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Benford's Law.ipynb
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"# Benford's law\n",
"\n",
"## Table of contents\n",
"\n",
"- [Benford's law](#benfords-law)\n",
" - [Table of contents](#table-of-contents)\n",
" - [Introduction](#introduction)\n",
" - [Where can Benford's Law be observed in real life?](#where-can-benfords-law-be-observed-in-real-life)\n",
" - [Example 1: 2021 Canadian Population Census](#example-1-2021-canadian-population-census)\n",
" - [Example 2: Economic statistics of Canada](#example-2-economic-statistics-of-canada)\n",
" - [Example 3: Against Benford! - Best-selling video games of all time](#example-3-against-benford---best-selling-video-games-of-all-time)\n",
" - [Conclusion](#conclusion)\n",
"\n",
"## Introduction\n",
"\n",
"Benford's law is a theory that states that the probability of the first non-zero digit, $d$, of a number, $n$, in a collection of numerical data, occurring decreases exponentially with $n$. The equation to get the probability of $d$ occurring at the start of $n$ can be determined by the equation:\n",
"\n",
"$$\n",
"P(d) = \\log(1 + {1 \\over d}),\\newline\n",
"\\newline\n",
"1 \\le d \\le 9, d \\in \\mathbb Z\n",
"$$\n",
"\n",
"By using this equation and inputting the set $D = \\{1, \\dots, 9\\}$ (all possible digits, assuming we are using base-10) we can get the probability that the corresponding digit will be the first digit."
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"\u001b[32m\u001b[1m Updating\u001b[22m\u001b[39m registry at `~/.julia/registries/General.toml`\n",
"\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.8/Project.toml`\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.8/Manifest.toml`\n",
"\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.8/Project.toml`\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.8/Manifest.toml`\n",
"\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.8/Project.toml`\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.8/Manifest.toml`\n"
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},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import Pkg\n",
"Pkg.add(\"IJulia\")\n",
"Pkg.add(\"DataFrames\")\n",
"Pkg.add(\"StatsPlots\")\n",
"\n",
"using IJulia\n",
"using StatsPlots\n",
"using DataFrames\n",
"\n",
"gr(size = (1024, 1024))\n",
"\n",
"#Benford's law digit probability function\n",
"P(d) = log10(1 + 1 / d)\n",
"\n",
"#x is a set from 1 to 9\n",
"x = 1 : 9\n",
"\n",
"# . is vector operations (I think that's what its called? It does operations on each item of the set)\n",
"benfords_values = P.(x) * 100\n",
"\n",
"g1 = plot(x, benfords_values, \n",
" title = \"Benford's law\",\n",
" ylabel = \"Frequency of the leading digit (%)\",\n",
" xlabel = \"Digit\",\n",
" label = \"Frequency (%)\",\n",
" lw = 5)\n",
"\n",
"g2 = bar(x, benfords_values,\n",
" ylabel = \"Frequency of the leading digit (%)\",\n",
" xlabel = \"Digit\",\n",
" label = \"Frequency (%)\")\n",
"\n",
"plot(g1, g2, layout = (2, 1))"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div><div style = \"float: left;\"><span>9×1 DataFrame</span></div><div style = \"clear: both;\"></div></div><div class = \"data-frame\" style = \"overflow-x: scroll;\"><table class = \"data-frame\" style = \"margin-bottom: 6px;\"><thead><tr class = \"header\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">Row</th><th style = \"text-align: left;\">Frequency (%)</th></tr><tr class = \"subheader headerLastRow\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\"></th><th title = \"Float64\" style = \"text-align: left;\">Float64</th></tr></thead><tbody><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1</td><td style = \"text-align: right;\">30.103</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">2</td><td style = \"text-align: right;\">17.6091</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">3</td><td style = \"text-align: right;\">12.4939</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">4</td><td style = \"text-align: right;\">9.691</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">5</td><td style = \"text-align: right;\">7.91812</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">6</td><td style = \"text-align: right;\">6.69468</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">7</td><td style = \"text-align: right;\">5.79919</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">8</td><td style = \"text-align: right;\">5.11525</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9</td><td style = \"text-align: right;\">4.57575</td></tr></tbody></table></div>"
],
"text/latex": [
"\\begin{tabular}{r|c}\n",
"\t& Frequency (\\%)\\\\\n",
"\t\\hline\n",
"\t& Float64\\\\\n",
"\t\\hline\n",
"\t1 & 30.103 \\\\\n",
"\t2 & 17.6091 \\\\\n",
"\t3 & 12.4939 \\\\\n",
"\t4 & 9.691 \\\\\n",
"\t5 & 7.91812 \\\\\n",
"\t6 & 6.69468 \\\\\n",
"\t7 & 5.79919 \\\\\n",
"\t8 & 5.11525 \\\\\n",
"\t9 & 4.57575 \\\\\n",
"\\end{tabular}\n"
],
"text/plain": [
"\u001b[1m9×1 DataFrame\u001b[0m\n",
"\u001b[1m Row \u001b[0m│\u001b[1m Frequency (%) \u001b[0m\n",
" │\u001b[90m Float64 \u001b[0m\n",
"─────┼───────────────\n",
" 1 │ 30.103\n",
" 2 │ 17.6091\n",
" 3 │ 12.4939\n",
" 4 │ 9.691\n",
" 5 │ 7.91812\n",
" 6 │ 6.69468\n",
" 7 │ 5.79919\n",
" 8 │ 5.11525\n",
" 9 │ 4.57575"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"DataFrame(\"Frequency (%)\" => benfords_values)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Above we can see how using the equation the frequency of the starting digits logarithmically decreases, starting at 1 having a frequency of 30.1%, then to 2 with a frequency of 17.6%, 3 at 12.5%, and beyond."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Where can Benford's Law be observed in real life?\n",
"\n",
"It seems very improbable that in most numerical data sets the leading digits have a pattern that can be determined, but there are many examples of this occurring in the real world, most famously used by tax agencies for the government to detect fraudulent information being submitted. That is just one example, but it occurs throughout most data sets which have a limit as will be shown."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Example 1: 2021 Canadian Population Census\n",
"\n",
"For the first example, the data will use the 2021 population census released by StatsCanada. Specicifically, the population of over 1000 population centres will be used to determine if Benford's law applies."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.8/Project.toml`\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.8/Manifest.toml`\n",
"\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.8/Project.toml`\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.8/Manifest.toml`\n"
]
},
{
"data": {
"text/html": [
