Optimal choice of granularity in commonsense estimation

From this paper, it follows that half-orders of magnitude (HOMs) are generally preferable to full orders of magnitude for estimation.

It is my belief that half-orders of magnitude provide a useful tool not only to estimate quantities but to mentally process quantities as well. Though remembering the exact spacial relations between celestial bodies in our solar system might require too much effort, it’s easy to remember their relations rounded to half an order of magnitude.

First, I will give a definition of half-orders of magnitude, and define HOM(*x*). Then I will apply them to the demographic size of countries. I hope the reader will excuse my non-TeX rendering of mathematical formulas.

# Defining Half-Orders of Magnitude

## Orders of Magnitude

An *order of magnitude *of a real number *x* > 0, is simply the power of ten closest to *x*. For example, if *x* = 210, we might write *x *~ 10^2, where the tilde (“~”) denotes “is of the same order of magnitude as.” Of course “closest” is not a mathematically rigorous concept. Two real non-negative numbers have the same order of magnitude if their *base ten* logarithms of two numbers round to the same integer. Their order of magnitude is then ten to the power of that integer. For example:

**log 210 = 2.322… ≈ 2 = log 10^2 ⇒ 210 ~ 10^2 (1)**

As 2.322… rounds to 2, the order of magnitude of 210 is 10^2. The logarithm of the smallest number of this order must be 1.5, as 1.5 is the smallest number that rounds to 2:

**log x = 1.5 ⇒ x = 10^1.5 = 31.623… (2)**

Hence the real numbers *x* with the same order of magnitude 10^N can be thought of as the numbers *x *such that:

** x ∈ [10^(N-0.5), 10^(N+0.5)), with N an integer (3)**

or equivalently, *x* such that:

**log x ∈ [N – 0.5, N + 0.5), with N an integer (4)**

For example, numbers with order of magnitude 10^2 fall inside the interval [31.623…, 316.23…). Notice how this interval is half-closed: 316.23… = 10^2.5 has order of magnitude 10^3, but any number smaller than 10^2.5 has order of magnitude 10^2.

## Half-Orders of Magnitude

Half-orders of magnitude can be defined in a similar way. Let *x *be real and non-zero. Then the *half-order of magnitude *of *x *is defined as 10^(*M*/2), such that

*x *∈ [10^(*M*/2*–*0.25), 10^(*M*/2+0.25)), with *M *an integer (5)

or equivalently, *x *such that:

**log x ∈ [M/2 – 0.25, M/2 + 0.25), with M an integer (6)**

Notice how an half-order of magnitude can be either ten to the power of an integer or ten to the power of half an integer. For example, *x *= 210 gives:

**log 210 = 2.322 ≈ 2.5 = log 10^2.5 ⇒ 210 ~’ 10^2.5 (7)**

Where I have used tilde prime (“~'”) to indicate that two numbers have the same half-order of magnitude. This divides the positive real numbers into intervals that differ (on average) by a factor of 10^0.5 = 3.162…; a better granularity than 10.

It’s straightforward to generalize half-orders of magnitude into quarter-orders of magnitude, 2^(-*N*)-orders of magnitude, and even *r*-orders of magnitude (with *r* a real number). We can even define similar concepts for logarithmic functions with base unequal to ten, or for different functions altogether. However, I will presently limit myself to half-orders of magnitude.

I will now define HOM(*x*) for positive real numbers *x *(or *x *in some other set), using our existing tilde-prime notation for numbers with the same half-order of magnitude:

**HOM( x) = X ⇔ x ~’ 10^X and X = M/2 for some integer M (8)**

That is, if the half-order of magnitude of x is 10^*X*, then its HOM is *X*. This definition is convenient as is omits the somewhat tedious “ten to the power of …” in the half-order of magnitude of a number. Furthermore, I will use “half-order of magnitude” also synonymously with HOM; for example, 210 can be said to have half-order of magnitude 2.5 or 10^2.5, of course with the same meaning. Finally,

**HOM-X := { x, such that x ~’ 10^X}, with X = M/2 for some integer M (9)**

Or equivalently,

**HOM-X = { x, such that x ∈ HOM(10^X)}; X = M/2 for some integer M (10)**

## Example HOMS

### First Natural Numbers

Here, HOM-X is restricted to some set, in this case, the non-zero natural numbers. Formally, one could write HOM-X ∩ ℕ≥0, but I trust my current notation is clear enough when context is considered.

