**I. Introduction, with Example in Population and Area of Countries and Country-Like Entities**

In this post, I introduce a way of looking at correlated data I will term “dual frontier analysis”.

What motivates this idea? Often, we like to compare entities via a certain “rate”, how much of one quantity there is for a unit amount of another quantity, across a set of entities. One example of this is population density. But if you, like me, have glanced at a population density chart of, say, the countries, you may have had one of the same first reactions as I have had: “the top of the chart is pretty much just a listing of city-states!” You might then proceed with questioning whether it really makes sense to compare this quantity for city-states versus for “more normal” countries. Maybe we want a way of looking at this data that better captures what our prior idea of what an “impressively high” or “impressively low” population density is: Bangladesh’s population density definitely “feels” more impressive, even if it’s not as numerically high as Bahrain’s.

There are probably solutions to this problem involving designing a prior distribution of likeliness of one variable in terms of the other, and then comparing percentiles along respective distributions, but going down this path requires crunching a lot of numbers and, more importantly, extensive knowledge in the ideas being analyzed already.

Here is another solution: output the data on the **dual frontiers**. If two attributes are somewhat correlated, a scatterplot for entities in these attributes probably looks something like this.

What we’re outputting is this.

That is, we’re outputting entities for which no other entity has both more of one attribute and less of the other attribute than this entity.

In this way, we would capture, for instance, the country with the highest population density *among countries of similar size*. (We could even extend this to become a quantitative metric for entities not on this frontier: the percentage of the way an entity is from one frontier to the other.)

One could also look at an entity in this data and compare it to neighboring entities and see how much larger in one attribute another entity must be to be larger in the other attribute as well (as otherwise, this entity would also be in the frontier), which shows how *prominently* impressive a particular entity is in the ratio.

If we do this analysis for the populations and areas of countries, here’s what we get.

Population Frontier

These are countries of relative maximum population density. No countries have both higher population and lower area.

**China**: 1388495000 people, 9640821 sq km (144 people/sq km)

→**India**: 1326199000 people, 3287240 sq km (403 people/sq km)

→**Indonesia**: 261891000 people, 1904569 sq km (138 people/sq km)

→**Pakistan**: 210157000 people, 803940 sq km (261 people/sq km)

→**Bangladesh**: 163775000 people, 143998 sq km (1137 people/sq km)

→**South Korea**: 51446000 people, 100210 sq km (513 people/sq km)

→**Taiwan**: 23557000 people, 36197 sq km (651 people/sq km)

→**Rwanda**: 11809000 people, 26338 sq km (448 people/sq km)

→**Israel**: 8797000 people, 22072 sq km (399 people/sq km)

→**El Salvador**: 6582000 people, 21040 sq km (313 people/sq km)

→**Lebanon**: 6082000 people, 10452 sq km (582 people/sq km)

→**Singapore**: 5612000 people, 720 sq km (7795 people/sq km)

→**Malta**: 429000 people, 315 sq km (1363 people/sq km)

→**Maldives**: 344000 people, 298 sq km (1154 people/sq km)

→**Marshall Islands**: 56086 people, 181 sq km (310 people/sq km)

→**Liechtenstein**: 37815 people, 160 sq km (236 people/sq km)

→**Monaco**: 37550 people, 2.0 sq km (18600 people/sq km)

→**Vatican City**: 800 people, 0.44 sq km (1800 people/sq km)

Burundi and Belgium are barely edged out by Rwanda (they have slightly larger area and slightly lower population).

Singapore bears the distinction of having more people than any country of up to 14 times its physical size and less area than any country of at least 8% its population. This becomes not as spanning a dominance if we allow country-like entities (like territories), though, because then our list becomes the following.

**China**: 1388495000 people, 9640821 sq km (144 people/sq km)

→**India**: 1326199000 people, 3287240 sq km (403 people/sq km)

→**Indonesia**: 261891000 people, 1904569 sq km (138 people/sq km)

→**Pakistan**: 210157000 people, 803940 sq km (261 people/sq km)

→**Bangladesh**: 163775000 people, 143998 sq km (1137 people/sq km)

→**South Korea**: 51446000 people, 100210 sq km (513 people/sq km)

→**Taiwan**: 23557000 people, 36197 sq km (651 people/sq km)

→**Rwanda**: 11809000 people, 26338 sq km (448 people/sq km)

→**Israel**: 8797000 people, 22072 sq km (399 people/sq km)

*→El Salvador: 6582000 people, 21040 sq km (313 people/sq km)*

*→Lebanon: 6082000 people, 10452 sq km (582 people/sq km)*

→**Hong Kong**: 7390000 people, 1106 sq km (6681 people/sq km)

→**Singapore**: 5612000 people, 720 sq km (7795 people/sq km)