"<div><div style = \"float: left;\"><span>1014×2 DataFrame</span></div><div style = \"float: right;\"><span style = \"font-style: italic;\">989 rows omitted</span></div><div style = \"clear: both;\"></div></div><div class = \"data-frame\" style = \"overflow-x: scroll;\"><table class = \"data-frame\" style = \"margin-bottom: 6px;\"><thead><tr class = \"header\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">Row</th><th style = \"text-align: left;\">Population centre name</th><th style = \"text-align: left;\">Population count</th></tr><tr class = \"subheader headerLastRow\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\"></th><th title = \"String\" style = \"text-align: left;\">String</th><th title = \"Int64\" style = \"text-align: left;\">Int64</th></tr></thead><tbody><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1</td><td style = \"text-align: left;\">Saint-Liboire</td><td style = \"text-align: right;\">1844</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">2</td><td style = \"text-align: left;\">Bracebridge</td><td style = \"text-align: right;\">9884</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">3</td><td style = \"text-align: left;\">Port-Alfred-Bagotville</td><td style = \"text-align: right;\">13350</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">4</td><td style = \"text-align: left;\">Souris</td><td style = \"text-align: right;\">1935</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">5</td><td style = \"text-align: left;\">Bourget</td><td style = \"text-align: right;\">1175</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">6</td><td style = \"text-align: left;\">Whistler</td><td style = \"text-align: right;\">9974</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">7</td><td style = \"text-align: left;\">Biggar</td><td style = \"text-align: right;\">2045</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">8</td><td style = \"text-align: left;\">Fox Creek</td><td style = \"text-align: right;\">1426</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9</td><td style = \"text-align: left;\">Parksville</td><td style = \"text-align: right;\">27330</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">10</td><td style = \"text-align: left;\">Oro Station - Hawkestone</td><td style = \"text-align: right;\">1806</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">11</td><td style = \"text-align: left;\">Shawnigan - Mill Bay</td><td style = \"text-align: right;\">7285</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">12</td><td style = \"text-align: left;\">Crossfield</td><td style = \"text-align: right;\">3579</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">13</td><td style = \"text-align: left;\">Terrace</td><td style = \"text-align: right;\">14633</td></tr><tr><td style = \"text-align: right;\">&vellip;</td><td style = \"text-align: right;\">&vellip;</td><td style = \"text-align: right;\">&vellip;</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1003</td><td style = \"text-align: left;\">Clarenville-Shoal Harbour</td><td style = \"text-align: right;\">6143</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1004</td><td style = \"text-align: left;\">Shippagan</td><td style = \"text-align: right;\">2189</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1005</td><td style = \"text-align: left;\">Chemawawin 2</td><td style = \"text-align: right;\">1163</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1006</td><td style = \"text-align: left;\">Lunenburg</td><td style = \"text-align: right;\">2405</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1007</td><td style = \"text-align: left;\">Brookside</td><td style = \"text-align: right;\">1439</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1008</td><td style = \"text-align: left;\">Deer Lake</td><td style = \"text-align: right;\">4346</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1009</td><td style = \"text-align: left;\">Parry Sound</td><td style = \"text-align: right;\">6788</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1010</td><td style = \"text-align: left;\">Saint-Jean-Port-Joli</td><td style = \"text-align: right;\">1381</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1011</td><td style = \"text-align: left;\">Deep River</td><td style = \"text-align: right;\">3860</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1012</td><td style = \"text-align: left;\">Saint-Michel</td><td style = \"text-align: right;\">1745</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1013</td><td style = \"text-align: left;\">Two Hills</td><td style = \"text-align: right;\">1379</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1014</td><td style = \"text-align: left;\">Meaford</td><td style = \"text-align: right;\">5078</td></tr></tbody></table></div>"
],
"text/latex": [
"\\begin{tabular}{r|cc}\n",
"\t& Population centre name & Population count\\\\\n",
"\t\\hline\n",
"\t& String & Int64\\\\\n",
"\t\\hline\n",
"\t1 & Saint-Liboire & 1844 \\\\\n",
"\t2 & Bracebridge & 9884 \\\\\n",
"\t3 & Port-Alfred-Bagotville & 13350 \\\\\n",
"\t4 & Souris & 1935 \\\\\n",
"\t5 & Bourget & 1175 \\\\\n",
"\t6 & Whistler & 9974 \\\\\n",
"\t7 & Biggar & 2045 \\\\\n",
"\t8 & Fox Creek & 1426 \\\\\n",
"\t9 & Parksville & 27330 \\\\\n",
"\t10 & Oro Station - Hawkestone & 1806 \\\\\n",
"\t11 & Shawnigan - Mill Bay & 7285 \\\\\n",
"\t12 & Crossfield & 3579 \\\\\n",
"\t13 & Terrace & 14633 \\\\\n",
"\t14 & Ponoka & 7032 \\\\\n",
"\t15 & Durham & 2755 \\\\\n",
"\t16 & New Dundee & 1174 \\\\\n",
"\t17 & Sarnia & 73944 \\\\\n",
"\t18 & Harrison Hot Springs & 1552 \\\\\n",
"\t19 & Roberval & 8056 \\\\\n",
"\t20 & Penhold & 3359 \\\\\n",
"\t21 & Cap-aux-Meules & 2516 \\\\\n",
"\t22 & Senneterre & 2212 \\\\\n",
"\t23 & Grand Centre & 7514 \\\\\n",
"\t24 & North Battleford & 13649 \\\\\n",
"\t25 & Rocky Mountain House & 6518 \\\\\n",
"\t26 & Melville & 4075 \\\\\n",
"\t27 & Normandin & 1880 \\\\\n",
"\t28 & Pictou & 2643 \\\\\n",
"\t29 & Lakefield & 2943 \\\\\n",
"\t30 & Birch Hills & 1066 \\\\\n",
"\t$\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
"\\end{tabular}\n"
],
"text/plain": [
"\u001b[1m1014×2 DataFrame\u001b[0m\n",
"\u001b[1m Row \u001b[0m│\u001b[1m Population centre name \u001b[0m\u001b[1m Population count \u001b[0m\n",
" │\u001b[90m String \u001b[0m\u001b[90m Int64 \u001b[0m\n",
"──────┼─────────────────────────────────────────────\n",
" 1 │ Saint-Liboire 1844\n",
" 2 │ Bracebridge 9884\n",
" 3 │ Port-Alfred-Bagotville 13350\n",
" 4 │ Souris 1935\n",
" 5 │ Bourget 1175\n",
" 6 │ Whistler 9974\n",
" 7 │ Biggar 2045\n",
" 8 │ Fox Creek 1426\n",
" 9 │ Parksville 27330\n",
" 10 │ Oro Station - Hawkestone 1806\n",
" 11 │ Shawnigan - Mill Bay 7285\n",
" ⋮ │ ⋮ ⋮\n",
" 1005 │ Chemawawin 2 1163\n",
" 1006 │ Lunenburg 2405\n",
" 1007 │ Brookside 1439\n",
