# Cube root law

The cube root law is an observation in political science that the number of members of a unicameral legislature, or of the lower house of a bicameral legislature, is about the cube root of the population being represented. The rule was devised by Estonian political scientist Rein Taagepera in his 1972 paper "The size of national assemblies".
The law has led to a proposal to increase the size of the United States House of Representatives so that the number of representatives would be the cube root of the US population as calculated in the most recent census. The House of Representatives has had 435 members since the Reapportionment Act of 1929 was passed; if the US followed the cube root rule, there would be 693 members of the House of Representatives based on the population at the 2020 Census.
This proposal was endorsed by the New York Times editorial board in 2018.

## Subsequent analysis

Giorgio Margaritondo argued that the experimental data, including the dataset originally used by Taagepera in 1972, actually fits better to a function with a higher exponent, and that there is sufficient deviation from the cube root rule to question its usefulness. In this regard, analysis by Margaritondo gives an optimal formula of: $A=0.1P^{E}$, where A is the size of the assembly, P is the population, and E = 0.45±0.03.
Applying this formula to the U.S. House of Representatives as of the 2020 Census would give a House of between 379 and 1231 members, while using an exponent of 0.4507 gives 693 (the same result using the cube root rule).

## Table comparing OECD nations in 2019 with EIU Democracy Index ranking

Out of the countries listed, Lithuania is the only one to exactly match the cube root rule. Moreover, Denmark, Canada, and Mexico come close to matching the rule.
Some of these countries (e.g. Germany) have overhang seats in a mixed member proportional system, as a result the size of their parliaments can vary significantly between elections.
| Country | Lower or unicameral house | Population (2019) | Lower house size (2019) | Cube root of population (nearest person) | Difference between lower house and cube root of population | Difference between lower house and cube root of population (%) | People per representative | People per representative (cube root lower house) | Democracy Index Ranking (2022) |
| Australia | House of Representatives | 25,364,307 | 151 | 294 | −143 | −49% | 167,976 | 86,327 | 15 |
| Austria | National Council | 8,877,067 | 183 | 207 | −24 | −12% | 48,509 | 42,873 | 20 |
| Belgium | Chamber of Representatives | 11,484,055 | 150 | 226 | −76 | −34% | 76,560 | 50,901 | 36 |
| Canada | House of Commons | 37,589,262 | 338 | 335 | +3 | +1% | 111,211 | 112,213 | 12 |
| Chile | Chamber of Deputies | 18,952,038 | 155 | 267 | −112 | −42% | 122,271 | 71,084 | 19 |
| Colombia | Chamber of Representatives | 50,339,443 | 172 | 363 | −191 | −53% | 303,250 | 136,334 | 53 |
| Czech Republic | Chamber of Deputies | 10,669,709 | 200 | 220 | −20 | −9% | 53,349 | 48,466 | 25 |
| Denmark | Folketing | 5,818,553 | 179 | 180 | −1 | −1% | 32,506 | 32,350 | 6 |
| Estonia | Riigikogu | 1,326,590 | 101 | 110 | −9 | −8% | 13,135 | 12,073 | 27 |
| Finland | Parliament | 5,520,314 | 200 | 177 | +23 | +13% | 27,602 | 31,235 | 5 |
| France | National Assembly | 67,059,887 | 577 | 406 | +171 | +42% | 116,222 | 165,060 | 22 |
| Germany | Bundestag | 83,132,799 | 734 | 436 | +298 | +68% | 113,260 | 190,480 | 14 |
| Greece | Parliament | 10,716,322 | 300 | 220 | +80 | +36% | 35,721 | 48,607 | 25 |
| Hungary | National Assembly | 9,769,949 | 199 | 214 | −15 | −7% | 49,095 | 45,701 | 56 |
| Iceland | Althing | 361,313 | 63 | 71 | −8 | −11% | 5,735 | 5,073 | 3 |
| Ireland | Dáil | 5,100,000 | 160 | 173 | −13 | −8% | 31,875 | 29,011 | 8 |
| Israel | Knesset | 9,053,300 | 120 | 208 | −88 | −42% | 75,444 | 43,438 | 29 |
| Italy | Chamber of Deputies | 60,297,396 | 400 | 392 | +8 | +2% | 150,743 | 153,768 | 34 |
| Japan | House of Representatives | 126,264,931 | 465 | 502 | −37 | −7% | 271,537 | 251,684 | 16 |
| Korea, Republic of | National Assembly | 51,709,098 | 300 | 373 | −73 | −20% | 172,384 | 138,796 | 24 |
| Latvia | Saeima | 1,912,789 | 100 | 124 | −24 | −19% | 19,218 | 15,409 | 38 |
| Lithuania | Seimas | 2,786,844 | 141 | 141 | 0 | 0% | 19,765 | 19,803 | 39 |
| Luxembourg | Chamber of Deputies | 619,896 | 60 | 85 | −25 | −29% | 10,332 | 7,270 | 13 |
| Mexico | Chamber of Deputies | 127,575,529 | 500 | 503 | −3 | −1% | 255,151 | 253,422 | 89 |
| Netherlands | House of Representatives | 17,332,850 | 150 | 259 | −109 | −42% | 115,552 | 66,975 | 9 |
| New Zealand | House of Representatives | 4,917,000 | 120 | 170 | −50 | −29% | 40,975 | 28,916 | 2 |
| Norway | Storting | 5,347,896 | 169 | 175 | −6 | −3% | 31,644 | 30,581 | 1 |
| Poland | Sejm | 37,970,874 | 460 | 336 | +124 | +37% | 82,545 | 112,971 | 46 |
| Portugal | Assembly of the Republic | 10,269,417 | 230 | 217 | +13 | +6% | 44,650 | 47,246 | 28 |
| Slovakia | National Council | 5,454,073 | 150 | 176 | −26 | −15% | 36,360 | 30,985 | 43 |
| Slovenia | National Assembly | 2,087,946 | 90 | 128 | −38 | −30% | 23,199 | 16,336 | 31 |
| Spain | Congress of Deputies | 47,076,781 | 350 | 361 | −11 | −3% | 134,505 | 130,378 | 22 |
| Sweden | Riksdag | 10,285,453 | 349 | 217 | +132 | +61% | 29,471 | 47,295 | 4 |
| Switzerland | National Council | 8,574,832 | 200 | 205 | −5 | −2% | 42,874 | 41,894 | 7 |
| Turkey | Grand National Assembly | 83,429,615 | 600 | 437 | +163 | +37% | 139,049 | 190,933 | 103 |
| United Kingdom | House of Commons | 66,834,405 | 650 | 406 | +244 | +60% | 102,822 | 164,690 | 18 |
| United States | House of Representatives | 328,239,523 | 435 | 690 | −255 | −37% | 754,574 | 475,840 | 30 |

