The question you opened
From 40 percent to 44 percent
A poll puts the mayor at 40 percent in June and 44 percent in July.
By what percent did that support rise?
10 percent
Four points on a base of 40 is one tenth of it, so the rise is 10 percent. The 4 is the gap in points, and 40 is what the gap is measured against.
Two units are hiding under one word. The mayor gained 4 percentage points, moving from 40 to 44. The relative rise is 4 divided by 40, which is 0.1, so support went up 10 percent. Both statements describe the same poll, and they are 6 apart. Percentage points measure the gap between two percentages.
Percent change measures that gap against where the number started. The first is a subtraction, the second is a division. A campaign that wants the news to sound big reports the 10. A rival that wants it to sound small reports the 4. Neither is lying. The gap between the two units widens as the base shrinks.
A candidate going from 2 percent to 4 percent gained 2 points and doubled, a rise of 100 percent. A candidate going from 80 to 82 also gained 2 points, a rise of 2.5 percent. Same points, wildly different stories, which is why the unit has to be stated. This shows up wherever rates are compared.
A drug that lifts a risk from 1 in 10,000 to 2 in 10,000 doubles the risk, a rise of 100 percent, while adding one case per ten thousand people. Interest rates moving from 4 percent to 5 percent are a one point move and a 25 percent increase in what the borrower pays. Central banks use points for exactly this reason.
The habit is to hear the word percent twice and ask which one is meant. If the sentence compares two percentages, points is the honest unit. If it compares a change to a starting level, percent is. Mixing them is the most common way a true number turns into a misleading headline.
Technique: Points and percent are different units