" 1008 │ Deer Lake 4346\n",
" 1009 │ Parry Sound 6788\n",
" 1010 │ Saint-Jean-Port-Joli 1381\n",
" 1011 │ Deep River 3860\n",
" 1012 │ Saint-Michel 1745\n",
" 1013 │ Two Hills 1379\n",
" 1014 │ Meaford 5078\n",
"\u001b[36m 993 rows omitted\u001b[0m"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Pkg.add(\"HTTP\")\n",
"Pkg.add(\"JSON\")\n",
"\n",
"using HTTP\n",
"using JSON\n",
"\n",
"# Gets the population census from data collection from 2021 (past 5 years)\n",
"data = JSON.parse(String(HTTP.request(\"GET\", \"https://api.statcan.gc.ca/census-recensement/profile/sdmx/rest/data/DF_POPCNTR/A5..1.1.1?format=jsondata\").body))\n",
"\n",
"#Blame the govt, not me\n",
"pop_centres = data[\"data\"][\"structures\"][1][\"dimensions\"][\"series\"][2][\"values\"]\n",
"populations = Dict{String, Int64}()\n",
"for i in range(1, length(pop_centres))\n",
" place = pop_centres[i][\"name\"]\n",
" pop_count = parse(Int64, data[\"data\"][\"dataSets\"][1][\"series\"][\"0:$(i - 1):0:0:0\"][\"observations\"][\"0\"][1])\n",
" populations[place] = pop_count\n",
"end\n",
"\n",
"centre_names = collect(keys(populations))\n",
"pop_counts = collect(values(populations))\n",
"\n",
"DataFrame(\"Population centre name\" => centre_names,\n",
" \"Population count\" => pop_counts)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now that we have the data, we must get the first digit of each of the population counts of the population centres. To do so, we will use this function\n",
"\n",
"$$\n",
"firstdigit(x) = {\\lfloor {{|x|} \\over {10^{\\lfloor \\log(|x|) \\rfloor}}} \\rfloor},\\newline \n",
"\\newline\n",
"{x \\ne 0}, {x \\in \\mathbb{R}}\n",
"$$\n",
"\n",
"(without symbols):\n",
"\n",
"$$\n",
"firstdigit(x) = {floor({{abs(x)} \\over {10^{floor(\\log(abs(x)))}}})},\\newline\n",
"\\newline\n",
"{x \\ne 0}, {x \\in \\mathbb{R}}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
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},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"firstdigit(x::Real)::Integer = floor(abs(x) / 10 ^ floor(log10(abs(x))))\n",
"\n",
"#Get the first digits of each population count\n",
"firstdigits = firstdigit.(pop_counts)\n",
"\n",
"#Create a vector of 9 0s\n",
"digitcount_population = zeros(Integer, 9)\n",
"\n",
"#Iterate through each digit\n",
"for digit in firstdigits\n",
" #Increment index where the digit is found\n",
" digitcount_population[digit] = digitcount_population[digit] + 1\n",
"end\n",
"\n",
"pg = plot(1 : 9, digitcount_population,\n",
" title = \"Benford's law in the Canadian population count\",\n",
" ylabel = \"Frequency\",\n",
" xlabel = \"Digit\",\n",
" label = \"Frequency of digit\",\n",
" lw = 5)\n",
"bg = bar(1 : 9, digitcount_population,\n",
" ylabel = \"Frequency\",\n",
" xlabel = \"Digit\",\n",
" label = \"Frequency of digit\")\n",
"\n",
"plot(pg, bg, \n",
" layout = (2, 1))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can see how once graphed the resulting graphs look familiar, albiet exaggerated compared to the original Benford's law equation.\n",
"Overlapping the graphs highlight the similarities.\n",
"\n",
"To find the percentage of each item, we can use the function\n",
"\n",
"$$\n",
"percentages(x) = {{{x} \\over \\sum\\limits_{s \\in S} s} \\cdot 100}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
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},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"percentages(x) = (x ./ sum(x)) .* 100\n",
"df = DataFrame(x = 1 : 9,\n",
" a = benfords_values,\n",
" b = percentages(digitcount_population))\n",
"\n",
"@df df plot(:x, [:a :b], \n",
" colour = [:red :blue], \n",
" names = [ \"Benford's law\", \"Canadian Population count\"],\n",
" ylabel = \"Frequency of the leading digit (%)\",\n",
" xlabel = \"Digit\",\n",
" lw = 5)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We observe even further similarities between the 2 data sets! By showing the raw data of the data collected from the census, we can understand how this occurs"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
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],
"text/latex": [
"\\begin{tabular}{r|ccc}\n",
"\t& Frequency & Leading digit percent & Benford's Percentages\\\\\n",
"\t\\hline\n",
"\t& Integer & Float64 & Float64\\\\\n",
"\t\\hline\n",
"\t1 & 472 & 46.5483 & 30.103 \\\\\n",
"\t2 & 188 & 18.5404 & 17.6091 \\\\\n",
"\t3 & 109 & 10.7495 & 12.4939 \\\\\n",
"\t4 & 60 & 5.91716 & 9.691 \\\\\n",
"\t5 & 57 & 5.6213 & 7.91812 \\\\\n",
"\t6 & 45 & 4.43787 & 6.69468 \\\\\n",
"\t7 & 33 & 3.25444 & 5.79919 \\\\\n",
"\t8 & 23 & 2.26824 & 5.11525 \\\\\n",
"\t9 & 27 & 2.66272 & 4.57575 \\\\\n",
"\\end{tabular}\n"
],
"text/plain": [
"\u001b[1m9×3 DataFrame\u001b[0m\n",
"\u001b[1m Row \u001b[0m│\u001b[1m Frequency \u001b[0m\u001b[1m Leading digit percent \u001b[0m\u001b[1m Benford's Percentages \u001b[0m\n",
" │\u001b[90m Integer \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\n",
"─────┼─────────────────────────────────────────────────────────\n",
" 1 │ 472 46.5483 30.103\n",
" 2 │ 188 18.5404 17.6091\n",
" 3 │ 109 10.7495 12.4939\n",
" 4 │ 60 5.91716 9.691\n",
" 5 │ 57 5.6213 7.91812\n",
" 6 │ 45 4.43787 6.69468\n",
" 7 │ 33 3.25444 5.79919\n",
" 8 │ 23 2.26824 5.11525\n",
" 9 │ 27 2.66272 4.57575"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"DataFrame(\n",
" \"Frequency\" => digitcount_population,\n",
" \"Leading digit percent\" => percentages(digitcount_population),\n",
" \"Benford's Percentages\" => benfords_values\n",
")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Comparing the data from the census and values from Benford's law shows how the data is more related than it seems, with a 15% max margin of error with the digit 1 appearing much more frequently than predicted.\n",
"\n",
"I think that the populations of population centres are an excellent way of testing Benford's law because of how wide the variety of values is present. Large cities are most likely to be in the 1-5 million mark with their populations, but as smaller and smaller population centres are introduced into the data set, they have a much larger range of leading digits.\n",
"\n",
"If the population centres had populations with leading digits, not 0, a strong argument could be made that the dataset does not follow the law. One restriction on the data observed is that it cannot be negative (you can't have negative people). The data seems to be completely accurate due to it being sourced directly from the StatsCanada census, and "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Example 2: Economic statistics of Canada\n",
"\n",
"Just as before, another dataset can be accessed which shows Benford's law in action. StatsCanada releases all economic indicators each business day, allowing us to use the data to find if Benford's law also applies here."
