Inflation is a difficult concept for many

Last friday, 6/28, new PCE (Personal Consumption Expenditures Price Index) data were released. The year-on-year inflation numbers decreased from 2.7% last month to 2.6% [1]: 

Let's see how the popular press reports this [2]:

The headline is just very wrong. Inflation was 2.6% but it did not rise by 2.6%. The PCE-based inflation number did actually decrease from 2.7% a month earlier. 

I see lots of mistakes in reporting and in posts about inflation and price indices. This is a good (or rather bad) example.

Here is a similar incorrect statement about CPI inflation:

CPI inflation was 3%. It did not rise 3%. It was actually down.

Background

There are a few different quantities here. Here is a summary:

PCE Index and Inflation
$x_t$	PCE index at month $t$
$i_t = \displaystyle\frac{x_t-x_{t-12}}{x_{t-12}}\cdot 100\%$	Inflation at month $t$
$i_t - i_{t-1}$	Change in inflation at month $t$

This hierarchy is often poorly understood by, let's say, amateur economists.

The inflation number is measured year-on-year (i.e. with a 12-month lag). That has the advantage that there are no seasonal effects. The theory behind price indices and their calculation is fascinating, see e.g. [3]. An interesting way to create an "instantaneous inflation" number based on kernel density estimation is shown in [4].

References

1. https://www.bea.gov/data/personal-consumption-expenditures-price-index
2. https://www.cnbc.com/2024/06/28/may-pce-inflation-report.html
3. Walter Erwin Diewert, John Greenlees, Charles R. Hulten, Price Index Concepts and Measurement. (2010). University of Chicago Press.
4. Jan Eekhout, Instantaneous Inflation, 2023.