**HOM-0 = {1}****HOM-0.5 = {2, 3, 4, 5}****HOM-1 = {6, 7, …, 17}****HOM-1.5 = {18, 19, …, 56}****HOM-2 = {57, …, 177}****HOM-2.5 = {178, …, 562}****HOM-3 = {563, …, 1778}**

### Numbers with Three Significant Digits

**HOM-(-0.5) = {0.178, …, 0.562}****HOM-0 = {0.563, 0.564, …, 1.77}****HOM-0.5 = {1.78, …, 5.62}****HOM-1 = {5.63, …, 17.7}****HOM-1.5 = {17.8, …, 56.2}****HOM-2 = {56.3, …, 177}****HOM-2.5 = {178, …, 562}****HOM-3 = {563, …, 1.77 * 10^3}**

The most important numbers to remember when using HOMs are 10^0.25 and 10^0.75 rounded to some place, as the upper boundaries of HOM-Xs follow their digit expansion.

**10^0.25 = 1.778279…****10^0.75 = 5.623413…**

For example, the *supremum* of HOM-7 restricted to the natural numbers must be 17782794; the first *eight* (7+1) digits of 10^7.25, while the *infimum *of HOM-7 over the natural numbers is 5623414; the first *seven *digits of 10^6.75 plus one. Finally, 10^0.5 equals:

**10^0.5 = 3.162277…**

# Demographic Sizes of Countries

For easy reference, I use Worldometers 2018-5-17 as my source (using 2018 population statistics). Each group of countries is named after the country with the population closest to 10^X, starting with X = 9, and then decreasing by 0.5 steps, until X = 3 for Group Holy See. Also available in color, see below.

**GROUP INDIA (9)****=**{China, India}**GROUP U.S. (8.5)**= {U.S., Indonesia, Brazil, Pakistan, Nigeria}**GROUP EGYPT (8)**= {Bangladesh, Russia, Mexico, Japan, Ethiopia, Philippines, Egypt, Vietnam, DR Congo, Germany, Iran, Turkey, Thailand, U.K., France, Italy, Tanzania, South Africa}**GROUP MALAYSIA (7.5)**= {Myanmar, South Korea, Kenya, Colombia, Spain, Argentina, Uganda, Ukraine, Algeria, Sudan, Iraq, Poland, Canada, Afghanistan, Morocco, Saudi Arabia, Peru, Venezuela, Uzbekistan, Malaysia, Angola, Mozambique, Nepal, Ghana, Yemen, Madagascar, North Korea, Ivory Coast, Australia, Cameroon, Taiwan, Niger, Sri Lanka, Burkina Faso, Romania, Malawi, Mali, Kazakhstan, Syria, Chile}**GROUP SWEDEN (7)**= {Zambia, Guatemala, Netherlands, Zimbabwe, Ecuador, Senegal, Cambodia, Chad, Somalia, Guinea, South Sudan, Rwanda, Tunisia, Belgium, Cuba, Benin, Burundi, Bolivia, Greece, Haiti, Dominican Republic, Czech Republic, Portugal, Sweden, Azerbaijan, Jordan, Hungary, United Arab Emirates, Belarus, Honduras, Tajikistan, Serbia, Austria, Switzerland, Israel, Papua New Guinea, Togo, Sierra Leone, Hong Kong, Bulgaria, Laos, Paraguay, Libya, El Salvador, Nicaragua, Kyrgyzstan, Lebanon, Turkmenistan, Singapore, Denmark}**GROUP MONGOLIA (6.5)**= {Finland, Slovakia, Congo, Norway, Eritrea, State of Palestine, Costa Rica, Liberia, Oman, Ireland, New Zealand, Central African Republic, Mauritania, Kuwait, Croatia, Panama, Moldova, Georgia, Puerto Rico, Bosnia & Herzegovina, Uruguay, Mongolia, Albania, Armenia, Jamaica, Lithuania, Qatar, Namibia, Botswana, Lesotho, Gambia, TFYR Macedonia, Slovenia, Gabon, Latvia, Guinea-Bissau}**GROUP DJIBOUTI (6)**= {Bahrain, Swaziland, Trinidad and Tobago, Timor-Leste, Equatorial Guinea, Estonia, Mauritius, Cyprus, Djibouti, Fiji, Réunion, Comoros, Bhutan, Guyana, Macao, Montenegro, Solomon Islands, Luxembourg, Suriname, Western Sahara}**GROUP ICELAND (5.5)**= {Capo Verde, Guadeloupe, Maldives, Brunei, Malta, Bahamas, Martinique, Belize, Iceland, French Guiana, Barbados, French Polynesia, Vanuatu, New Caledonia, Mayotte, Sao Tome & Principe, Samoa, Saint Lucia}**GROUP ANTIGUA AND BARBUDA (5)**= {Channel Islands, Guam, Curaçao, Kiribati, St. Vincent & Grenadines, Tonga, Grenada, Micronesia, Aruba, U.S. Virgin Islands, Antigua and Barbuda, Seychelles, Isle of Man, Andorra, Dominica, Cayman Islands, Bermuda, Greenland}**GROUP BRITISH VIRGIN ISLANDS (4.5)**= {Saint Kitts & Nevis, American Samoa, Northern Mariana Islands, Marshall Islands, Faroe Islands, Sint Maarten, Monaco, Liechtenstein, Turks and Caicos, Gibraltar, San Marino, British Virgin Islands, Caribbean Netherlands, Palau}**GROUP TUVALU (4)**= {Cook Islands, Anguilla, Wallis & Futuna, Nauru, Tuvalu, Saint Pierre & Miquelon}**GROUP FALKLAND ISLANDS (3.5)**= {Montserrat, Saint Helena, Falkland Islands}**GROUP HOLY SEE (3)**= {Niue, Tokelau, Holy See}.