]
},
{
"cell_type": "code",
"execution_count": 7,
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"data": {
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"<div><div style = \"float: left;\"><span>163×2 DataFrame</span></div><div style = \"float: right;\"><span style = \"font-style: italic;\">138 rows omitted</span></div><div style = \"clear: both;\"></div></div><div class = \"data-frame\" style = \"overflow-x: scroll;\"><table class = \"data-frame\" style = \"margin-bottom: 6px;\"><thead><tr class = \"header\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">Row</th><th style = \"text-align: left;\">Category</th><th style = \"text-align: left;\">Value</th></tr><tr class = \"subheader headerLastRow\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\"></th><th title = \"String\" style = \"text-align: left;\">String</th><th title = \"String\" style = \"text-align: left;\">String</th></tr></thead><tbody><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1</td><td style = \"text-align: left;\">Worldwide sales by Canadian multinationals</td><td style = \"text-align: left;\">$2.1 trillion</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">2</td><td style = \"text-align: left;\">Total capital expenditures</td><td style = \"text-align: left;\">$25.8 billion</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">3</td><td style = \"text-align: left;\">Median employment income five years after graduation for college-level diploma graduates</td><td style = \"text-align: left;\">$43,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">4</td><td style = \"text-align: left;\">Farm cash receipts</td><td style = \"text-align: left;\">$18,713.0 million</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">5</td><td style = \"text-align: left;\">Annual labour productivity (chained (2012) dollars)</td><td style = \"text-align: left;\">$44.00 per hour</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">6</td><td style = \"text-align: left;\">Operating revenue, motion picture and video exhibition</td><td style = \"text-align: left;\">$552.9 million</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">7</td><td style = \"text-align: left;\">Domestic tourism spending on culture products, Canada</td><td style = \"text-align: left;\">$1.2 billion</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">8</td><td style = \"text-align: left;\">Operating revenue, non-residential leasing</td><td style = \"text-align: left;\">$62.7 billion</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9</td><td style = \"text-align: left;\">Operating revenue, architectural services and landscape architectural services</td><td style = \"text-align: left;\">$5.4 billion</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">10</td><td style = \"text-align: left;\">Average annual wages and salaries by males, natural resources sector</td><td style = \"text-align: left;\">$100,143</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">11</td><td style = \"text-align: left;\">Rail transportation operating revenues</td><td style = \"text-align: left;\">$16.5 billion</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">12</td><td style = \"text-align: left;\">Operating revenue, record production and distribution</td><td style = \"text-align: left;\">$ 614.0 million</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">13</td><td style = \"text-align: left;\">Operating revenue, consumer goods rental</td><td style = \"text-align: left;\">$2.4 billion</td></tr><tr><td style = \"text-align: right;\">&vellip;</td><td style = \"text-align: right;\">&vellip;</td><td style = \"text-align: right;\">&vellip;</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">152</td><td style = \"text-align: left;\">Unfilled orders</td><td style = \"text-align: left;\">$106.3 billion</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">153</td><td style = \"text-align: left;\">Book publishers operating revenue</td><td style = \"text-align: left;\">$1,495.8 million</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">154</td><td style = \"text-align: left;\">Average weekly earnings</td><td style = \"text-align: left;\">$1,511.58</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">155</td><td style = \"text-align: left;\">Capital expenditures: Non-residential construction</td><td style = \"text-align: left;\">$2,566.9 million</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">156</td><td style = \"text-align: left;\">Imports of services</td><td style = \"text-align: left;\">$14.5 billion</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">157</td><td style = \"text-align: left;\">Residential stock</td><td style = \"text-align: left;\">$2,314 billion &lt;span class=&quot;sd-indicator-value-details&quot;&gt;(2012 dollars)&lt;/span&gt;</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">158</td><td style = \"text-align: left;\">Operating revenue, drinking places (alcoholic beverages)</td><td style = \"text-align: left;\">$1.5 billion</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">159</td><td style = \"text-align: left;\">Operating revenue, film, television and video production</td><td style = \"text-align: left;\">$ 9.2 billion</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">160</td><td style = \"text-align: left;\">Canadian General Government expense on environmental protection</td><td style = \"text-align: left;\">$12.6 billion </td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">161</td><td style = \"text-align: left;\">Investment in non-residential building construction</td><td style = \"text-align: left;\">$14.4 billion</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">162</td><td style = \"text-align: left;\">Imports of goods and services</td><td style = \"text-align: left;\">$78.4 billion</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">163</td><td style = \"text-align: left;\">Wholesale sales</td><td style = \"text-align: left;\">$36.8 million</td></tr></tbody></table></div>"
],
"text/latex": [
"\\begin{tabular}{r|cc}\n",
"\t& Category & \\\\\n",
"\t\\hline\n",
"\t& String & \\\\\n",
"\t\\hline\n",
"\t1 & Worldwide sales by Canadian multinationals & $\\dots$ \\\\\n",
"\t2 & Total capital expenditures & $\\dots$ \\\\\n",
"\t3 & Median employment income five years after graduation for college-level diploma graduates & $\\dots$ \\\\\n",
"\t4 & Farm cash receipts & $\\dots$ \\\\\n",
"\t5 & Annual labour productivity (chained (2012) dollars) & $\\dots$ \\\\\n",
"\t6 & Operating revenue, motion picture and video exhibition & $\\dots$ \\\\\n",
"\t7 & Domestic tourism spending on culture products, Canada & $\\dots$ \\\\\n",
"\t8 & Operating revenue, non-residential leasing & $\\dots$ \\\\\n",
"\t9 & Operating revenue, architectural services and landscape architectural services & $\\dots$ \\\\\n",
"\t10 & Average annual wages and salaries by males, natural resources sector & $\\dots$ \\\\\n",
"\t11 & Rail transportation operating revenues & $\\dots$ \\\\\n",
"\t12 & Operating revenue, record production and distribution & $\\dots$ \\\\\n",
"\t13 & Operating revenue, consumer goods rental & $\\dots$ \\\\\n",
"\t14 & International exports of sport products & $\\dots$ \\\\\n",
"\t15 & Canada’s net international investment position & $\\dots$ \\\\\n",
"\t16 & Realized net farm income & $\\dots$ \\\\\n",
"\t17 & Operating revenue, promoters (presenters) of performing arts, sports and similar events & $\\dots$ \\\\\n",
"\t18 & Operating revenue, automotive equipment rental and leasing & $\\dots$ \\\\\n",
"\t19 & Operating revenue, full-service restaurants & $\\dots$ \\\\\n",
"\t20 & Median household after-tax income & $\\dots$ \\\\\n",
"\t21 & Total investment in building construction & $\\dots$ \\\\\n",
"\t22 & Retail sales & $\\dots$ \\\\\n",
"\t23 & Greenhouse vegetable and fruit sales & $\\dots$ \\\\\n",
"\t24 & Operating revenue, software publishers & $\\dots$ \\\\\n",
"\t25 & Average hourly wage, environmental and clean technology products sector—part-time jobs & $\\dots$ \\\\\n",
"\t26 & Food services and drinking places sales & $\\dots$ \\\\\n",
"\t27 & Operating revenue, engineering services & $\\dots$ \\\\\n",
"\t28 & Operating revenue, total performing arts companies & $\\dots$ \\\\\n",
"\t29 & Operating revenue, independent artists, writers and performers & $\\dots$ \\\\\n",
"\t30 & Operating revenue, tour operators & $\\dots$ \\\\\n",
"\t$\\dots$ & $\\dots$ & \\\\\n",
"\\end{tabular}\n"
],
"text/plain": [
"\u001b[1m163×2 DataFrame\u001b[0m\n",
"\u001b[1m Row \u001b[0m│\u001b[1m Category \u001b[0m\u001b[1m Value \u001b[0m\n",
" │\u001b[90m String \u001b[0m\u001b[90m String \u001b[0m\n",
"─────┼──────────────────────────────────────────────────────────────────────\n",
" 1 │ Worldwide sales by Canadian mult… $2.1 trillion\n",
" 2 │ Total capital expenditures $25.8 billion\n",
" 3 │ Median employment income five ye… $43,000\n",
" 4 │ Farm cash receipts $18,713.0 million\n",
" 5 │ Annual labour productivity (chai… $44.00 per hour\n",
" 6 │ Operating revenue, motion pictur… $552.9 million\n",
" 7 │ Domestic tourism spending on cul… $1.2 billion\n",
