.execute_reply.started": "2021-12-18T19:58:57.808078Z"
        },
        "papermill": {
          "duration": 0.056586,
          "end_time": "2022-02-27T23:26:58.575097",
          "exception": false,
          "start_time": "2022-02-27T23:26:58.518511",
          "status": "completed"
        },
        "tags": [],
        "id": "652cd17e"
      },
      "outputs": [],
      "source": [
        "size = 100\n",
        "samples = np.random.choice(all_data, 10000*size).reshape(10000,size) #sample 10 000 x 100 times, and divide into 10 000 samples of size 100\n",
        "sample_means = samples.mean(axis=1)"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "f11fb68a",
      "metadata": {
        "papermill": {
          "duration": 0.015422,
          "end_time": "2022-02-27T23:26:58.606528",
          "exception": false,
          "start_time": "2022-02-27T23:26:58.591106",
          "status": "completed"
        },
        "tags": [],
        "id": "f11fb68a"
      },
      "source": [
        "## Part 6: Visualize sample means"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "id": "8a308d7a",
      "metadata": {
        "execution": {
          "iopub.execute_input": "2022-02-27T23:26:58.642448Z",
          "iopub.status.busy": "2022-02-27T23:26:58.641811Z",
          "iopub.status.idle": "2022-02-27T23:26:58.900950Z",
          "shell.execute_reply": "2022-02-27T23:26:58.901495Z",
          "shell.execute_reply.started": "2021-12-18T19:58:57.842687Z"
        },
        "papermill": {
          "duration": 0.279261,
          "end_time": "2022-02-27T23:26:58.901735",
          "exception": false,
          "start_time": "2022-02-27T23:26:58.622474",
          "status": "completed"
        },
        "tags": [],
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 314
        },
        "id": "8a308d7a",
        "outputId": "4d37bb84-695d-4aab-ffc7-cb4bd9d225c2"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 1500x300 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ],
      "source": [
        "plt.figure(figsize=(15,3))\n",
        "plt.hist(sample_means, bins=50)\n",
        "plt.yticks([])\n",
        "plt.grid()\n",
        "plt.title('Distribution of sample means')\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "c8fbf288",
      "metadata": {
        "papermill": {
          "duration": 0.016274,
          "end_time": "2022-02-27T23:26:58.935136",
          "exception": false,
          "start_time": "2022-02-27T23:26:58.918862",
          "status": "completed"
        },
        "tags": [],
        "id": "c8fbf288"
      },
      "source": [
        "Hey! It looks very normal to me!"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "f6c30253",
      "metadata": {
        "papermill": {
          "duration": 0.016597,
          "end_time": "2022-02-27T23:26:58.968559",
          "exception": false,
          "start_time": "2022-02-27T23:26:58.951962",
          "status": "completed"
        },
        "tags": [],
        "id": "f6c30253"
      },
      "source": [
        "## Part 7: Mean and Standard Deviation of sample means"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 8,
      "id": "8dbc4edd",
      "metadata": {
        "execution": {
          "iopub.execute_input": "2022-02-27T23:26:59.010620Z",
          "iopub.status.busy": "2022-02-27T23:26:59.009035Z",
          "iopub.status.idle": "2022-02-27T23:26:59.017639Z",
          "shell.execute_reply": "2022-02-27T23:26:59.018279Z",
          "shell.execute_reply.started": "2021-12-18T19:58:58.09856Z"
        },
        "papermill": {
          "duration": 0.032608,
          "end_time": "2022-02-27T23:26:59.018479",
          "exception": false,
          "start_time": "2022-02-27T23:26:58.985871",
          "status": "completed"
        },
        "tags": [],
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "8dbc4edd",
        "outputId": "98ee3d63-14ba-457b-a47d-49d15adb0020"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The expected mean of sample means is 57.24\n",
            "The expected standard deviation of sample means is 2.455\n"
          ]
        }
      ],
      "source": [
        "print('The expected mean of sample means is', np.round(np.mean(all_data),2))\n",
        "print('The expected standard deviation of sample means is', np.round(np.std(all_data)/np.sqrt(100), 3))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "97ef1ac7",
      "metadata": {
        "execution": {
          "iopub.execute_input": "2022-02-27T23:26:59.057277Z",
          "iopub.status.busy": "2022-02-27T23:26:59.056416Z",
          "iopub.status.idle": "2022-02-27T23:26:59.062149Z",
          "shell.execute_reply": "2022-02-27T23:26:59.062710Z",
          "shell.execute_reply.started": "2021-12-18T19:58:58.111305Z"
        },
        "papermill": {
          "duration": 0.026947,
          "end_time": "2022-02-27T23:26:59.062917",
          "exception": false,
          "start_time": "2022-02-27T23:26:59.035970",
          "status": "completed"
        },
        "tags": [],
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "97ef1ac7",
        "outputId": "3f923dfd-4313-4bc6-c218-7b828b55e4f9"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The mean of sample means is 57.2\n",
            "The standard deviation of sample means is 2.43\n"
          ]
        }
      ],
      "source": [
        "print('The mean of sample means is', np.round(np.mean(sample_means),2))\n",
        "print('The standard deviation of sample means is', np.round(np.std(sample_means),2))"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "10a52ced",
      "metadata": {
        "papermill": {
          "duration": 0.017308,
          "end_time": "2022-02-27T23:26:59.097538",
          "exception": false,
          "start_time": "2022-02-27T23:26:59.080230",
          "status": "completed"
        },
        "tags": [],
        "id": "10a52ced"
      },
      "source": [
        "Very close!"
      ]
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.7.12"
    },
    "papermill": {
      "default_parameters": {},
      "duration": 13.252291,
      "end_time": "2022-02-27T23:26:59.825588",
      "environment_variables": {},
      "exception": null,
      "input_path": "__notebook__.ipynb",
      "output_path": "__notebook__.ipynb",
      "parameters": {},
      "start_time": "2022-02-27T23:26:46.573297",
      "version": "2.3.3"
    },
    "colab": {
      "provenance": []
    }
  },
  "nbformat": 4,
  "nbformat_minor": 5
}