In the following chart, the number of countries in each group, as well as its total share of world population is displayed. Top three groups in demographic size: Group India, Group Egypt, Group Malaysia. Top three groups with most countries: Group Sweden, Group Malaysia, Group Mongolia. Over 99.5% of the world population belongs to one of the first seven groups.

The same groups with different colors for each geographic region: Asia (Southern ~, Eastern ~, South-Eastern ~, Central ~, Western ~), Africa (Western ~, Eastern ~, Northern ~, Southern ~, Middle ~), Northern America, Oceania (Australia & New Zealand, Melanesia, Polynesia, Micronesia), Latin America & Caribbean (South America, Central America, Caribbean), Europe (Western ~, Eastern ~, Northern ~, Southern ~).

**GROUP INDIA (9)****=**{China, India}**GROUP U.S. (8.5)**= {U.S., Indonesia, Brazil, Pakistan, Nigeria}**GROUP EGYPT (8)**= {Bangladesh, Russia, Mexico, Japan, Ethiopia, Philippines, Egypt, Vietnam, DR Congo, Germany, Iran, Turkey, Thailand, U.K., France, Italy, Tanzania, South Africa}**GROUP MALAYSIA (7.5)**= {Myanmar, South Korea, Kenya, Colombia, Spain, Argentina, Uganda, Ukraine, Algeria, Sudan, Iraq, Poland, Canada, Afghanistan, Morocco, Saudi Arabia, Peru, Venezuela, Uzbekistan, Malaysia, Angola, Mozambique, Nepal, Ghana, Yemen, Madagascar, North Korea, Ivory Coast, Australia, Cameroon, Taiwan, Niger, Sri Lanka, Burkina Faso, Romania, Malawi, Mali, Kazakhstan, Syria, Chile}**GROUP SWEDEN (7)**= {Zambia, Guatemala, Netherlands, Zimbabwe, Ecuador, Senegal, Cambodia, Chad, Somalia, Guinea, South Sudan, Rwanda, Tunisia, Belgium, Cuba, Benin, Burundi, Bolivia, Greece, Haiti, Dominican Republic, Czech Republic, Portugal, Sweden, Azerbaijan, Jordan, Hungary, United Arab Emirates, Belarus, Honduras, Tajikistan, Serbia, Austria, Switzerland, Israel, Papua New Guinea, Togo, Sierra Leone, Hong Kong, Bulgaria, Laos, Paraguay, Libya, El Salvador, Nicaragua, Kyrgyzstan, Lebanon, Turkmenistan, Singapore, Denmark}**GROUP MONGOLIA (6.5)**= {Finland, Slovakia, Congo, Norway, Eritrea, State of Palestine, Costa Rica, Liberia, Oman, Ireland, New Zealand, Central African Republic, Mauritania, Kuwait, Croatia, Panama, Moldova, Georgia, Puerto Rico, Bosnia & Herzegovina, Uruguay, Mongolia, Albania, Armenia, Jamaica, Lithuania, Qatar, Namibia, Botswana, Lesotho, Gambia, TFYR Macedonia, Slovenia, Gabon, Latvia, Guinea-Bissau}**GROUP DJIBOUTI (6)**= {Bahrain, Swaziland, Trinidad and Tobago, Timor-Leste, Equatorial Guinea, Estonia, Mauritius, Cyprus, Djibouti, Fiji, Réunion, Comoros, Bhutan, Guyana, Macao, Montenegro, Solomon Islands, Luxembourg, Suriname, Western Sahara}**GROUP ICELAND (5.5)**= {Capo Verde, Guadeloupe, Maldives, Brunei, Malta, Bahamas, Martinique, Belize, Iceland, French Guiana, Barbados, French Polynesia, Vanuatu, New Caledonia, Mayotte, Sao Tome & Principe, Samoa, Saint Lucia}**GROUP ANTIGUA AND BARBUDA (5)**= {Channel Islands, Guam, Curaçao, Kiribati, St. Vincent & Grenadines, Tonga, Grenada, Micronesia, Aruba, U.S. Virgin Islands, Antigua and Barbuda, Seychelles, Isle of Man, Andorra, Dominica, Cayman Islands, Bermuda, Greenland}**GROUP BRITISH VIRGIN ISLANDS (4.5)**= {Saint Kitts & Nevis, American Samoa, Northern Mariana Islands, Marshall Islands, Faroe Islands, Sint Maarten, Monaco, Liechtenstein, Turks and Caicos, Gibraltar, San Marino, British Virgin Islands, Caribbean Netherlands, Palau}**GROUP TUVALU (4)**= {Cook Islands, Anguilla, Wallis & Futuna, Nauru, Tuvalu, Saint Pierre & Miquelon}**GROUP FALKLAND ISLANDS (3.5)**= {Montserrat, Saint Helena, Falkland Islands}**GROUP HOLY SEE (3)**= {Niue, Tokelau, Holy See}.

# GDP/Capita (PPP) in US$; HOM-9 – HOM-7 (98% of World Population)

Source: CIA World Factbook 2018-5-17. Also available in color below.

**GROUP INDIA****=**{China 4, India 4}**GROUP U.S.**= {U.S. 5, Indonesia 4, Brazil 4, Pakistan 3.5, Nigeria 4}**GROUP EGYPT**= {Bangladesh 3.5, Russia 4.5, Mexico 4.5, Japan 4.5, Ethiopia 3.5, Philippines 4, Egypt 4, Vietnam 4, DR Congo 3, Germany 4.5, Iran 4.5, Turkey 4.5, Thailand 4.5, U.K. 4.5, France 4.5, Italy 4.5, Tanzania 3.5, South Africa 4}**GROUP MALAYSIA**= {Myanmar 4, South Korea 4.5, Kenya 3.5, Colombia 4, Spain 4.5, Argentina 4.5, Uganda 3.5, Ukraine 4, Algeria 4, Sudan 3.5, Iraq 4, Poland 4.5, Canada 4.5, Afghanistan 3.5, Morocco 4, Saudi Arabia 4.5, Peru 4, Venezuela 4, Uzbekistan 4, Malaysia 4.5, Angola 4, Mozambique 3, Nepal 3.5, Ghana 3.5, Yemen 3.5, Madagascar 3, North Korea 3, Cote D’Ivoire 3.5, Australia 4.5, Cameroon 3.5, Taiwan 4.5, Niger 3, Sri Lanka 4, Burkina Faso 3.5, Romania 4.5, Malawi 3, Mali 3.5, Kazakhstan 4.5, Syria 3.5, Chile 4.5}**GROUP SWEDEN**= {Zambia 3.5, Guatemala 4, Netherlands 4.5, Zimbabwe 3.5, Ecuador 4, Senegal 3.5, Cambodia 3.5, Chad 3.5, Somalia N/A, Guinea 3.5, South Sudan 3, Rwanda 3.5, Tunisia 4, Belgium 4.5, Cuba 4, Benin 3.5, Burundi 3, Bolivia 4, Greece 4.5, Haiti 3.5, Dominican Republic 4, Czech Republic 4.5, Portugal 4.5, Sweden 4.5, Azerbaijan 4, Jordan 4, Hungary 4.5, United Arab Emirates 5, Belarus 4.5, Honduras 3.5, Tajikistan 3.5, Serbia 4, Austria 4.5, Switzerland 5, Israel 4.5, Papua New Guinea 3.5, Togo 3, Sierra Leone 3.5, Hong Kong 5, Bulgaria 4.5, Laos 4, Paraguay 4, Libya 4, El Salvador 4, Nicaragua 4, Kyrgyzstan 3.5, Lebanon 4.5, Turkmenistan 4.5, Singapore 5, Denmark 4.5}

