At the government level the CPI is a perfect example of How to Lie With Statistics. The CPI can be manipulated in an infinite number of ways.
An easy way is to calculate back to front instead of front to back, like the cartoon above. If things cost $10,000 in 2010 and $11,000 in 2012 you have an annual inflation of 5% (1000 divided by 10000). However, if you calculate it in reverse (1000 divided by 11000) you get an inflation rate of only 4.5%.
The Chained CPI uses this trick then averages the two results. But since time only moves forward, never backwards, this trick is only a trick designed to artificially reduce the CPI percent by 0.25% per year.
Another way is to manipulate the "market basket" used to calculate the CPI. The CPI assumes that people buy so many apples, so many cars, so many houses, and so many aspirin in a year. The Chained CPI assumes that when the price of something goes up people buy less of it. To a certain extent this is true; bargain hunters are real. But the Chained CPI relies on rigid formula
Like this one invented by a Finnish banker in 1936to make grand assumptions about how real live people will react, ignoring how being on fixed incomes limits choices. The Chained CPI is much like Marie Antoinette who when told that the people were too poor to buy bread replied, "let them eat cake." It seriously under-considers inflation in education and healthcare on the assumption that people would choice ignorance and death over tuition and medicine.
The Chained CPI is being favored not because it is any more accurate than any of the many other available formula. It is being favored by politicians because of all the formula it routinely produces the tiniest calculation for inflation. This will allow for real cuts in Social Security payments without actually voting to cut Social Security.
P.S. I have such a headache from reading up on this shit.