-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathindex.html
149 lines (126 loc) · 27 KB
/
index.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Pareto principle</title>
<meta name="description" content="Interactive chart illustrating Pareto principle (also known as “80-20 rule”). Pareto principle can be broadly defined as “80% of consequences come from 20% of causes” or “80% of results come from 20% of efforts”.">
<link rel="icon" href="./favicon.ico" sizes="any">
<link rel="icon" href="./icon.svg" type="image/svg+xml">
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/[email protected]/normalize.css">
<link rel="stylesheet" href="style.css">
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/katex.min.css" integrity="sha384-MlJdn/WNKDGXveldHDdyRP1R4CTHr3FeuDNfhsLPYrq2t0UBkUdK2jyTnXPEK1NQ" crossorigin="anonymous">
<script src="script.js"></script>
</head>
<body>
<main class="main">
<header class="header">
<h1>Pareto principle</h1>
<a href="#short-description">What is this?</a>
<ul class="lang-links">
<li><a href="index.ru.html">RU</a></li>
<li class="current-lang">EN</li>
</ul>
</header>
<div class="chart-section">
<div id="chart">
<svg preserveAspectRatio="none">
<defs>
<polyline id="right-arrow" points="-6,-4 0,0 -6,4"/>
<polyline id="left-arrow" points="6,-4 0,0 6,4"/>
<g id="divider-handle" class="divider-handle">
<polyline points="8,10 20,18 8,26"/>
<polyline points="-8,10 -20,18 -8,26"/>
</g>
</defs>
</svg>
</div>
</div>
</main>
<section class="content" id="short-description">
<h2>What is this?</h2>
<p>The above graph showcases <a href="https://en.wikipedia.org/wiki/Pareto_distribution" rel="noreferrer">Pareto distribution</a>
(see <a href="#maths">the math behind the graph</a>)
in order to illustrate the Pareto principle (also known as the "80-20 rule").</p>
</section>
<section class="content" id="full-description">
<h2>Pareto principle</h2>
<p>The principle was named after economist Vilfredo Pareto, who noticed
that approximately 80% of the land in Italy
was owned by 20% of the population at the end of the 19th century
<a href="#link-1">[1]</a>.</p>
<p>Later, similar patterns began to be noticed in other
areas. The principle can be broadly defined as
“80% of consequences come from 20% of causes” or
“80% of results come from 20% of efforts”.</p>
<p>It is important to remember that all these patterns are
approximate observations, not proven scientific laws.</p>
</section>
<section class="content" id="meaning">
<h2>What does this mean for me?</h2>
<p>This distribution is a mathematical model which means nothing by itself.</p>
<p>There are a lot of different phenomena and patterns which
approximately follow this distribution (and this diversity is fascinating!),
and there are some assumptions which can be made about similar future events.</p>
</section>
<section class="content" id="maths">
<!-- LaTeX is pre-generated with `npx katex [-d]` -->
<h2>Mathematical details of the graph</h2>
<p>It is worth noting that the horizontal axis goes from 1 (100%) to 0, so the graph is mirrored relative to the normal direction of the axis.</p>
<p>The graph is the inverse cumulative distribution function (also known as the quantile function <a href="#link-2">[2]</a>)</p>
<!-- x(p)\,=\,Q(p)\,=\,\inf \left\{x\in {\mathbb {R}}:p\leq F(x)\right\}=\frac{x_\mathrm{m}}{(1-p)^{\frac{1}{\alpha}}} -->
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mtext> </mtext><mo>=</mo><mtext> </mtext><mi>Q</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mtext> </mtext><mo>=</mo><mtext> </mtext><mi>inf</mi><mo></mo><mrow><mo fence="true">{</mo><mi>x</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi><mo>:</mo><mi>p</mi><mo>≤</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo fence="true">}</mo></mrow><mo>=</mo><mfrac><msub><mi>x</mi><mi mathvariant="normal">m</mi></msub><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mfrac><mn>1</mn><mi>α</mi></mfrac></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">x(p)\,=\,Q(p)\,=\,\inf \left\{x\in {\mathbb {R}}:p\leq F(x)\right\}=\frac{x_\mathrm{m}}{(1-p)^{\frac{1}{\alpha}}}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">Q</span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">in<span style="margin-right:0.07778em;">f</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">{</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathbb">R</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mclose delimcenter" style="top:0em;">}</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1871em;vertical-align:-1.0796em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.1704em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">p</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9396em;"><span style="top:-3.3486em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathrm mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.0796em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>
<p>where
<!-- \alpha = log_45 -->
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>=</mo><mi>l</mi><mi>o</mi><msub><mi>g</mi><mn>4</mn></msub><mn>5</mn></mrow><annotation encoding="application/x-tex">\alpha = log_45
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">5</span></span></span></span>
is <a href="https://en.wikipedia.org/wiki/Shape_parameter" rel="noreferrer">a shape parameter</a> of the Pareto distribution (Pareto index),<br>
<!-- x_m = 1 -->
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>m</mi></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x_m = 1
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>
is <a href="https://en.wikipedia.org/wiki/Scale_parameter" rel="noreferrer">a scale parameter</a>.