Bear in mind that countries in the same group can differ in GDP/Capita with a factor of at least three, depending on the uncertainties in the data. Some countries might “switch categories” in other estimates, e.g. Saudi Arabia being 5, and the U.S. 4.5.

With colors

**GROUP INDIA****=**{China, India}**GROUP U.S.**= {U.S., Indonesia, Brazil, Pakistan, Nigeria}**GROUP EGYPT**= {Bangladesh, Russia, Mexico, Japan, Ethiopia, Philippines, Egypt, Vietnam,**DR Congo**, Germany, Iran, Turkey, Thailand, U.K., France, Italy, Tanzania, South Africa}**GROUP MALAYSIA**= {Myanmar, South Korea, Kenya, Colombia, Spain, Argentina, Uganda, Ukraine, Algeria, Sudan, Iraq, Poland, Canada, Afghanistan, Morocco, Saudi Arabia, Peru, Venezuela, Uzbekistan, Malaysia, Angola,**Mozambique**, Nepal, Ghana, Yemen,**Madagascar**,**North Korea**, Cote D’Ivoire, Australia, Cameroon, Taiwan,**Niger**, Sri Lanka, Burkina Faso, Romania,**Malawi**, Mali, Kazakhstan 4.5, Syria, Chile}**GROUP SWEDEN**= {Zambia, Guatemala, Netherlands, Zimbabwe, Ecuador, Senegal, Cambodia, Chad, Somalia N/A, Guinea,**South Sudan**, Rwanda, Tunisia 4, Belgium, Cuba, Benin,**Burundi**, Bolivia, Greece, Haiti, Dominican Republic, Czech Republic, Portugal, Sweden, Azerbaijan, Jordan, Hungary, United Arab Emirates, Belarus, Honduras, Tajikistan, Serbia, Austria, Switzerland, Israel, Papua New Guinea,**Togo**, Sierra Leone, Hong Kong, Bulgaria, Laos, Paraguay, Libya, El Salvador, Nicaragua, Kyrgyzstan, Lebanon, Turkmenistan, Singapore, Denmark}

From this perspective, the Democratic Republic of Congo seems the most tragic case, with ~100,000,000 citizens living on an average ~$1.000/year (~84,000,000 on ~$800 according to the sources); only the Mongolia Group’s Central African Republic has a lower (estimated) GDP/capita of ~$700/year.

Nevertheless, GDP/capita is only one rough statistic. South Sudan and Yemen, despite their higher GDP/capita, face humanitarian crises just like DR Congo.

# Humanitarian Crises: Countries on the 2018-5-17 ReliefWeb Homepage – HOM-9 – HOM-6.5

**GROUP INDIA**= ∅**GROUP U.S.**= {Nigeria}**GROUP EGYPT**= {DR Congo, Bangladesh, Ethiopia}**GROUP MALAYSIA**= {Syria, Myanmar, Yemen, Iraq}**GROUP SWEDEN**= {Somalia, South Sudan}**GROUP MONGOLIA**= {Central African Republic}

It gets worse. Two other countries in the Egypt group face humanitarian crises. Nigeria in the U.S. group does too. So do tens of millions in smaller countries.

Of course, the more relevant statistic would be the size(s) (or size*intensity(‘s)) of the individual humanitarian conflicts in obtaining a first orientation. Nevertheless, I think the groups give some weights when thinking about countries. This doesn’t mean small countries do not matter! It only means that events in those groups tend to have less impact, as they tend to affect fewer people.

To be continued…