Rounding in the context of "Quantization error"

In mathematics and fair division, apportionment problems involve dividing (apportioning) a whole number of identical goods fairly across several parties with real-valued entitlements. The original, and best-known, example of an apportionment problem involves distributing seats in a legislature between different federal states or political parties. However, apportionment methods can be applied to other situations as well, including bankruptcy problems, inheritance law (e.g. dividing animals), manpower planning (e.g. demographic quotas), and rounding percentages.

Mathematically, an apportionment method is just a method of rounding real numbers to natural numbers. Despite the simplicity of this problem, every method of rounding suffers one or more paradoxes, as proven by the Balinski–Young theorem. The mathematical theory of apportionment identifies what properties can be expected from an apportionment method.

Data binning, also called data discrete binning or data bucketing, is a data pre-processing technique used to reduce the effects of minor observation errors. The original data values which fall into a given small interval, a bin, are replaced by a value representative of that interval, often a central value (mean or median). It is related to quantization: data binning operates on the abscissa axis while quantization operates on the ordinate axis. Binning is a generalization of rounding.

Statistical data binning is a way to group numbers of more-or-less continuous values into a smaller number of "bins". For example, if you have data about a group of people, you might want to arrange their ages into a smaller number of age intervals (for example, grouping every five years together). It can also be used in multivariate statistics, binning in several dimensions at once.

A fen (Chinese: 分; pinyin: fēn) (Cantonese: sin [Chinese: 仙]), is a unit of currency used in Greater China, including the People's Republic of China, the Republic of China (Taiwan), Hong Kong (called a cent in English) and Macao (called an avo in Portuguese). One fen is equal to 1⁄100 of a yuan or 1⁄10 of a Chinese jiao.

• Renminbi were issued in coin denominations of 1, 2, and 5 fen and also banknote denominations of 1, 2, and 5 fen. The fen banknotes have stopped circulation since 1 April 2007, while the fen coins are still legal tenders by de jure, but generally limited to interests settlement by banks or exchange of foreign remittances, and rarely used in normal shopping purpose, as shops usually rounded it to jiao (e.g. ¥4.55 is rounded to ¥4.60).
• The character 分 is also used to translate "cent" in other currencies. A euro cent is called 欧分; 歐分; Ōufēn in Chinese.

Cash rounding or Swedish rounding (New Zealand English) occurs when the minimum unit of account is smaller than the lowest physical denomination of currency. The amount payable for a cash transaction is rounded to the nearest multiple of the minimum currency unit available, whereas transactions paid in other ways are not rounded (for example electronic funds transfer such as with payment cards, or negotiable instruments such as cheques). Cash rounding typically occurs when low-denomination coins are removed from circulation owing to inflation. Cash rounding may be a compulsory legal requirement if such coins are no longer legal tender, or a voluntary practice where they remain in circulation but are scarce or impractical.

Cash rounding (Swedish: öresavrundning, lit. 'öre rounding') was introduced in Sweden in 1972 when 1 and 2 öre coins were withdrawn from circulation, and has continued to be applied at incremental levels as smaller denomination coins have been withdrawn. The current level of cash rounding in Sweden is to the closest whole krona, after the 50 öre coin was withdrawn in 2010. The Reserve Bank of New Zealand used the name "Swedish rounding" in 1990 when withdrawing their 1- and 2-cent coins. In Canada, cash rounding to the nearest nickel (5 cents) due to the elimination of the penny in 2013 is also called penny rounding.