" 8 │ Operating revenue, non-residenti… $62.7 billion\n",
" 9 │ Operating revenue, architectural… $5.4 billion\n",
" 10 │ Average annual wages and salarie… $100,143\n",
" 11 │ Rail transportation operating re… $16.5 billion\n",
" ⋮ │ ⋮ ⋮\n",
" 154 │ Average weekly earnings $1,511.58\n",
" 155 │ Capital expenditures: Non-reside… $2,566.9 million\n",
" 156 │ Imports of services $14.5 billion\n",
" 157 │ Residential stock $2,314 billion <span class=\"sd-i…\n",
" 158 │ Operating revenue, drinking plac… $1.5 billion\n",
" 159 │ Operating revenue, film, televis… $ 9.2 billion\n",
" 160 │ Canadian General Government expe… $12.6 billion\n",
" 161 │ Investment in non-residential bu… $14.4 billion\n",
" 162 │ Imports of goods and services $78.4 billion\n",
" 163 │ Wholesale sales $36.8 million\n",
"\u001b[36m 142 rows omitted\u001b[0m"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rawdata = JSON.parse(String(HTTP.request(\"GET\", \"https://www150.statcan.gc.ca/n1/dai-quo/ssi/homepage/ind-all.json\").body))\n",
"data = Dict{String, String}()\n",
"\n",
"\n",
"for indicator in rawdata[\"results\"][\"indicators\"]\n",
" val = indicator[\"value\"][\"en\"]\n",
"\n",
" #Skip non-economic data\n",
" if !startswith(val, \"\\$\") continue end\n",
"\n",
" data[indicator[\"title\"][\"en\"]] = val\n",
"end\n",
"\n",
"econ_categories = collect(keys(data))\n",
"econ_values = collect(values(data))\n",
"\n",
"DataFrame(\n",
" \"Category\" => econ_categories,\n",
" \"Value\" => econ_values\n",
")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The dataset has values that range from format and unit, but that should not matter for analyzing for the Benford's law. Now we can get the first digits, and graph it"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
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},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"#Dirty trick to get the first character in the values provided\n",
"#We cannot do the mathematical way because it's hard to parse \"$78.4 billion\", etc. into a number ;(\n",
"function firstdigit(str::String)\n",
" for c in str\n",
" if Int(c) >= Int('1') && Int(c) <= Int('9')\n",
" return parse(Int, c)\n",
" end\n",
" end\n",
"end\n",
"\n",
"econ_firstdigits = firstdigit.(econ_values)\n",
"\n",
"econ_digitcount = zeros(Integer, 9)\n",
"\n",
"#Iterate through each digit\n",
"for digit in econ_firstdigits\n",
" #Skip over digit if it is nothing (no numbers in the value)\n",
" if digit == nothing continue end\n",
"\n",
" econ_digitcount[digit] = econ_digitcount[digit] + 1\n",
"end\n",
"\n",
"pg = plot(1 : 9, econ_digitcount,\n",
" title = \"Benford's law in the Economic Indicators of Canada\",\n",
" ylabel = \"Frequency\",\n",
" xlabel = \"Digit\",\n",
" label = \"Frequency of digit\",\n",
" lw = 5)\n",
"bg = bar(1 : 9, econ_digitcount,\n",
" ylabel = \"Frequency\",\n",
" xlabel = \"Digit\",\n",
" label = \"Frequency of digit\")\n",
"\n",
"plot(pg, bg, \n",
" layout = (2, 1))\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The graphs above further support Benford's law by showing the frequency digits (except for 6) logarithmically decreasing starting at one. The only outlier seems to be 6. Overlaying the line graph of Benford's law and the values we got with this data set further supports this claim."
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
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},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"df = DataFrame(x = 1 : 9,\n",
" a = benfords_values,\n",
" b = percentages(econ_digitcount))\n",
"\n",
"@df df plot(:x, [:a :b], \n",
" colour = [:red :blue], \n",
" names = [ \"Benford's law\", \"Economic Indicators of Canada\"],\n",
" ylabel = \"Frequency of the leading digit (%)\",\n",
" xlabel = \"Digit\",\n",
" lw = 5)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Our line is even more identical to Benford's law than the previous data set, excluding 6. If we analyze the raw data we can see an enhanced insight on how this is possible."
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div><div style = \"float: left;\"><span>9×3 DataFrame</span></div><div style = \"clear: both;\"></div></div><div class = \"data-frame\" style = \"overflow-x: scroll;\"><table class = \"data-frame\" style = \"margin-bottom: 6px;\"><thead><tr class = \"header\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">Row</th><th style = \"text-align: left;\">Frequency</th><th style = \"text-align: left;\">Leading digit percent</th><th style = \"text-align: left;\">Benford&apos;s Percentages</th></tr><tr class = \"subheader headerLastRow\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\"></th><th title = \"Integer\" style = \"text-align: left;\">Integer</th><th title = \"Float64\" style = \"text-align: left;\">Float64</th><th title = \"Float64\" style = \"text-align: left;\">Float64</th></tr></thead><tbody><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1</td><td style = \"text-align: right;\">58</td><td style = \"text-align: right;\">35.5828</td><td style = \"text-align: right;\">30.103</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">2</td><td style = \"text-align: right;\">28</td><td style = \"text-align: right;\">17.1779</td><td style = \"text-align: right;\">17.6091</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">3</td><td style = \"text-align: right;\">16</td><td style = \"text-align: right;\">9.81595</td><td style = \"text-align: right;\">12.4939</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">4</td><td style = \"text-align: right;\">14</td><td style = \"text-align: right;\">8.58896</td><td style = \"text-align: right;\">9.691</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">5</td><td style = \"text-align: right;\">10</td><td style = \"text-align: right;\">6.13497</td><td style = \"text-align: right;\">7.91812</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">6</td><td style = \"text-align: right;\">17</td><td style = \"text-align: right;\">10.4294</td><td style = \"text-align: right;\">6.69468</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">7</td><td style = \"text-align: right;\">9</td><td style = \"text-align: right;\">5.52147</td><td style = \"text-align: right;\">5.79919</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">8</td><td style = \"text-align: right;\">5</td><td style = \"text-align: right;\">3.06748</td><td style = \"text-align: right;\">5.11525</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9</td><td style = \"text-align: right;\">6</td><td style = \"text-align: right;\">3.68098</td><td style = \"text-align: right;\">4.57575</td></tr></tbody></table></div>"
],
"text/latex": [
"\\begin{tabular}{r|ccc}\n",
"\t& Frequency & Leading digit percent & Benford's Percentages\\\\\n",
"\t\\hline\n",
"\t& Integer & Float64 & Float64\\\\\n",
"\t\\hline\n",
"\t1 & 58 & 35.5828 & 30.103 \\\\\n",
"\t2 & 28 & 17.1779 & 17.6091 \\\\\n",
"\t3 & 16 & 9.81595 & 12.4939 \\\\\n",
"\t4 & 14 & 8.58896 & 9.691 \\\\\n",
"\t5 & 10 & 6.13497 & 7.91812 \\\\\n",
"\t6 & 17 & 10.4294 & 6.69468 \\\\\n",
"\t7 & 9 & 5.52147 & 5.79919 \\\\\n",
"\t8 & 5 & 3.06748 & 5.11525 \\\\\n",
"\t9 & 6 & 3.68098 & 4.57575 \\\\\n",
"\\end{tabular}\n"
],
"text/plain": [
"\u001b[1m9×3 DataFrame\u001b[0m\n",
"\u001b[1m Row \u001b[0m│\u001b[1m Frequency \u001b[0m\u001b[1m Leading digit percent \u001b[0m\u001b[1m Benford's Percentages \u001b[0m\n",
" │\u001b[90m Integer \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\n",
"─────┼─────────────────────────────────────────────────────────\n",
" 1 │ 58 35.5828 30.103\n",
" 2 │ 28 17.1779 17.6091\n",
" 3 │ 16 9.81595 12.4939\n",
" 4 │ 14 8.58896 9.691\n",
" 5 │ 10 6.13497 7.91812\n",
" 6 │ 17 10.4294 6.69468\n",
" 7 │ 9 5.52147 5.79919\n",
" 8 │ 5 3.06748 5.11525\n",
" 9 │ 6 3.68098 4.57575"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"DataFrame(\n",
" \"Frequency\" => econ_digitcount,\n",
" \"Leading digit percent\" => percentages(econ_digitcount),\n",
" \"Benford's Percentages\" => benfords_values\n",
")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"With a maximum of 5% margin of error, it is confirmed that this dataset follows Benford's law more than the previous one. \n",
"\n",
"If more data was added to the graph, it would be even closer to approaching Benford's law, but I also believe that the spike in numbers with a leading number of 6 will stay because when including all data from the entire StatsCanada dataset, not just the economic ones, the spike is still shown and even more amplified. One reason for this could be that the same value is being repeated, for example, the percentage of people employed aligning with the percentage of people over working age could have a very prominent overlap, making the leading digits repeat."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Example 3: Against Benford! - Best-selling video games of all time\n",
"\n",
"The examples shown beforehand display datasets where Benford's law is evident, but not all datasets follow this pattern.\n",
"\n",
"The top 50 best-selling video games of all time, retrieved from Wikipedia is an example of a dataset which does not follow Benford's law."