</p>
<p>You could have noticed that the graph is not shown fully
(it is bounded on the vertical axis),
since closer to 1 it tends to infinity. </p>
<p>The percentage under the graph is
<a href="https://en.wikipedia.org/wiki/Lorenz_curve" rel="noreferrer">the Lorenz (curve) function</a> <a href="#link-3">[3]</a>.
It shows part of the area under the graph that is occupied by the selected area:
</p>
<!-- L(F)={\frac {\int _{0}^{x}t\,f(t)\,dt}{\int _{0}^{\infty }t\,f(t)\,dt}}={\frac {\int _{0}^{F}x(F_{1})\,dF_{1}}{\int _{0}^{1}x(F_{1})\,dF_{1}}} -->
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>x</mi></msubsup><mi>t</mi><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>t</mi></mrow><mrow><msubsup><mo>∫</mo><mn>0</mn><mi mathvariant="normal">∞</mi></msubsup><mi>t</mi><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>F</mi></msubsup><mi>x</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><msub><mi>F</mi><mn>1</mn></msub></mrow><mrow><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mi>x</mi><mo stretchy="false">(</mo><msub><mi>F</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><msub><mi>F</mi><mn>1</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">L(F)={\frac {\int _{0}^{x}t\,f(t)\,dt}{\int _{0}^{\infty }t\,f(t)\,dt}}={\frac {\int _{0}^{F}x(F_{1})\,dF_{1}}{\int _{0}^{1}x(F_{1})\,dF_{1}}}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.7102em;vertical-align:-1.1051em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6051em;"><span style="top:-2.2507em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8593em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3558em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.7458em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8593em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3558em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.1051em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.0369em;vertical-align:-1.2548em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7821em;"><span style="top:-2.1372em;"><span class="pstrut" style="height:3.0362em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.009em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3558em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.2662em;"><span class="pstrut" style="height:3.0362em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.7821em;"><span class="pstrut" style="height:3.0362em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0362em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3558em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2548em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span>
</section>
<section class="content" id="resources">
<h2>Links</h2>
<ol>
<li id="link-1">Original publication by Vilfredo Pareto, in which his observation is mentioned:<br>
Vilfredo Pareto, Cours d'économie politique professé à l'université de Lausanne, 3 volumes, 1896–7
</li>
<li id="link-2">A good explanation of a quantile and quantile functions:<br>
Taboga, Marco (2021). "Quantile of a probability distribution", Lectures on probability theory and
mathematical statistics. Kindle Direct Publishing. Online appendix.
<a href="https://www.statlect.com/fundamentals-of-probability/quantile" rel="noreferrer">https://www.statlect.com/fundamentals-of-probability/quantile</a>.
</li>
<li id="link-3">Lorenz curve:<br>
Lorenz, M. O. (1905). "Methods of measuring the concentration of wealth". <em>Publications of the
American Statistical Association</em>. Publications of the American Statistical Association, Vol. 9, No.
70. 9 (70): 209–219.
</li>
<li id="link-4">Popular science video:<br>
The Zipf Mystery - VSauce - YouTube.
<a href="https://youtu.be/fCn8zs912OE" rel="noreferrer">https://youtu.be/fCn8zs912OE</a>
</li>
<li id="link-5">Pareto principle - Wikipedia.
<a href="https://en.wikipedia.org/wiki/Pareto_principle" rel="noreferrer">https://en.wikipedia.org/wiki/Pareto_principle</a>
</li>
</ol>
</section>
<section class="content" id="author">
<p>Made by <a href="https://www.sprkweb.dev">Vadim Saprykin</a>.</p>
<p>The text on this page is licensed under a <a rel="license noreferrer" href="http://creativecommons.org/licenses/by/4.0/">Creative Commons Attribution 4.0 International License</a>.</p>
</section>
</body>
</html>