Roles include:

Controlling the operations of commercial banks through the reserve rate. 

Short term lending to commercial banks, which provides the central bank with control of interest rates in the economy - presuming that the high street banks then pass those funds onto the consumer.

Control of monetary policy by maintenance of low interest rates.

Securing the nation's gold reserves - the last resort should a nation require funds.

Maintain inflation within the target set by national governments.

Maintenance of target inflation - implying that this is the key objective for any central bank.

Many central banks are required to consider a range of macroeconomic objectives when deciding on their policy decisions, for example the Bank of England in 2016 maintained interest rates at historically low levels despite above target inflation, because of concerns that higher rates might destabilise other areas of the economy that were thought to be at risk.  Banks that prioritise low inflation, however, do so because unlike politicians they do not need to be elected and so are therefore less likely to be swayed by short term expansionary policies aimed at securing popularity before an election campaign.

There would appear to be a direct correlation - the three highest rates of inflation belong to three nations where the central bank does not enjoy full independence.  That said, many would argue that no central bank has complete independence because they are appointed by the same politicians that they represent.

Central banks, unlike national governments, do not seek election every 4-5 years, making them less likely to be swayed by short term expansionary policies aimed at securing popularity before an election campaign.

The table would certainly support the notion that independent central banks are more effective at keeping inflation in check but not necessarily more effective at stimulating economic growth.

Average inflation rate pre-independence was 6.04, compared to 1.88% afterwards.

Average growth rates pre-independence was 2.51, compared to 2.11% afterwards.

Prior to 2008 the world's major financial centres were largely unregulated with governments relying on the sector to regulate itself - which it failed to do.  This was because the sector had become increasingly sophisticated in the range of products that it sold and governments, as well as the regulatory agencies they set up did not have a proper understanding of these products either.  For example, in the short clip the products that the team were betting against contained some of the riskiest loans on the market, which were packaged together and sold on to investment banks all over the world.