Percentages are taught early enough that most people assume they have them covered, which is precisely why the two mistakes below appear routinely in news reports, investment summaries, and business dashboards.
Percentages of different bases are not comparable
A percentage is always a proportion of something, and the something can change between the two figures you are comparing. This produces the most counter-intuitive result in everyday arithmetic.
Take an investment of 1,000 that falls 50%. It is now worth 500. It then rises 50%. Most people expect to be back at 1,000; the actual figure is 750. The fall was 50% of 1,000, while the rise was 50% of 500 — different bases, so they do not cancel.
Recovering from a 50% loss requires a 100% gain. From an 80% loss it takes a 400% gain. This asymmetry is why avoiding large drawdowns matters more to long-term returns than capturing large gains, and it is not intuitive to most people.
The same trap in discounts
A shop marks an item up 20%, then advertises 20% off. A 100 item becomes 120, then 96 — below the original. The retailer has made a loss relative to the starting price, not broken even. Stacked discounts behave the same way: 20% off then a further 10% off is not 30% off, it is 28%, because the second discount applies to the already reduced price.
Percentage points are a separate unit
This distinction causes real misreporting. If an interest rate moves from 4% to 6%, it has risen by 2 percentage points — but by 50% in relative terms. Both are correct; they measure different things, and which one gets quoted changes the impression substantially.
The convention is straightforward once you know it. Use percentage points for the arithmetic difference between two percentages. Use percent for the relative change. A survey moving from 40% to 44% support rose 4 points, or 10%. Reporting that as a 4% rise is simply wrong, and the reverse error — reporting a 10-point rise — overstates it wildly.
Averaging percentages
Averaging percentages directly is valid only when the underlying groups are the same size. If one department has a 90% completion rate across 10 tasks and another has 50% across 1,000, the true overall rate is close to 50%, not the 70% you get by averaging the two figures. Weight by the size of each group, or work from the raw counts.
A quick checklist
- Ask what the base is. Percentage of what, exactly?
- When comparing two percentages, decide whether you mean points or relative change, and say which.
- Never assume a fall and an equal rise cancel out.
- Do not average percentages from groups of different sizes.