notebook.ipynb

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      "name": "Benford's-Law.ipynb",
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      "source": [
        "import math\n",
        "import pandas as pd\n",
        "import matplotlib.pyplot as plt\n"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "df = pd.read_csv('sample_data/googleplaystore.csv')\n",
        "df.head()"
      ],
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              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>App</th>\n",
              "      <th>Category</th>\n",
              "      <th>Rating</th>\n",
              "      <th>Reviews</th>\n",
              "      <th>Size</th>\n",
              "      <th>Installs</th>\n",
              "      <th>Type</th>\n",
              "      <th>Price</th>\n",
              "      <th>Content Rating</th>\n",
              "      <th>Genres</th>\n",
              "      <th>Last Updated</th>\n",
              "      <th>Current Ver</th>\n",
              "      <th>Android Ver</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>Photo Editor &amp; Candy Camera &amp; Grid &amp; ScrapBook</td>\n",
              "      <td>ART_AND_DESIGN</td>\n",
              "      <td>4.1</td>\n",
              "      <td>159</td>\n",
              "      <td>19M</td>\n",
              "      <td>10,000+</td>\n",
              "      <td>Free</td>\n",
              "      <td>0</td>\n",
              "      <td>Everyone</td>\n",
              "      <td>Art &amp; Design</td>\n",
              "      <td>January 7, 2018</td>\n",
              "      <td>1.0.0</td>\n",
              "      <td>4.0.3 and up</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>Coloring book moana</td>\n",
              "      <td>ART_AND_DESIGN</td>\n",
              "      <td>3.9</td>\n",
              "      <td>967</td>\n",
              "      <td>14M</td>\n",
              "      <td>500,000+</td>\n",
              "      <td>Free</td>\n",
              "      <td>0</td>\n",
              "      <td>Everyone</td>\n",
              "      <td>Art &amp; Design;Pretend Play</td>\n",
              "      <td>January 15, 2018</td>\n",
              "      <td>2.0.0</td>\n",
              "      <td>4.0.3 and up</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>U Launcher Lite – FREE Live Cool Themes, Hide ...</td>\n",
              "      <td>ART_AND_DESIGN</td>\n",
              "      <td>4.7</td>\n",
              "      <td>87510</td>\n",
              "      <td>8.7M</td>\n",
              "      <td>5,000,000+</td>\n",
              "      <td>Free</td>\n",
              "      <td>0</td>\n",
              "      <td>Everyone</td>\n",
              "      <td>Art &amp; Design</td>\n",
              "      <td>August 1, 2018</td>\n",
              "      <td>1.2.4</td>\n",
              "      <td>4.0.3 and up</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>Sketch - Draw &amp; Paint</td>\n",
              "      <td>ART_AND_DESIGN</td>\n",
              "      <td>4.5</td>\n",
              "      <td>215644</td>\n",
              "      <td>25M</td>\n",
              "      <td>50,000,000+</td>\n",
              "      <td>Free</td>\n",
              "      <td>0</td>\n",
              "      <td>Teen</td>\n",
              "      <td>Art &amp; Design</td>\n",
              "      <td>June 8, 2018</td>\n",
              "      <td>Varies with device</td>\n",
              "      <td>4.2 and up</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>Pixel Draw - Number Art Coloring Book</td>\n",
              "      <td>ART_AND_DESIGN</td>\n",
              "      <td>4.3</td>\n",
              "      <td>967</td>\n",
              "      <td>2.8M</td>\n",
              "      <td>100,000+</td>\n",
              "      <td>Free</td>\n",
              "      <td>0</td>\n",
              "      <td>Everyone</td>\n",
              "      <td>Art &amp; Design;Creativity</td>\n",
              "      <td>June 20, 2018</td>\n",
              "      <td>1.1</td>\n",
              "      <td>4.4 and up</td>\n",
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              "                                                     [key], {});\n",
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              "            '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
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            "text/plain": [
              "                                                 App        Category  Rating  \\\n",
              "0     Photo Editor & Candy Camera & Grid & ScrapBook  ART_AND_DESIGN     4.1   \n",
              "1                                Coloring book moana  ART_AND_DESIGN     3.9   \n",
              "2  U Launcher Lite – FREE Live Cool Themes, Hide ...  ART_AND_DESIGN     4.7   \n",
              "3                              Sketch - Draw & Paint  ART_AND_DESIGN     4.5   \n",
