Three correct answers to one question
Ask how far apart 4% and 5% are and there are three defensible answers, all arithmetically correct and all describing something different. The confusion is not in the maths; it is in the naming.
1. The percentage-point difference
points = second − first
A raw subtraction: 5 − 4 = 1 percentage point. This is the right measure when the two percentages are shares of the same total, such as an unemployment rate or a party's vote share, because the gap is itself meaningful in the original units.
2. The relative change
relative change = ((second − first) ÷ first) × 100
(5 − 4) ÷ 4 = 25%. This is the right measure when one figure came from the other — a rate that rose, a conversion that improved — because it says how much bigger the second is than the first.
3. The percentage difference
percentage difference = |second − first| ÷ ((first + second) ÷ 2) × 100
1 ÷ 4.5 = 22.2%. This one is symmetric: it gives the same answer whichever number you put first, because it divides by the average rather than by one of them. Use it when neither figure is the baseline — two independent measurements, two competing products — and the direction is arbitrary.
The three answers to "4% versus 5%" are 1 point, 25% and 22.2%. Anyone reporting one of them as "the difference" without saying which has left out the part that makes it interpretable.
Where the confusion does damage
Interest rates
A mortgage rate moving from 4% to 5% is one percentage point, but your interest bill rises by 25%. A headline saying rates rose 1% and one saying they rose 25% can describe exactly the same change, and only the first is using the conventional meaning.
Polling
A candidate going from 40% to 44% has gained four points and about 10% of their previous support. Polls quote points because the margin of error is also in points — a "3-point lead with a 3-point margin of error" is a statement about points, and converting it to a relative change makes it uninterpretable.
Conversion and click-through rates
This is where the two get mixed most often, and usually in the flattering direction. A conversion rate improving from 2% to 2.5% is half a percentage point — and a 25% improvement. Both are true; only the second gets put in the deck.
Tax rates and margins
A tax rate rising from 20% to 22% is two points. A business margin falling from 12% to 9% is three points, and a quarter of the margin. When both numbers are already percentages, the word "percent" on its own is ambiguous, which is exactly why the phrase "percentage point" exists.
| Change | In points | Relative change | Percentage difference |
|---|---|---|---|
| 4% → 5% | +1 pp | +25.0% | 22.2% |
| 2% → 2.5% | +0.5 pp | +25.0% | 22.2% |
| 40% → 44% | +4 pp | +10.0% | 9.5% |
| 12% → 9% | −3 pp | −25.0% | 28.6% |
| 50% → 75% | +25 pp | +50.0% | 40.0% |
| 1% → 2% | +1 pp | +100.0% | 66.7% |
| 80% → 85% | +5 pp | +6.3% | 6.1% |
The second and sixth rows are the ones to remember. A half-point move and a one-point move can both be "a 25% increase" or "a doubling" depending only on where they started, which is why the relative figure alone never tells you how big the change actually was.
Choosing which to report
A short test that settles most cases: ask whether the first number is a baseline that the second came from, or whether the two are simply being compared.
- The second figure came from the first, over time — report the relative change, and give the points as well if the reader needs the scale.
- The two figures are shares of the same whole at the same moment — report percentage points.
- Neither figure is privileged and the order is arbitrary — report the percentage difference, which is symmetric.
- You are writing for a general audience — give both the points and the relative change. They are cheap to state together and impossible to reconstruct from one another.
The reversal test
If swapping the two numbers should change the answer, you want the relative change, which is asymmetric — 4 to 5 is +25% while 5 to 4 is −20%. If swapping them should not change the answer, you want the percentage difference, which gives 22.2% either way. This one property is usually enough to pick the right measure.
When the first value is zero
Relative change is undefined when the starting figure is zero, since it divides by it. A rate going from 0% to 3% is a three-point rise and nothing more can honestly be said about it in relative terms. The calculator reports the relative change as undefined rather than showing an infinity, because the honest answer is that the question has no numeric answer.
How to compare two percentages
- Enter the starting percentage. Type the earlier or baseline figure into the first field, without the percent sign.
- Enter the second percentage. Type the later or comparison figure into the second field.
- Read all three measures. The panel shows the percentage-point gap, the relative change and the symmetric percentage difference, each labelled.
- Pick the one that fits. Use points for shares of the same whole, relative change for a figure that moved over time, and percentage difference when neither value is the baseline.
Frequently Asked Questions
- What is the difference between a percent and a percentage point?
- A percentage point is the raw gap between two percentages; a percent is that gap relative to the starting value. A rate moving from 4% to 5% rose one percentage point and 25 percent.
- How do I calculate percentage points?
- Subtract one percentage from the other. There is no division involved, which is what makes the measure a "point" rather than a percentage.
- What is percentage difference, as opposed to percentage change?
- Percentage difference divides the gap by the average of the two values, so it is symmetric. Percentage change divides by the first value, so reversing the order changes the answer.
- Why is 4% to 5% a 25% increase?
- Because the increase of one point is measured against the starting value of four: 1 ÷ 4 = 0.25. The same one-point rise from 40% to 41% is only a 2.5% increase.
- What happens when the first percentage is zero?
- The relative change is undefined, because it would divide by zero. Only the percentage-point difference can be reported.
- Which measure should a news article use?
- Both, ideally. Points give the scale of the move and the relative change gives its significance, and neither can be worked out from the other without knowing the starting value.
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