*→Malta: 429000 people, 315 sq km (1363 people/sq km)*

*→Maldives: 344000 people, 298 sq km (1154 people/sq km)*

*→Marshall Islands: 56086 people, 181 sq km (310 people/sq km)*

*→Liechtenstein: 37815 people, 160 sq km (236 people/sq km)*

→**Macau**: 648500 people, 30 sq km (21300 people/sq km)

→**Monaco**: 37550 people, 2.0 sq km (18600 people/sq km)

→**Vatican City**: 800 people, 0.44 sq km (1800 people/sq km)

If you, instead, do not recognize Taiwan as an independent country, the list becomes this:

**China**: 1388495000 people, 9640821 sq km (144 people/sq km)

→**India**: 1326199000 people, 3287240 sq km (403 people/sq km)

→**Indonesia**: 261891000 people, 1904569 sq km (138 people/sq km)

→**Pakistan**: 210157000 people, 803940 sq km (261 people/sq km)

→**Bangladesh**: 163775000 people, 143998 sq km (1137 people/sq km)

→**South Korea**: 51446000 people, 100210 sq km (513 people/sq km)

→**Sri Lanka**: 21444000 people, 65610 sq km (327 people/sq km)

→**Netherlands**: 17186000 people, 41526 sq km (327 people/sq km)

*→Taiwan: 23557000 people, 36197 sq km (651 people/sq km)*

→**Rwanda**: 11809000 people, 26338 sq km (448 people/sq km)

→**Israel**: 8797000 people, 22072 sq km (399 people/sq km)

→**El Salvador**: 6582000 people, 21040 sq km (313 people/sq km)

→**Lebanon**: 6082000 people, 10452 sq km (582 people/sq km)

→**Singapore**: 5612000 people, 720 sq km (7795 people/sq km)

→**Malta**: 429000 people, 315 sq km (1363 people/sq km)

→**Maldives**: 344000 people, 298 sq km (1154 people/sq km)

→**Marshall Islands**: 56086 people, 181 sq km (310 people/sq km)

→**Liechtenstein**: 37815 people, 160 sq km (236 people/sq km)

→**Monaco**: 37550 people, 2.0 sq km (18600 people/sq km)

→**Vatican City**: 800 people, 0.44 sq km (1800 people/sq km)

Whether Palestine is recognized or not, it comes close but is edged out by Singapore. With the extreme case of Singapore’s population and area, though, it should be noted that Palestine’s population density is still remarkably impressive.

Area Frontier

These are countries of relative minimum population density. No countries have both lower population and higher area.

**Russia**: 146810000 people, 17125242 sq km (8.57 people/sq km)

→**Canada**: 36980000 people, 9984670 sq km (3.70 people/sq km)

→**Australia**: 24771000 people, 7692024 sq km (3.22 people/sq km)

→**Kazakhstan**: 18097000 people, 2724900 sq km (6.64 people/sq km)

→**Libya**: 6375000 people, 1770060 sq km (3.60 people/sq km)

→**Mongolia**: 3000000 people, 1564100 sq km (1.92 people/sq km)

→**Namibia**: 2113000 people, 825118 sq km (2.56 people/sq km)

→**Botswana**: 2025000 people, 581730 sq km (3.48 people/sq km)

→**Guyana**: 785000 people, 214999 sq km (3.65 people/sq km)

→**Suriname**: 534000 people, 163820 sq km (3.26 people/sq km)

→**Iceland**: 347000 people, 102775 sq km (3.37 people/sq km)

→**Vanuatu**: 272000 people, 12190 sq km (22.4 people/sq km)

→**Samoa**: 188000 people, 2831 sq km (66.3 people/sq km)

→**Sao Tome & Principe**: 187000 people, 1001 sq km (187 people/sq km)

→**Dominica**: 71293 people, 739 sq km (96.5 people/sq km)

→**Palau**: 20901 people, 488 sq km (42.8 people/sq km)

→**Tuvalu**: 11323 people, 26 sq km (436 people/sq km)

→**Nauru**: 10084 people, 21 sq km (480 people/sq km)

→**Vatican City**: 800 people, 0.44 sq km (1800 people/sq km)

Notice that as we near the small end of this list, numeric population densities exceed those of entries near the large end of the Population Frontier.

Mauritania is just slightly edged out by Mongolia.

If we allow non-Antarctica country-like entities to join, the list drastically shrinks.