## Historical US House sizes

The following table describes how the US House of Representatives would have looked historically under the cube root rule according to the Huntington–Hill method.
| Census, Year | Size | AL | AK | AZ | AR | CA | CO | CT | DE | DC | FL | GA | HI | ID | IL | IN | IA | KS | KY | LA | ME | MD | MA | MI | MN | MS | MO | MT | NE | NV | NH | NJ | NM | NY | NC | ND | OH | OK | OR | PA | RI | SC | SD | TN | TX | UT | VT | VA | WA | WV | WI | WY |
| 1st, 1790 | 158 | | | | | | | 10 | 3 | | | 3 | | | | | | | | | | 13 | 20 | | | | | | | | 6 | 8 | | 14 | 16 | | | | | 18 | 3 | 10 | | | | | | 34 | | | | |
| 2nd, 1800 | 175 | | | | | | | 8 | 2 | | | 5 | | | | | | | 7 | | | 12 | 19 | | | | | | | | 6 | 7 | | 20 | 16 | | | | | 20 | 2 | 12 | | 4 | | | 5 | 30 | | | | |
| 3rd, 1810 | 194 | | | | | | | 7 | 2 | | | 7 | | | | | | | 11 | | | 11 | 19 | | | | | | | | 6 | 7 | | 27 | 15 | | 6 | | | 22 | 2 | 12 | | 7 | | | 6 | 27 | | | | |
| 4th, 1820 | 213 | 3 | | | | | | 6 | 2 | | | 8 | | | 1 | 3 | | | 13 | 3 | 7 | 9 | 12 | | | 2 | | | | | 5 | 6 | | 31 | 14 | | 13 | | | 24 | 2 | 11 | | 9 | | | 5 | 24 | | | | |
| 5th, 1830 | 235 | 6 | | | | | | 5 | 1 | | | 10 | | | 3 | 6 | | | 13 | 4 | 7 | 8 | 11 | | | 3 | 3 | | | | 5 | 6 | | 35 | 14 | | 17 | | | 25 | 2 | 11 | | 13 | | | 5 | 22 | | | | |
| 6th, 1840 | 258 | 9 | | | 2 | | | 5 | 1 | | | 11 | | | 7 | 10 | | | 12 | 5 | 8 | 7 | 11 | 3 | | 6 | 6 | | | | 4 | 6 | | 37 | 12 | | 23 | | | 26 | 2 | 9 | | 13 | | | 4 | 19 | | | | |
| 7th, 1850 | 286 | 10 | | | 3 | 1 | | 5 | 1 | | 1 | 11 | | | 10 | 12 | 2 | | 12 | 6 | 7 | 7 | 12 | 5 | | 8 | 8 | | | | 4 | 6 | | 38 | 11 | | 25 | | | 29 | 2 | 8 | | 12 | 3 | | 4 | 18 | | | 4 | |
| 8th, 1860 | 316 | 10 | | | 4 | 4 | | 5 | 1 | | 2 | 11 | | | 17 | 14 | 7 | | 12 | 7 | 6 | 7 | 13 | 8 | 2 | 8 | 12 | | | | 3 | 7 | | 39 | 10 | | 24 | | 1 | 29 | 2 | 7 | | 11 | 6 | | 3 | 16 | | | 8 | |
| 9th, 1870 | 338 | 9 | | | 4 | 5 | | 5 | 1 | | 2 | 10 | | | 22 | 15 | 11 | 3 | 12 | 6 | 6 | 7 | 13 | 10 | 4 | 7 | 15 | | 1 | 1 | 3 | 8 | | 39 | 10 | | 24 | | 1 | 31 | 2 | 6 | | 11 | 7 | | 3 | 11 | | 4 | 9 | |
