With changing the Social Security benefits formula in the news, it's worth just noting all the different inflation indexes the government currently uses.
Social Security is currently based on the somewhat old-school CPI-W which is a Lapseyres Price Index based on the basket of goods consumed by urban (which in this case means metropolitan areas, suburbs included) households who receive more than half of their income from clerical or wage occupations and have one earner employed for at least 37 weeks during the previous 12 months. Roughly speaking, it's the price index of "working people" in metropolitan areas—not retirees or people living off investment income. It covers about 32 percent of the population and it's a little odd since its main official use is cost-of-living adjustments for Social Security beneficiaries who aren't working.
You more frequently hear about the broader CPI-U which again is a Laspeyres index, this time based on the basket of goods consumed by all households in metropolitan areas.
The proposal is to switch off the CPI-W and start using the Chained CPI-U. C-CPI-U assumes households engage in substitution in response to relative price shifts and thus shows a lower overall rate of inflation.
An increasingly influential inflation index is the Personal Consumption Expenditure Deflator, which the Federal Reserve uses to measure inflation. The PCE Deflator is a Fisher Index, which means it's "chained" just like the C-CPI-U. The difference is that all the different flavors of CPI only measure prices that consumers actually pay. The PCE Deflator measures the price of everything that's consumed, regardless of who pays. In practical terms the most important difference is that the CPI considers the price of a doctors' visit to be what patients pay out of pocket, while the PCE Deflator measures the price doctors charge to insurance companies, the government, patients, or whoever it is that's footing the bill.
Last but by no means least there's the inflation index that the Fed ought to target, the GDP Deflator. The GDP Deflator is, in my opinion, the only index that even purports to measure what's most properly meant by "inflation." It does that by including the price of everything, including capital goods and other stuff that isn't "consumed" by households. Obviously people care most about the price of goods they consumer. But inflation—price shifts induced by monetary phenomena—should happen everywhere. It's an interesting and important fact that the price of consumption goods is rising relative to the price of capital goods, but it all deserves to be in the inflation box.
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