**Russia**: 146810000 people, 17125242 sq km (8.57 people/sq km)

→**Canada**: 36980000 people, 9984670 sq km (3.70 people/sq km)

→**Australia**: 24771000 people, 7692024 sq km (3.22 people/sq km)

→**Kazakhstan**: 18097000 people, 2724900 sq km (6.64 people/sq km)

→**Greenland**: 55984 people, 2166000 sq km (0.03 people/sq km)

*→Libya: 6375000 people, 1770060 sq km (3.60 people/sq km)*

*→Mongolia: 3000000 people, 1564100 sq km (1.92 people/sq km)*

*→Namibia: 2113000 people, 825118 sq km (2.56 people/sq km)*

*→Botswana: 2025000 people, 581730 sq km (3.48 people/sq km)*

*→Guyana: 785000 people, 214999 sq km (3.65 people/sq km)*

*→Suriname: 534000 people, 163820 sq km (3.26 people/sq km)*

*→Iceland: 347000 people, 102775 sq km (3.37 people/sq km)*

*→Vanuatu: 272000 people, 12190 sq km (22.4 people/sq km)*

*→Samoa: 188000 people, 2831 sq km (66.3 people/sq km)*

*→Sao Tome & Principe: 187000 people, 1001 sq km (187 people/sq km)*

*→Dominica: 71293 people, 739 sq km (96.5 people/sq km)*

→**Svalbard and Jan Mayen**: 2655 people, 61399 sq km (0.04 people/sq km)

*→Palau: 20901 people, 488 sq km (42.8 people/sq km)*

*→Tuvalu: 11323 people, 26 sq km (436 people/sq km)*

*→Nauru: 10084 people, 21 sq km (480 people/sq km)*

→**Falkland Islands**: 2563 people, 12173 sq km (0.21 people/sq km)

→**Niue**: 1613 people, 261 sq km (6.18 people/sq km)

*→Vatican City: 800 people, 0.44 sq km (1800 people/sq km)*

→**Pitcairn Islands**: 56 people, 47 sq km (1.19 people/sq km)

And if we let Antarctica barge in, our list utterly collapses.

**Russia**: 146810000 people, 17125242 sq km (8.57 people/sq km)

→**Antarctica**: 1106 people, 14000000 sq km (0.00008 people/sq km)

*→Greenland: 55984 people, 2166000 sq km (0.03 people/sq km)*

*→Svalbard and Jan Mayen: 2655 people, 61399 sq km (0.04 people/sq km)*

*→Falkland Islands: 2563 people, 12173 sq km (0.21 people/sq km)*

*→Niue: 1613 people, 261 sq km (6.18 people/sq km)*

→**Pitcairn Islands**: 56 people, 47 sq km (1.19 people/sq km)

(Although, when did we manage to get a thousand people onto Antarctica?)

Another great thing you can do with these lists is to compare items across the lists to get an idea of span of variety in ratio, and also to realize the starkest comparisons.

Here we go.

**Russia** has less population than **Bangladesh**.

**Canada** has less population than **South Korea**.

**Australia** has only slightly more population than **Taiwan**.

Combining the two above, Canada and Australia combined have less population than South Korea and Taiwan combined. The Canada-Australia combination also falls short of the population of the United Kingdom.

**Kazakhstan** has less population than **Sri Lanka**.

**Libya** has less population than **Rwanda**.

**Mongolia** has less population than either **Singapore** or **Hong Kong**.

**Namibia** and **Botswana** combined have less population than **Singapore**.

**Botswana** has less population than **Lesotho**. (Yeah, I know, Lesotho isn’t on an above list.)

**Iceland** has less population than **Malta**.

And, of course,

**Antarctica** has less population than **Nauru**.

Okay, let’s do a dual frontier analysis for a few more data sets.

**II. Population and Area of US States**

We don’t have a city-states problem or a tiny island nations problem anymore, but there’s still quite a span of sizes. Let’s see what we get.

Population Frontier

**California**: 39250000 people, 423967 sq km (93 people/sq km)

→**Florida**: 20612000 people, 170312 sq km (121 people/sq km)

→**New York**: 19745000 people, 141297 sq km (140 people/sq km)

→**Pennsylvania**: 12784000 people, 119280 sq km (107 people/sq km)

→**Ohio**: 11614000 people, 116098 sq km (100. people/sq km)

→**New Jersey**: 8944000 people, 22591 sq km (396 people/sq km)

→**Connecticut**: 3576000 people, 14357 sq km (249 people/sq km)

→**Rhode Island**: 1056000 people, 4001 sq km (264 people/sq km)

If **Puerto Rico** becomes a state, it would take Connecticut’s place.

Area Frontier

**Alaska**: 742000 people, 1723337 sq km (0.43 people/sq km)

→**Wyoming**: 586000 people, 253335 sq km (2.3 people/sq km)

That was fast.

**III. Population and Area of US Cities**

(And towns and other incorporated areas, I guess, since this will eventually take us to the tiny.)