| 10th, 1880 | 369 | 9 | | | 6 | 6 | 2 | 5 | 1 | | 2 | 12 | | | 23 | 15 | 12 | 7 | 12 | 7 | 5 | 7 | 13 | 12 | 6 | 8 | 16 | | 3 | 1 | 3 | 8 | | 38 | 11 | | 24 | | 1 | 32 | 2 | 7 | | 12 | 12 | | 3 | 11 | | 5 | 10 | |
| 11th, 1890 | 398 | 10 | | | 7 | 8 | 3 | 5 | 1 | | 3 | 12 | | 1 | 25 | 14 | 12 | 9 | 12 | 7 | 4 | 7 | 14 | 13 | 8 | 8 | 17 | 1 | 7 | 1 | 2 | 9 | | 39 | 10 | 1 | 24 | | 2 | 34 | 2 | 7 | 2 | 11 | 14 | | 2 | 11 | 2 | 5 | 11 | 1 |
| 12th, 1900 | 426 | 10 | | | 7 | 8 | 3 | 5 | 1 | 2 | 3 | 13 | | 1 | 27 | 14 | 13 | 8 | 12 | 8 | 4 | 7 | 16 | 14 | 10 | 9 | 18 | 1 | 6 | 1 | 2 | 11 | | 41 | 11 | 2 | 24 | | 2 | 36 | 2 | 8 | 2 | 11 | 17 | 2 | 2 | 11 | 3 | 5 | 12 | 1 |
| 13th, 1910 | 453 | 11 | | | 8 | 12 | 4 | 6 | 1 | 2 | 4 | 13 | | 2 | 28 | 13 | 11 | 8 | 11 | 8 | 4 | 6 | 17 | 14 | 10 | 9 | 16 | 2 | 6 | 1 | 2 | 12 | | 45 | 11 | 3 | 23 | 8 | 3 | 38 | 3 | 7 | 3 | 11 | 19 | 2 | 2 | 10 | 6 | 6 | 12 | 1 |
| 14th, 1920 | 475 | 11 | | 2 | 8 | 15 | 4 | 6 | 1 | 2 | 4 | 13 | | 2 | 29 | 13 | 11 | 8 | 11 | 8 | 3 | 6 | 17 | 16 | 11 | 8 | 15 | 3 | 6 | 1 | 2 | 14 | 2 | 46 | 11 | 3 | 26 | 9 | 4 | 39 | 3 | 8 | 3 | 10 | 21 | 2 | 2 | 10 | 6 | 7 | 12 | 1 |
| 15th, 1930 | 500 | 11 | | 2 | 7 | 23 | 4 | 6 | 1 | 2 | 6 | 12 | | 2 | 31 | 13 | 10 | 8 | 11 | 8 | 3 | 7 | 17 | 20 | 10 | 8 | 15 | 2 | 6 | 1 | 2 | 16 | 2 | 51 | 13 | 3 | 27 | 10 | 4 | 39 | 3 | 7 | 3 | 11 | 23 | 2 | 2 | 10 | 6 | 7 | 12 | 1 |
| 16th, 1940 | 512 | 11 | | 2 | 8 | 27 | 4 | 7 | 1 | 3 | 7 | 12 | | 2 | 31 | 13 | 10 | 7 | 11 | 9 | 3 | 7 | 17 | 20 | 11 | 9 | 15 | 2 | 5 | 1 | 2 | 16 | 2 | 52 | 14 | 3 | 27 | 9 | 4 | 39 | 3 | 7 | 3 | 11 | 25 | 2 | 1 | 10 | 7 | 7 | 12 | 1 |
| 17th, 1950 | 536 | 11 | | 3 | 7 | 38 | 5 | 7 | 1 | 3 | 10 | 12 | | 2 | 31 | 14 | 9 | 7 | 10 | 10 | 3 | 8 | 17 | 23 | 11 | 8 | 14 | 2 | 5 | 1 | 2 | 17 | 2 | 53 | 14 | 2 | 28 | 8 | 5 | 37 | 3 | 8 | 2 | 12 | 27 | 2 | 1 | 12 | 8 | 7 | 12 | 1 |
