The Headline That Could Mean Two Things
A news line says a rate “rose by 1%”. A reader has no way to tell whether the rate went from 4% to 4.04%, or from 4% to 5%. Both are reasonable readings, and they are very different outcomes. This confusion costs people real money, from savings rates and loan offers to medical statistics and election polls.
Readers who meet this problem usually want three things: a clear rule for which phrase to use, a quick way to convert between the two, and a way to spot a misleading claim. This guide covers each, with figures computed from scratch.
Two Definitions, One Base
Percentage point. A percentage point is the plain subtraction of one percentage from another. A rate moving from 4% to 5% has risen by 1 percentage point.
Percentage change. Percentage change, also called relative change, measures the movement as a share of the starting value: (new − old) ÷ old × 100. The same move from 4% to 5% is 1 ÷ 4 = 25%.
The key to everything is the base. A percentage point has no base because it is a subtraction. A percentage change always has one, and it is the old value. That is why the two numbers differ, and why the difference grows as the starting rate shrinks.
The Two Formulas and a Worked Example
Percentage points: new rate − old rate. Percentage change: (new rate − old rate) ÷ old rate × 100.
Example. A savings account moves from 3% to 4% interest. The change is 4 − 3 = 1 percentage point. As a relative change, it is 1 ÷ 3 = 33.3%. Both statements are correct and describe the same event. A provider advertising “33% more interest” and a bank saying “up one point” are not disagreeing.
The percentage calculator handles the relative change, and a quick subtraction gives the points.
Why the Same One-Point Move Looks Different
Mathvelo computed what a rise of exactly 1 percentage point means in relative terms at different starting rates.
| Starting rate | New rate | Change in points | Relative change |
|---|---|---|---|
| 1% | 2% | +1 | +100% |
| 2% | 3% | +1 | +50% |
| 5% | 6% | +1 | +20% |
| 10% | 11% | +1 | +10% |
| 25% | 26% | +1 | +4% |
| 50% | 51% | +1 | +2% |
The relative change is simply 100 ÷ starting rate. A one-point move is “doubling” at 1% and a rounding error at 50%. Headlines that quote the relative figure for a tiny base are the usual source of dramatic claims.
A Common Misstatement
An explainer at Macroption points to a typical broadcast error: reporting that inflation “increased by 0.5 percent” when it moved from 2.1% to 2.6%. The subtraction is right, but the unit is wrong. The move is 0.5 percentage points. In relative terms, 0.5 ÷ 2.1 = 23.8%, so “inflation rose by 23.8%” would also be true, though it sounds like a very different story.
Neither phrase is dishonest by itself. The problem arises when a reader cannot tell which one is meant, so the safe habit is to state both the old and the new value: “from 2.1% to 2.6%”.
Where It Matters Most: Risk and Medicine
Medical claims often quote relative risk reduction because it sounds larger. If a treatment lowers the chance of an event from 2% to 1%, the absolute reduction is 1 percentage point, while the relative risk reduction is 50%. Both are accurate, and a decision needs both. The relative figure says how large the effect is for those at risk. The absolute figure says how many people it helps. In 1,000 people, the change is from 20 events to 10, so 10 people benefit.
A useful question for any such claim is: “50% of what?” If the base is not given, the number cannot be judged.
Worked Case: A Tax Rate Rise
A sales tax rate rises from 20% to 25%. That is 5 percentage points, and a relative change of 5 ÷ 20 = 25% in the rate. It is tempting to say prices rise 25%, but the rate is not the price. An item costing 100 before tax cost 120 at the old rate and 125 at the new one. The shelf price rose by 5 ÷ 120 = 4.2%.
One event produced three different numbers: 5 points, 25% in the rate, and 4.2% in the price paid. Each one answers a different question, and mixing them is how a reasonable policy gets described as either trivial or enormous.
Worked Case: Reading a Poll
Suppose a poll shows support for a proposal moving from 48% to 51%, with a stated margin of error of ±3 points. The move is 3 percentage points, or a 6.25% relative increase. Because the margin of error is expressed in points, the comparison is direct: a 3-point shift is about the size of the uncertainty, so it should not be described as a clear change. Quoting the relative figure alone would hide that comparison entirely.
Percent Change Versus Percent Difference
Percent change has a direction. From 40 to 50 is a 25% rise, and from 50 back to 40 is a 20% fall. When neither value is the “original”, a symmetric measure is fairer: the absolute difference divided by the average. Here that is 10 ÷ 45 = 22.2% in both directions. Use percent change when there is a clear before and after, and percent difference when comparing two things side by side.