Population Frontier

**New York, NY**: 8175000 people, 783.7 sq km (10431 people/sq km)

→**Chicago, IL**: 2696000 people, 588.3 sq km (4583 people/sq km)

→**Philadelphia, PA**: 1518000 people, 349.9 sq km (4338 people/sq km)

→**San Francisco, CA**: 806000 people, 120.9 sq km (6665 people/sq km)

→**Miami, FL**: 399000 people, 92.4 sq km (4320 people/sq km)

→**Santa Ana, CA**: 338000 people, 70.3 sq km (4810 people/sq km)

→**Newark, NJ**: 274000 people, 61.6 sq km (4440 people/sq km)

→**Jersey City, NJ**: 240000 people, 38.6 sq km (6220 people/sq km)

→**Paterson, NJ**: 146000 people, 21.9 sq km (6680 people/sq km)

→**Cambridge, MA**: 105000 people, 16.6 sq km (6340 people/sq km)

→**Cicero, IL**: 85616 people, 15.2 sq km (5630 people/sq km)

→**Somerville, MA**: 75754 people, 10.6 sq km (7150 people/sq km)

→**Passaic, NJ**: 69781 people, 8.05 sq km (8670 people/sq km)

→**Union City, NJ**: 66455 people, 3.32 sq km (20100 people/sq km)

→**West New York, NJ**: 49708 people, 2.61 sq km (19000 people/sq km)

→**Cliffside Park, NJ**: 23594 people, 2.49 sq km (9480 people/sq km)

→**Weehawken, NJ**: 13501 people, 2.20 sq km (6140 people/sq km)

→**Sweetwater, FL**: 13499 people, 2.12 sq km (6370 people/sq km)

→**Guttenberg, NJ**: 11176 people, 0.51 sq km (22000 people/sq km)

→**Kaser, NY**: 4724 people, 0.44 sq km (11000 people/sq km)

→**East Newark, NJ**: 2406 people, 0.26 sq km (9300 people/sq km)

→**Millbourne, PA**: 1159 people, 0.18 sq km (6400 people/sq km)

→**Pennsbury Village, PA**: 738 people, 0.16 sq km (4600 people/sq km)

→**Poplar Hills, KY**: 362 people, 0.052 sq km (7000 people/sq km)

San Francisco edges out Boston; Paterson edges out El Monte and Berkeley; Cambridge edges out Daly City.

And good grief, New York metro area.

Area Frontier

All the area frontier list really tells us in this case is where city limits can be drawn ridiculously far out where there aren’t actually people living.

**Sitka, AK**: 8881 people, 2870 sq km (3.09 people/sq km)

→**Wrangell, AK**: 2369 people, 2542 sq km (0.93 people/sq km)

→**Babbitt, MN**: 1475 people, 105.9 sq km (13.9 people/sq km)

→**Nightmute, AK**: 280 people, 96.9 sq km (2.89 people/sq km)

→this frontier becomes increasingly neither helpful nor interesting as we reach the small end

**IV. Mass and Volume of Solar System Bodies**

Mass Frontier

**Sun**→**Jupiter**→**Saturn**→**Neptune**→**Earth**→**Venus**→**Mars**→**Mercury**→**Callisto**→**Io**→

**Moon**→**Europa**→**Triton**→**Eris**→**Haumea**→**Charon**→**Dione**→**Ceres**→**Varuna**→…

Volume Frontier

**Sun**→**Jupiter**→**Saturn**→**Uranus**→**Earth**→**Venus**→**Mars**→**Ganymede**→**Titan**→

**Callisto**→**Io**→**Moon**→**Europa**→**Triton**→**Pluto**→**Titania**→**Rhea**→**Iapetus**→**Charon**→…

Notice how similar these lists are in this case.

**V. Melting Points and Boiling Points of Elements**

Here, we’ll apply this method to attributes for which the ratio shouldn’t actually be taken, but for which we’re just interested in high or low separations.

Elements with atomic number at most 96, with the exceptions of At and Fr, are included in this data.

Boiling Point Frontier

These elements have generally long spans of temperature in the liquid state at atmospheric pressure, compared to elements of similar phase transition points.

**Tungsten**→**Rhenium**→**Tantalum**→**Thorium**→**Uranium**→**Neptunium**→**Plutonium**→

**Tin**→**Gallium**→**Cesium**→**Mercury**→**Chlorine**→**Xenon**→**Krypton**→**Oxygen**→**Neon**→

**Hydrogen**→**Helium**

Melting Point Frontier

And these elements are only liquid in small temperature ranges.

**Carbon**→**Boron**→**Rhodium**→**Vanadium**→**Chromium**→**Thulium**→**Samarium**→

**Calcium**→**Ytterbium**→**Arsenic**→**Sulfur**→**Iodine**→**Bromine**→**Radon**→**Xenon**→

**Krypton**→**Argon**→**Nitrogen**→**Neon**→**Hydrogen**→**Helium**

Notice many noble gases managing to make it onto both lists. All of their liquid temperatures ranges are tiny; it’s just that most of them have no competition in their transition temperature ranges.