| 18th, 1960 | 566 | 10 | 1 | 4 | 6 | 49 | 6 | 8 | 1 | 2 | 16 | 12 | 2 | 2 | 32 | 15 | 9 | 7 | 10 | 10 | 3 | 10 | 16 | 25 | 11 | 7 | 14 | 2 | 4 | 1 | 2 | 19 | 3 | 53 | 14 | 2 | 31 | 7 | 6 | 36 | 3 | 8 | 2 | 11 | 30 | 3 | 1 | 12 | 9 | 6 | 12 | 1 |
| 19th, 1970 | 590 | 10 | 1 | 5 | 6 | 58 | 6 | 9 | 2 | 2 | 20 | 13 | 2 | 2 | 32 | 15 | 8 | 7 | 9 | 11 | 3 | 11 | 17 | 26 | 11 | 6 | 14 | 2 | 4 | 2 | 2 | 21 | 3 | 53 | 15 | 2 | 31 | 7 | 6 | 34 | 3 | 8 | 2 | 11 | 32 | 3 | 1 | 13 | 10 | 5 | 13 | 1 |
| 20th, 1980 | 612 | 11 | 1 | 7 | 6 | 64 | 8 | 8 | 2 | 2 | 26 | 15 | 3 | 3 | 31 | 15 | 8 | 6 | 10 | 11 | 3 | 11 | 16 | 25 | 11 | 7 | 13 | 2 | 4 | 2 | 3 | 20 | 4 | 48 | 16 | 2 | 29 | 8 | 7 | 32 | 3 | 8 | 2 | 12 | 38 | 4 | 1 | 14 | 11 | 5 | 13 | 1 |
| 21st, 1990 | 631 | 10 | 1 | 9 | 6 | 75 | 8 | 8 | 2 | 2 | 33 | 16 | 3 | 3 | 29 | 14 | 7 | 6 | 9 | 11 | 3 | 12 | 15 | 24 | 11 | 7 | 13 | 2 | 4 | 3 | 3 | 20 | 4 | 46 | 17 | 2 | 27 | 8 | 7 | 30 | 3 | 9 | 2 | 12 | 43 | 4 | 2 | 16 | 12 | 5 | 12 | 1 |
| 22nd, 2000 | 657 | 10 | 2 | 12 | 6 | 79 | 10 | 8 | 2 | 1 | 37 | 19 | 3 | 3 | 29 | 14 | 7 | 6 | 9 | 10 | 3 | 12 | 15 | 23 | 11 | 7 | 13 | 2 | 4 | 5 | 3 | 20 | 4 | 44 | 19 | 2 | 27 | 8 | 8 | 29 | 2 | 9 | 2 | 13 | 49 | 5 | 2 | 17 | 14 | 4 | 13 | 1 |
| 23rd, 2010 | 677 | 10 | 2 | 14 | 6 | 82 | 11 | 8 | 2 | 1 | 41 | 21 | 3 | 3 | 28 | 14 | 7 | 6 | 10 | 10 | 3 | 13 | 14 | 22 | 12 | 7 | 13 | 2 | 4 | 6 | 3 | 19 | 5 | 43 | 21 | 2 | 25 | 8 | 8 | 28 | 2 | 10 | 2 | 14 | 55 | 6 | 1 | 18 | 15 | 4 | 12 | 1 |
| 24th, 2020 | 695 | 11 | 2 | 15 | 6 | 83 | 12 | 8 | 2 | 2 | 45 | 22 | 3 | 4 | 27 | 14 | 7 | 6 | 9 | 10 | 3 | 13 | 15 | 21 | 12 | 6 | 13 | 2 | 4 | 7 | 3 | 19 | 4 | 42 | 22 | 2 | 25 | 8 | 9 | 27 | 2 | 11 | 2 | 15 | 61 | 7 | 1 | 18 | 16 | 4 | 12 | 1 |
| Census, Year | Size | AL | AK | AZ | AR | CA | CO | CT | DE | DC | FL | GA | HI | ID | IL | IN | IA | KS | KY | LA | ME | MD | MA | MI | MN | MS | MO | MT | NE | NV | NH | NJ | NM | NY | NC | ND | OH | OK | OR | PA | RI | SC | SD | TN | TX | UT | VT | VA | WA | WV | WI | WY |