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.8/Project.toml`\n",
"\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.8/Manifest.toml`\n"
]
},
{
"data": {
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"<div><div style = \"float: left;\"><span>50×2 DataFrame</span></div><div style = \"float: right;\"><span style = \"font-style: italic;\">25 rows omitted</span></div><div style = \"clear: both;\"></div></div><div class = \"data-frame\" style = \"overflow-x: scroll;\"><table class = \"data-frame\" style = \"margin-bottom: 6px;\"><thead><tr class = \"header\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">Row</th><th style = \"text-align: left;\">Game</th><th style = \"text-align: left;\">Sales</th></tr><tr class = \"subheader headerLastRow\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\"></th><th title = \"String\" style = \"text-align: left;\">String</th><th title = \"String15\" style = \"text-align: left;\">String15</th></tr></thead><tbody><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1</td><td style = \"text-align: left;\">Super Mario Bros.</td><td style = \"text-align: left;\">58,000,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">2</td><td style = \"text-align: left;\">Pokémon Gold / Silver / Crystal</td><td style = \"text-align: left;\">29,490,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">3</td><td style = \"text-align: left;\">Tetris (EA)</td><td style = \"text-align: left;\">100,000,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">4</td><td style = \"text-align: left;\">Wii Sports</td><td style = \"text-align: left;\">82,900,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">5</td><td style = \"text-align: left;\">Human: Fall Flat</td><td style = \"text-align: left;\">30,000,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">6</td><td style = \"text-align: left;\">Pokémon Sun / Moon / Ultra Sun / Ultra Moon</td><td style = \"text-align: left;\">25,370,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">7</td><td style = \"text-align: left;\">Mario Kart 8 / Deluxe</td><td style = \"text-align: left;\">55,280,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">8</td><td style = \"text-align: left;\">Overwatch</td><td style = \"text-align: left;\">50,000,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9</td><td style = \"text-align: left;\">New Super Mario Bros.</td><td style = \"text-align: left;\">30,800,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">10</td><td style = \"text-align: left;\">Grand Theft Auto IV</td><td style = \"text-align: left;\">25,000,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">11</td><td style = \"text-align: left;\">PUBG: Battlegrounds</td><td style = \"text-align: left;\">75,000,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">12</td><td style = \"text-align: left;\">Call of Duty: Modern Warfare 2</td><td style = \"text-align: left;\">22,700,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">13</td><td style = \"text-align: left;\">Wii Sports Resort</td><td style = \"text-align: left;\">33,140,000</td></tr><tr><td style = \"text-align: right;\">&vellip;</td><td style = \"text-align: right;\">&vellip;</td><td style = \"text-align: right;\">&vellip;</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">39</td><td style = \"text-align: left;\">Grand Theft Auto: San Andreas</td><td style = \"text-align: left;\">27,500,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">40</td><td style = \"text-align: left;\">Pokémon Red / Green / Blue / Yellow</td><td style = \"text-align: left;\">47,520,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">41</td><td style = \"text-align: left;\">Duck Hunt</td><td style = \"text-align: left;\">28,300,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">42</td><td style = \"text-align: left;\">Sonic the Hedgehog</td><td style = \"text-align: left;\">23,982,960</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">43</td><td style = \"text-align: left;\">The Witcher 3 / Hearts of Stone / Blood and Wine</td><td style = \"text-align: left;\">40,000,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">44</td><td style = \"text-align: left;\">Super Smash Bros. Ultimate</td><td style = \"text-align: left;\">28,820,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">45</td><td style = \"text-align: left;\">Super Mario Bros. 3</td><td style = \"text-align: left;\">24,430,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">46</td><td style = \"text-align: left;\">Call of Duty: Modern Warfare 3</td><td style = \"text-align: left;\">26,500,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">47</td><td style = \"text-align: left;\">Call of Duty: Modern Warfare</td><td style = \"text-align: left;\">30,000,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">48</td><td style = \"text-align: left;\">Diablo III / Reaper of Souls</td><td style = \"text-align: left;\">30,000,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">49</td><td style = \"text-align: left;\">Mario Kart Wii</td><td style = \"text-align: left;\">37,380,000</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">50</td><td style = \"text-align: left;\">Wii Play</td><td style = \"text-align: left;\">28,020,000</td></tr></tbody></table></div>"
],
"text/latex": [
"\\begin{tabular}{r|cc}\n",
"\t& Game & Sales\\\\\n",
"\t\\hline\n",
"\t& String & String15\\\\\n",
"\t\\hline\n",
"\t1 & Super Mario Bros. & 58,000,000 \\\\\n",
"\t2 & Pokémon Gold / Silver / Crystal & 29,490,000 \\\\\n",
"\t3 & Tetris (EA) & 100,000,000 \\\\\n",
"\t4 & Wii Sports & 82,900,000 \\\\\n",
"\t5 & Human: Fall Flat & 30,000,000 \\\\\n",
"\t6 & Pokémon Sun / Moon / Ultra Sun / Ultra Moon & 25,370,000 \\\\\n",
"\t7 & Mario Kart 8 / Deluxe & 55,280,000 \\\\\n",
"\t8 & Overwatch & 50,000,000 \\\\\n",
"\t9 & New Super Mario Bros. & 30,800,000 \\\\\n",
"\t10 & Grand Theft Auto IV & 25,000,000 \\\\\n",
"\t11 & PUBG: Battlegrounds & 75,000,000 \\\\\n",
"\t12 & Call of Duty: Modern Warfare 2 & 22,700,000 \\\\\n",
"\t13 & Wii Sports Resort & 33,140,000 \\\\\n",
"\t14 & New Super Mario Bros. Wii & 30,320,000 \\\\\n",
"\t15 & Pac-Man & 42,071,635 \\\\\n",
"\t16 & FIFA 18 & 24,000,000 \\\\\n",
"\t17 & Pokémon Sword / Shield & 24,500,000 \\\\\n",
"\t18 & Kinect Adventures! & 24,000,000 \\\\\n",
"\t19 & Tetris (1989) & 43,000,000 \\\\\n",
"\t20 & Super Mario 64 / DS & 22,960,000 \\\\\n",
"\t21 & Super Mario World & 26,662,500 \\\\\n",
"\t22 & The Legend of Zelda: Breath of the Wild & 28,840,000 \\\\\n",
"\t23 & Animal Crossing: New Horizons & 39,380,000 \\\\\n",
"\t24 & Pokémon Diamond / Pearl / Platinum & 24,730,000 \\\\\n",
"\t25 & Red Dead Redemption & 23,000,000 \\\\\n",
"\t26 & The Walking Dead & 28,000,000 \\\\\n",
"\t27 & Borderlands 2 & 26,000,000 \\\\\n",
"\t28 & Call of Duty: Black Ops II & 24,200,000 \\\\\n",
"\t29 & Red Dead Redemption 2 & 45,000,000 \\\\\n",
"\t30 & Mario Kart DS & 23,600,000 \\\\\n",
"\t$\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
"\\end{tabular}\n"
],
"text/plain": [
"\u001b[1m50×2 DataFrame\u001b[0m\n",
"\u001b[1m Row \u001b[0m│\u001b[1m Game \u001b[0m\u001b[1m Sales \u001b[0m\n",
" │\u001b[90m String \u001b[0m\u001b[90m String15 \u001b[0m\n",
"─────┼────────────────────────────────────────────────\n",
" 1 │ Super Mario Bros. 58,000,000\n",
" 2 │ Pokémon Gold / Silver / Crystal 29,490,000\n",
" 3 │ Tetris (EA) 100,000,000\n",
" 4 │ Wii Sports 82,900,000\n",
" 5 │ Human: Fall Flat 30,000,000\n",
" 6 │ Pokémon Sun / Moon / Ultra Sun /… 25,370,000\n",
" 7 │ Mario Kart 8 / Deluxe 55,280,000\n",
" 8 │ Overwatch 50,000,000\n",
" 9 │ New Super Mario Bros. 30,800,000\n",
" 10 │ Grand Theft Auto IV 25,000,000\n",
" 11 │ PUBG: Battlegrounds 75,000,000\n",
" ⋮ │ ⋮ ⋮\n",
" 41 │ Duck Hunt 28,300,000\n",
" 42 │ Sonic the Hedgehog 23,982,960\n",
" 43 │ The Witcher 3 / Hearts of Stone … 40,000,000\n",
" 44 │ Super Smash Bros. Ultimate 28,820,000\n",
" 45 │ Super Mario Bros. 3 24,430,000\n",
" 46 │ Call of Duty: Modern Warfare 3 26,500,000\n",
" 47 │ Call of Duty: Modern Warfare 30,000,000\n",
" 48 │ Diablo III / Reaper of Souls 30,000,000\n",
" 49 │ Mario Kart Wii 37,380,000\n",
" 50 │ Wii Play 28,020,000\n",
"\u001b[36m 29 rows omitted\u001b[0m"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Pkg.add(\"CSV\")\n",
"\n",
"using CSV\n",
"\n",
"data = CSV.File(\"best-selling-videogames.csv\") |> Dict\n",
"\n",
"sale_count = collect(values(data))\n",
"DataFrame(\n",
" \"Game\" => collect(keys(data)),\n",
" \"Sales\" => sale_count\n",
")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Using the data, we can graph and compare with Benford's law."