              "4              Pixel Draw - Number Art Coloring Book  ART_AND_DESIGN     4.3   \n",
              "\n",
              "  Reviews  Size     Installs  Type Price Content Rating  \\\n",
              "0     159   19M      10,000+  Free     0       Everyone   \n",
              "1     967   14M     500,000+  Free     0       Everyone   \n",
              "2   87510  8.7M   5,000,000+  Free     0       Everyone   \n",
              "3  215644   25M  50,000,000+  Free     0           Teen   \n",
              "4     967  2.8M     100,000+  Free     0       Everyone   \n",
              "\n",
              "                      Genres      Last Updated         Current Ver  \\\n",
              "0               Art & Design   January 7, 2018               1.0.0   \n",
              "1  Art & Design;Pretend Play  January 15, 2018               2.0.0   \n",
              "2               Art & Design    August 1, 2018               1.2.4   \n",
              "3               Art & Design      June 8, 2018  Varies with device   \n",
              "4    Art & Design;Creativity     June 20, 2018                 1.1   \n",
              "\n",
              "    Android Ver  \n",
              "0  4.0.3 and up  \n",
              "1  4.0.3 and up  \n",
              "2  4.0.3 and up  \n",
              "3    4.2 and up  \n",
              "4    4.4 and up  "
            ]
          },
          "metadata": {},
          "execution_count": 5
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "digit_counts = df[\"Reviews\"].dropna().astype(str).str[0].value_counts(normalize=True).drop('0')\n",
        "digit_counts"
      ],
      "metadata": {
        "colab": {
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        },
        "id": "2eyHQg8PBXLk",
        "outputId": "f574d906-7758-43ac-fb39-d9f5e3dcde90"
      },
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "1    0.280786\n",
              "2    0.177659\n",
              "3    0.117148\n",
              "4    0.094087\n",
              "5    0.073056\n",
              "6    0.063740\n",
              "7    0.052578\n",
              "8    0.046675\n",
              "9    0.039295\n",
              "Name: Reviews, dtype: float64"
            ]
          },
          "metadata": {},
          "execution_count": 6
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "digit_count_df = pd.DataFrame({\"Leading Digit\": digit_counts.index, \"Observed\": digit_counts.values})\n",
        "digit_count_df"
      ],
      "metadata": {
        "colab": {
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          "height": 332
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        "id": "o5ZkQnOTHTpr",
        "outputId": "94398f4a-ede6-4eae-9126-3ab3d96dc0ab"
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      "execution_count": 7,
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              "\n",
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              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>Leading Digit</th>\n",
              "      <th>Observed</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>1</td>\n",
              "      <td>0.280786</td>\n",
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              "      <th>1</th>\n",
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              "      <td>0.177659</td>\n",
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              "      <th>2</th>\n",
              "      <td>3</td>\n",
              "      <td>0.117148</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>4</td>\n",
              "      <td>0.094087</td>\n",
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              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>5</td>\n",
              "      <td>0.073056</td>\n",
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              "    <tr>\n",
              "      <th>5</th>\n",
              "      <td>6</td>\n",
              "      <td>0.063740</td>\n",
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              "    <tr>\n",
              "      <th>6</th>\n",
              "      <td>7</td>\n",
              "      <td>0.052578</td>\n",
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              "      <th>7</th>\n",
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              "          google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
              "        async function convertToInteractive(key) {\n",
              "          const element = document.querySelector('#df-3926285e-9fcf-4608-8de9-a9143746b15d');\n",
              "          const dataTable =\n",
              "            await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                     [key], {});\n",
              "          if (!dataTable) return;\n",
              "\n",
              "          const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "            '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
