The fact that people rarely take the last step is a strong indication that they can't challenge the new number on the basis of statistics or mathematics. Thus the simple, and transparent, efforts to create uncertainty without doing any work.
The failure to do the 'simple' and transparent work to establish the validity of stats is on the front end, and lies with those who produce the stats. It's not the responsiblity of the general public to prove the validity of stats.
I recall the ridiculous graphs which plotted nothing but international trade as a percentage of GDP, and the conclusions that that proved Smoot-Hawley was some major factor in the Great Depression. In that case, practically any element of GDP could have been plotted against the total with the same result, or same trend line.
This practice of taking one element of GDP, and plotting it against the total over several decades, then claiming a definite valid relationship is nonsense. The relationship might be valid, but not because the statistical method was valid. There are far too many other factors involved.