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
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},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sales_firstdigits = []\n",
"\n",
"for sale in sale_count\n",
" if typeof(sale) == Missing continue end\n",
" push!(sales_firstdigits, firstdigit(String(sale)))\n",
"end\n",
"\n",
"sales_digitcount = zeros(Integer, 9)\n",
"\n",
"for digit in sales_firstdigits\n",
" if digit == nothing continue end\n",
"\n",
" sales_digitcount[digit] = sales_digitcount[digit] + 1\n",
"end\n",
"\n",
"pg = plot(1 : 9, sales_digitcount,\n",
" title = \"Benford's law (not) in the best-selling video games of all time\",\n",
" ylabel = \"Frequency\",\n",
" xlabel = \"Digit\",\n",
" label = \"Frequency of digit\",\n",
" lw = 5)\n",
"bg = bar(1 : 9, sales_digitcount,\n",
" ylabel = \"Frequency\",\n",
" xlabel = \"Digit\",\n",
" label = \"Frequency of digit\")\n",
"\n",
"plot(pg, bg, \n",
" layout = (2, 1))\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Right off the bat, it is extremely clear that the data does not fit Benford's law, with digit 1 being one of the least frequent starting digits, with digit 2 having an overwhelming majority over the other digits.\n",
"\n",
"Graphing it with Benford's data shows an extreme disparity"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
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},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"df = DataFrame(x = 1 : 9,\n",
" a = benfords_values,\n",
" b = percentages(sales_digitcount))\n",
"\n",
"@df df plot(:x, [:a :b], \n",
" colour = [:red :blue], \n",
" names = [ \"Benford's law\", \"Best selling games of all time\"],\n",
" ylabel = \"Frequency of the leading digit (%)\",\n",
" xlabel = \"Digit\",\n",
" lw = 5)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Comparing the raw data of the 2 datasets provides concrete evidence supporting how this dataset does not follow the law."
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div><div style = \"float: left;\"><span>9×3 DataFrame</span></div><div style = \"clear: both;\"></div></div><div class = \"data-frame\" style = \"overflow-x: scroll;\"><table class = \"data-frame\" style = \"margin-bottom: 6px;\"><thead><tr class = \"header\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">Row</th><th style = \"text-align: left;\">Frequency</th><th style = \"text-align: left;\">Leading digit percent</th><th style = \"text-align: left;\">Benford&apos;s Percentages</th></tr><tr class = \"subheader headerLastRow\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\"></th><th title = \"Integer\" style = \"text-align: left;\">Integer</th><th title = \"Float64\" style = \"text-align: left;\">Float64</th><th title = \"Float64\" style = \"text-align: left;\">Float64</th></tr></thead><tbody><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1</td><td style = \"text-align: right;\">2</td><td style = \"text-align: right;\">4.0</td><td style = \"text-align: right;\">30.103</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">2</td><td style = \"text-align: right;\">27</td><td style = \"text-align: right;\">54.0</td><td style = \"text-align: right;\">17.6091</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">3</td><td style = \"text-align: right;\">9</td><td style = \"text-align: right;\">18.0</td><td style = \"text-align: right;\">12.4939</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">4</td><td style = \"text-align: right;\">7</td><td style = \"text-align: right;\">14.0</td><td style = \"text-align: right;\">9.691</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">5</td><td style = \"text-align: right;\">3</td><td style = \"text-align: right;\">6.0</td><td style = \"text-align: right;\">7.91812</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">6</td><td style = \"text-align: right;\">0</td><td style = \"text-align: right;\">0.0</td><td style = \"text-align: right;\">6.69468</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">7</td><td style = \"text-align: right;\">1</td><td style = \"text-align: right;\">2.0</td><td style = \"text-align: right;\">5.79919</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">8</td><td style = \"text-align: right;\">1</td><td style = \"text-align: right;\">2.0</td><td style = \"text-align: right;\">5.11525</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">9</td><td style = \"text-align: right;\">0</td><td style = \"text-align: right;\">0.0</td><td style = \"text-align: right;\">4.57575</td></tr></tbody></table></div>"
],
"text/latex": [
"\\begin{tabular}{r|ccc}\n",
"\t& Frequency & Leading digit percent & Benford's Percentages\\\\\n",
"\t\\hline\n",
"\t& Integer & Float64 & Float64\\\\\n",
"\t\\hline\n",
"\t1 & 2 & 4.0 & 30.103 \\\\\n",
"\t2 & 27 & 54.0 & 17.6091 \\\\\n",
"\t3 & 9 & 18.0 & 12.4939 \\\\\n",
"\t4 & 7 & 14.0 & 9.691 \\\\\n",
"\t5 & 3 & 6.0 & 7.91812 \\\\\n",
"\t6 & 0 & 0.0 & 6.69468 \\\\\n",
"\t7 & 1 & 2.0 & 5.79919 \\\\\n",
"\t8 & 1 & 2.0 & 5.11525 \\\\\n",
"\t9 & 0 & 0.0 & 4.57575 \\\\\n",
"\\end{tabular}\n"
],
"text/plain": [
"\u001b[1m9×3 DataFrame\u001b[0m\n",
"\u001b[1m Row \u001b[0m│\u001b[1m Frequency \u001b[0m\u001b[1m Leading digit percent \u001b[0m\u001b[1m Benford's Percentages \u001b[0m\n",
" │\u001b[90m Integer \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\n",
"─────┼─────────────────────────────────────────────────────────\n",
" 1 │ 2 4.0 30.103\n",
" 2 │ 27 54.0 17.6091\n",
" 3 │ 9 18.0 12.4939\n",
" 4 │ 7 14.0 9.691\n",
" 5 │ 3 6.0 7.91812\n",
" 6 │ 0 0.0 6.69468\n",
" 7 │ 1 2.0 5.79919\n",
" 8 │ 1 2.0 5.11525\n",
" 9 │ 0 0.0 4.57575"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"DataFrame(\n",
" \"Frequency\" => sales_digitcount,\n",
" \"Leading digit percent\" => percentages(sales_digitcount),\n",
" \"Benford's Percentages\" => benfords_values\n",