              "            + ' to learn more about interactive tables.';\n",
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            "text/plain": [
              "  Leading Digit  Observed\n",
              "0             1  0.280786\n",
              "1             2  0.177659\n",
              "2             3  0.117148\n",
              "3             4  0.094087\n",
              "4             5  0.073056\n",
              "5             6  0.063740\n",
              "6             7  0.052578\n",
              "7             8  0.046675\n",
              "8             9  0.039295"
            ]
          },
          "metadata": {},
          "execution_count": 7
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "expected = {\n",
        "    \"1\": [math.log10((1+1)/1)],\n",
        "    \"2\": [math.log10((2+1)/2)],\n",
        "    \"3\": [math.log10((3+1)/3)],\n",
        "    \"4\": [math.log10((4+1)/4)],\n",
        "    \"5\": [math.log10((5+1)/5)],\n",
        "    \"6\": [math.log10((6+1)/6)],\n",
        "    \"7\": [math.log10((7+1)/7)],\n",
        "    \"8\": [math.log10((8+1)/8)],\n",
        "    \"9\": [math.log10((9+1)/9)]    \n",
        "    }\n",
        "\n",
        "expected_df = pd.DataFrame.from_dict(expected, orient='index',\n",
        "                                     columns=['Benford']).set_index(\n",
        "                                         pd.Index([0, 1, 2, 3, 4, 5, 6, 7, 8]))\n",
        "expected_df"
      ],
      "metadata": {
        "colab": {
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      "execution_count": 8,
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              "      <th></th>\n",
              "      <th>Benford</th>\n",
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              "      <td>0.301030</td>\n",
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            "text/plain": [
              "    Benford\n",
              "0  0.301030\n",
              "1  0.176091\n",
              "2  0.124939\n",
              "3  0.096910\n",
              "4  0.079181\n",
              "5  0.066947\n",
              "6  0.057992\n",
              "7  0.051153\n",
              "8  0.045757"
            ]
          },
          "metadata": {},
          "execution_count": 8
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "comparison_df = pd.concat([digit_count_df, expected_df], axis=1)\n",
        "comparison_df"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 332
        },
        "id": "3quc-sKXHFBe",
        "outputId": "7e0b1f55-b9f3-47d0-8811-3d26cd5092da"
      },
      "execution_count": 9,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/html": [
              "\n",
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              "      <th></th>\n",
              "      <th>Leading Digit</th>\n",
              "      <th>Observed</th>\n",
              "      <th>Benford</th>\n",
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              "  </thead>\n",
              "  <tbody>\n",
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              "    </tr>\n",
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              "      <th>6</th>\n",
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              "      <td>0.052578</td>\n",
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              "    <tr>\n",
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              "      <td>0.046675</td>\n",
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              "      <td>9</td>\n",
              "      <td>0.039295</td>\n",
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              "\n",
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              "  </style>\n",
              "\n",
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              "        const buttonEl =\n",
              "          document.querySelector('#df-d9042e66-e9b1-4271-9714-ced59c648e72 button.colab-df-convert');\n",
              "        buttonEl.style.display =\n",
              "          google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
              "        async function convertToInteractive(key) {\n",
              "          const element = document.querySelector('#df-d9042e66-e9b1-4271-9714-ced59c648e72');\n",
              "          const dataTable =\n",
              "            await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                     [key], {});\n",
              "          if (!dataTable) return;\n",
              "\n",
              "          const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "            '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
              "            + ' to learn more about interactive tables.';\n",
              "          element.innerHTML = '';\n",
