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Where do web sites get foreign exchange currency rate / quote information?
The prices quoted are for currency pairs traded on the foreign exchange market. For currencies traded on these exchanges, the exchange rates of a given currency pair are determined by the market, so supply and demand, investor confidence, etc. all play a role. EBS and Reuters are the two primary trading platforms in the foreign exchange market, and much of the data on exchange rates comes from them. Websites will usually get their data either from these sources directly or from a data provider that in turn gets it from EBS, Reuters, or another data source like Bloomberg or Haver Analytics. These data sources aren't free, however.  In the US, many contracts, transactions, etc. that involve exchange rates use the exchange rate data published by the Federal Reserve. You might see this in contracts that specify to use "the exchange rate published by the Federal Reserve at 12 pm (noon) on date --some date--". You can also look at the Federal Reserve Economic Data, which maintains data series of historical daily, weekly, and monthly exchange rates for major currency pairs. These data are free, although they aren't realtime. Data for each business day is mostly updated the next business day.