Why inflation numbers aren't always comparable

Often people (admittedly. even I have occasionally in the past sometimes done it) calculate relative real appreciation of currencies by adding the nominal exchange rate change with the inflation differential (or, more correctly mathematically, by multiplying 1+the nominal exchange rate change with 1+the domestic inflation rate divided by 1+the foreign inflation rate). That is however often misleading for several reasons.
First of all, the methodology used isn't always the same, as different countries use various adjustments designed to lower the reported inflation rate, such as "chain-linking" and "hedonic adjustment" to different extent.
And secondly and perhaps even more importantly, the goods and services that are part of the consumer price indexes and the relative weight of the different goods and services tend to differ.
For example, alcoholic beverages are a few percent of the consumer price index in most countries, but in for example Saudi Arabia and Iran they are absent. And while food is a part of all consumer price indexes in all countries, they are a much higher percentage of the index in third world countries because people there spend a lot higher percentage of their income on food. This is one of the reasons why soaring prices of food commodities raises inflation rates in poorer countries more than in richer countries.
The other reason why higher prices of food commodities raises inflation more in poorer countries is because food purchased in poorer countries are processed in food factories to a much lower extent. In poor countries, people usually buy only raw rice or potato or other food commodities, meaning that higher prices of food commodities raises food prices for consumers as much or almost as much. By contrast, in richer countries, consumers usually often buy food items that have to varying extent been prepared or processed in food factories- or in the supermarkets or restaurants. The production cost of food in richer countries therefore too a far larger extent include non-commodity components like the labor cost of those that prepare or process the food. As a result, higher food commodity prices raises food prices a lot less in richer countries than in poorer countries.
A 20% increase in food commodity prices therefore increase consumer food prices by perhaps around 15% in poorer countries but perhaps by say only 5% in richer countries. And because food is perhaps 40% of the consumer price index in poorer countries but say only 15% in richer countries, the increase in the overall consumer price index from the increase in food commodity prices will therefore be 6% (0.15*0.4) in poorer countries but only 0.75% (0.05*0.15) in richer countries.
Note that this differential arose despite assuming identical prices of identical food goods, meaning that even if one assumes that "the law of one price" holds fully, that data collection is flawless and that the same methodology for calculating consumer price inflation is used, increases in the prices of commodities will have greatly different effects on official consumer price inflation in different countries.
But because the law of one price is assumed to hold, this inflation differential will not have any effect at all on "ralative competitiveness" or trade flows. Thus, the fact that for example reported consumer price inflation in China is a lot higher than in the United States, probably doesn't mean that the cost of specific items have risen more, and that thus, the "relative competitiveness" of the U.S. hasn't improved much more than the nominal exchange rate depreciation (And if you also take the Penn effect into account, it might have increased less).