The Spreadsheet Trap
Spreadsheets add one more source of confusion. Subtract a cell showing 5% from a cell showing 6% and the result is shown as 1%, which really means 1 percentage point. Dividing that result by the 5% cell gives 20%, which is the relative change. A column labelled “Change” should say which of the two it holds, since the format alone does not. Labelling columns “pp change” and “% change” is a simple fix that removes the guesswork for anyone reading the file later.
Percentages Do Not Cancel Out
A related trap is assuming that an increase and an equal decrease cancel. They do not, because the second percentage applies to a new base. A 10% rise followed by a 10% fall gives 1.10 × 0.90 = 0.99, a net loss of 1%. A 50% fall followed by a 50% rise gives 0.5 × 1.5 = 0.75, a net loss of 25%.
To recover from a 50% fall, the value must rise by 100%, because 0.5 × 2 = 1. Anyone tracking investments, prices or discounts meets this constantly. The fraction calculator is useful for checking the multipliers as fractions, such as 3/2 and 1/2.
Basis Points: The Clean Alternative
Finance avoids the confusion with a separate unit. One basis point is one hundredth of a percentage point, so 100 basis points equal 1 percentage point. A rate that moves from 4.25% to 4.50% has risen by 25 basis points. Nobody can mistake that for a relative change, which is why central banks and lenders use it.
In everyday writing, the equivalent habit is to say “percentage points” whenever the quantity is itself a percentage, and “percent” only when describing a relative change.
A Four-Step Method for Reading Any Claim
First, identify whether the quantity is a rate (already a percentage) or an amount. Second, find the old and new values; if only one is given, treat the claim as incomplete. Third, compute both the difference and the relative change. Fourth, ask which one the writer chose, and why. A school poll that moves from 40% to 44% is up 4 points, or 10% in relative terms, and a headline can choose either. The same discipline helps in the guide on how to study math, where one study’s 21-point gain is also a 52.5% relative gain.
Try Four Quick Conversions
Test the idea with four moves, then check against the answers. (1) A rate goes from 8% to 10%. (2) A rate goes from 20% to 15%. (3) A discount rises from 10% to 30%. (4) A rate goes from 0.5% to 0.6%.
Answers. (1) +2 points, +25%. (2) −5 points, −25%. (3) +20 points, +200%, so the discount tripled. (4) +0.1 points, +20%, which is 10 basis points. Notice that cases (1) and (2) share the same relative size with different point values, because the bases differ. The percentage calculator verifies each one in seconds.
How to Write It Clearly
Give the old and new values first, then the point difference, then the relative change only if it adds something. “Unemployment rose from 4.0% to 5.0%, a rise of 1 percentage point” cannot be misread. Avoid a bare “up 1%” for any quantity that is already a percentage, and avoid “percent” as shorthand for “points” in titles, charts and captions, where space pressure causes most of the errors.
What Is Overrated, Underrated and Often Wrong
Overrated: relative change as a headline number. It is easy to inflate by choosing a small base. Underrated: basis points and the plain phrase “from X to Y”, which remove every ambiguity at no cost. Often wrong: assuming percentage changes are symmetric, and writing “%” when “percentage points” is meant. The last error is so common that spotting it in a published article is not a sign of a pedantic reader but of a careful one.
Quick Answers
What is the difference between percent and percentage points?
Percentage points are a subtraction of two percentages. Percent change is that difference divided by the starting value.
How do I convert percentage points to percent change?
Divide the point difference by the old rate and multiply by 100. A rise from 4% to 5% is 1 ÷ 4 × 100 = 25%.
Is going from 10% to 15% a 5% increase?
It is a 5 percentage point increase and a 50% relative increase. Calling it “5%” alone is ambiguous.
How many basis points are in one percentage point?
One hundred.
Why do a rise and an equal fall not cancel?
The second percentage applies to the new value, not the original, so 10% up then 10% down leaves 99% of the start.
Related Mathvelo Tools
Further Reading and Sources
- Percentage point — Definition and usage
- Relative change and difference — Formulas and caveats
- Basis point — The finance convention
- Macroption: percentage point vs percent — The inflation example above
A note on these guides. All figures were computed from the stated formulas. Examples with rates and risks are illustrative, not data about a real product or treatment. This page is educational and not financial or medical advice.