")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The reasoning behind such a disparity takes form in many factors, starting with the data itself, which is approximate of how many purchases each game has but is mostly accurate coming from reputable sources (the company themselves, well-known news outlets). The second factor that can influence the data and its accuracy is the number of inputs given (sales of each game) is much lesser than in the previous examples with only 50 values. I believe that with more data points the result would tend to be more like Benford's data but still be unique enough to not follow the law. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Conclusion\n",
"\n",
"Benford's law is an interesting phenomenon that seems to be present in nearly every data set. It is interesting to note the correlation of the base of the logarithm in the equation to calculate the frequencies of the first digit in the law has a really important meaning, that being defining the base of the counting system that the digits are a part of. The logarithm in the law is base 10, corresponding with our base 10 counting system, however, this means that Benford's law also works for non-base 10 counting systems, such as binary, octal, and hexadecimal.\n",
"\n",
"Recall back to the original equation\n",
"$$\n",
"P_{10}(d) = \\log_{10}(1 + {1 \\over d}),\\newline\n",
"\\newline\n",
"1 \\e d \\le 9, \\in \\mathbb Z\n",
"$$\n",
"\n",
"This produced the frequency that a number from 1 - 9 will occur, which are all the non-zero integers in the base 10 counting system.\n",
"\n",
"Changing the base of the logarithm will produce results for the base supplied, for example for hexadecimal (base 8) which is commonly used in computing\n",
"\n",
"$$\n",
"P_8(d) = \\log_{8}(1 + {1 \\over d}),\\newline\n",
"\\newline\n",
"d \\in \\{n \\lvert n \\in \\mathbb Z, 1 \\le n \\le 7\\}\n",
"$$\n",
"\n",
"This produces the frequencies of"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div><div style = \"float: left;\"><span>7×1 DataFrame</span></div><div style = \"clear: both;\"></div></div><div class = \"data-frame\" style = \"overflow-x: scroll;\"><table class = \"data-frame\" style = \"margin-bottom: 6px;\"><thead><tr class = \"header\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">Row</th><th style = \"text-align: left;\">Frequency (%)</th></tr><tr class = \"subheader headerLastRow\"><th class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\"></th><th title = \"Float64\" style = \"text-align: left;\">Float64</th></tr></thead><tbody><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">1</td><td style = \"text-align: right;\">48.0</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">2</td><td style = \"text-align: right;\">28.0782</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">3</td><td style = \"text-align: right;\">19.9218</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">4</td><td style = \"text-align: right;\">15.4525</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">5</td><td style = \"text-align: right;\">12.6257</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">6</td><td style = \"text-align: right;\">10.6748</td></tr><tr><td class = \"rowNumber\" style = \"font-weight: bold; text-align: right;\">7</td><td style = \"text-align: right;\">9.24696</td></tr></tbody></table></div>"
],
"text/latex": [
"\\begin{tabular}{r|c}\n",
"\t& Frequency (\\%)\\\\\n",
"\t\\hline\n",
"\t& Float64\\\\\n",
"\t\\hline\n",
"\t1 & 48.0 \\\\\n",
"\t2 & 28.0782 \\\\\n",
"\t3 & 19.9218 \\\\\n",
"\t4 & 15.4525 \\\\\n",
"\t5 & 12.6257 \\\\\n",
"\t6 & 10.6748 \\\\\n",
"\t7 & 9.24696 \\\\\n",
"\\end{tabular}\n"
],
"text/plain": [
"\u001b[1m7×1 DataFrame\u001b[0m\n",
"\u001b[1m Row \u001b[0m│\u001b[1m Frequency (%) \u001b[0m\n",
" │\u001b[90m Float64 \u001b[0m\n",
"─────┼───────────────\n",
" 1 │ 48.0\n",
" 2 │ 28.0782\n",
" 3 │ 19.9218\n",
" 4 │ 15.4525\n",
" 5 │ 12.6257\n",
" 6 │ 10.6748\n",
" 7 │ 9.24696"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"P_8(d) = log(8, 1 + 1 / d)\n",
"\n",
"# Base 8 uses the digits 0 - 7, so 8_octal = 10, and 9_octal = 11\n",
"x = 1 : 7\n",
"\n",
"#144 base 8 is 100 decimal\n",
"octal_values = P_8.(x) .* 144\n",
"\n",
"DataFrame(\"Frequency (%)\" => octal_values)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"From this example, we can then conclude that you can achieve an equation to get the frequencies of the leading digit of any base by using the equation\n",
"\n",
"$$\n",
"P_b(n) = log_b(1 + 1 / n),\\newline\n",
"\\newline\n",
"b \\ge 2, b \\in \\mathbb Z,\\newline\n",
"\\newline\n",
"b - 1 \\ge n \\ge 1, b \\in \\mathbb Z\n",
"$$\n",
"\n"
]
}
],
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Raw
best-selling-videogames.csv
Game name Sales
Minecraft 238,000,000
Grand Theft Auto V 169,000,000
Tetris (EA) 100,000,000
Wii Sports 82,900,000
PUBG: Battlegrounds 75,000,000
Super Mario Bros. 58,000,000
Mario Kart 8 / Deluxe 55,280,000
Overwatch 50,000,000
Pokémon Red / Green / Blue / Yellow 47,520,000
Red Dead Redemption 2 45,000,000
Terraria 44,500,000
Wii Fit / Plus 43,800,000
Tetris (1989) 43,000,000
Pac-Man 42,071,635
The Witcher 3 / Hearts of Stone / Blood and Wine 40,000,000
Animal Crossing: New Horizons 39,380,000
Mario Kart Wii 37,380,000
Wii Sports Resort 33,140,000
New Super Mario Bros. 30,800,000
New Super Mario Bros. Wii 30,320,000
Call of Duty: Modern Warfare 30,000,000
Diablo III / Reaper of Souls 30,000,000
Human: Fall Flat 30,000,000
The Elder Scrolls V: Skyrim 30,000,000
Pokémon Gold / Silver / Crystal 29,490,000
The Legend of Zelda: Breath of the Wild 28,840,000
Super Smash Bros. Ultimate 28,820,000
Duck Hunt 28,300,000
Wii Play 28,020,000
The Walking Dead 28,000,000
Grand Theft Auto: San Andreas 27,500,000
Super Mario World 26,662,500
Call of Duty: Modern Warfare 3 26,500,000
Call of Duty: Black Ops 26,200,000
Borderlands 2 26,000,000
Pokémon Sun / Moon / Ultra Sun / Ultra Moon 25,370,000
Grand Theft Auto IV 25,000,000
Pokémon Diamond / Pearl / Platinum 24,730,000
Pokémon Sword / Shield 24,500,000
Super Mario Bros. 3 24,430,000
Call of Duty: Black Ops II 24,200,000
FIFA 18 24,000,000
Kinect Adventures! 24,000,000
Sonic the Hedgehog 23,982,960
Nintendogs 23,960,000
Super Mario Odyssey 23,930,000
Mario Kart DS 23,600,000
Red Dead Redemption 23,000,000
Super Mario 64 / DS 22,960,000
Call of Duty: Modern Warfare 2 22,700,000
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