              "          dataTable['output_type'] = 'display_data';\n",
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              "        }\n",
              "      </script>\n",
              "    </div>\n",
              "  </div>\n",
              "  "
            ],
            "text/plain": [
              "  Leading Digit  Observed   Benford\n",
              "0             1  0.280786  0.301030\n",
              "1             2  0.177659  0.176091\n",
              "2             3  0.117148  0.124939\n",
              "3             4  0.094087  0.096910\n",
              "4             5  0.073056  0.079181\n",
              "5             6  0.063740  0.066947\n",
              "6             7  0.052578  0.057992\n",
              "7             8  0.046675  0.051153\n",
              "8             9  0.039295  0.045757"
            ]
          },
          "metadata": {},
          "execution_count": 9
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "title = \"Application Ratings Compared to Benford's Law\"\n",
        "comparison_df.plot(x=\"Leading Digit\", kind=\"bar\")\n",
        "plt.title(title, fontsize=16, fontweight=\"bold\")\n",
        "plt.xlabel(\"Leading Digit\", fontsize=16)\n",
        "plt.ylabel(\"Proportion\", fontsize=16)\n",
        "plt.xticks(rotation=0)\n",
        "plt.legend(fontsize=14)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 320
        },
        "id": "wwiobViGBkeK",
        "outputId": "f6a398c1-582d-444f-cc08-df5ae1a36e10"
      },
      "execution_count": 10,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7f7e31483590>"
            ]
          },
          "metadata": {},
          "execution_count": 10
        },
        {
          "output_type": "display_data",
          "data": {
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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "chi_square = 0\n",
        "for index, row in comparison_df.iterrows():\n",
        "      single = (((row[\"Observed\"] - row[\"Benford\"])**2)/row[\"Benford\"])\n",
        "      chi_square = chi_square + single\n",
        "\n",
        "chi_square\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "fubDJ7_nDDP-",
        "outputId": "3350b1f3-31be-4f05-ee3a-5d52d1c2cd2c"
      },
      "execution_count": 11,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "0.0043809164664262"
            ]
          },
          "metadata": {},
          "execution_count": 11
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "import scipy.stats\n",
        "\n",
        "#Chi-Square critical value\n",
        "critical_value = scipy.stats.chi2.ppf(1-.05, df=8)\n",
        "critical_value"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "nH3Q6sIbS3MY",
        "outputId": "a457adcc-260e-4540-bc75-fd6c12f48c59"
      },
      "execution_count": 41,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "15.50731305586545"
            ]
          },
          "metadata": {},
          "execution_count": 41
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "from scipy.stats import chi2\n",
        "\n",
        "x = np.arange(0, 20, 0.001)\n",
        "\n",
        "plt.plot(x, chi2.pdf(x, df=8))\n",
        "plt.axvline(x=critical_value, color='r')\n",
        "plt.text(15.89,0.045,'Critical Value')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 284
        },
        "id": "gzf2DFFRT3gr",
        "outputId": "60f7a8fe-ffa5-43b0-c504-dc7dbc36c5f6"
      },
      "execution_count": 44,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Text(15.89, 0.045, 'Critical Value')"
            ]
          },
          "metadata": {},
          "execution_count": 44
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "###Another Way"
      ],
      "metadata": {
        "id": "Ta0u_uSbSTaD"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "%pip install scipy"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "iB4rbHn9PsUm",
        "outputId": "0b5251a5-b96f-41df-8fbb-9b3fe126fac5"
      },
      "execution_count": 12,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Requirement already satisfied: scipy in /usr/local/lib/python3.7/dist-packages (1.4.1)\n",
            "Requirement already satisfied: numpy>=1.13.3 in /usr/local/lib/python3.7/dist-packages (from scipy) (1.21.5)\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "from scipy.stats import chisquare\n",
        "\n",
        "obs = comparison_df['Observed'].to_list()\n",
        "exp = comparison_df['Benford'].to_list()\n",
        "chisquare(obs, f_exp=exp)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "VXDRM59-Pwth",
        "outputId": "f763cf27-ce47-415d-e093-1ce13bc779fb"
      },
      "execution_count": 43,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Power_divergenceResult(statistic=0.0043809164664262, pvalue=0.9999999999990424)"
            ]
          },
          "metadata": {},
          "execution_count": 43
        }
      ]
    }
  ]
}