Did you know that a school result can improve by 15 percentage points and by 25 percent at the same time? The numbers may sound contradictory until you ask one simple question: are we subtracting two percentages, or comparing the change with the starting percentage?
Percentage points measure the arithmetic gap between two percentages; percent change measures that gap relative to the original value. If a score rises from 60% to 75%, the increase is 15 percentage points, while the relative percent increase is 25%, because 15 divided by 60 is 0.25. A learner who can name the comparison is less likely to misread marks, surveys and school reports.
Start with the question, not the formula
Imagine a parent reading, “The class improved by 15%.” Did the share of correct answers rise from 60% to 75%, or did the score grow by 15 percent of its starting level? Those are different claims. Before calculating, write down the original percentage, the new percentage and the exact unit the sentence asks for. If the wording is ambiguous, report both figures rather than choosing silently.
This is a small but powerful learning habit: diagnose what a number means before trying to make it look impressive. Percentages describe proportions, but a change between percentages needs an additional label.
A complete worked example: 24 correct answers become 30
Consider an illustrative 40-question practice quiz. A student first answers 24 questions correctly, then answers 30 correctly on a later comparable quiz. The first score is 24 ÷ 40 × 100 = 60%. The second is 30 ÷ 40 × 100 = 75%. The raw gain is six correct answers.
- Percentage-point gain: 75% − 60% = 15 percentage points.
- Relative percent increase: (75 − 60) ÷ 60 × 100% = 25%.
- Verification: 60% × 1.25 = 75%; the new percentage is one quarter greater than the old percentage.
- Careful conclusion: on these two comparable 40-question quizzes, the score rose by six marks, 15 percentage points, or 25% relative to the starting percentage.
These calculations describe the two scores. They do not prove that a particular tuition method caused the change. Topic mix, test difficulty, conditions, support and chance can all affect a pair of results. Good education trains children to separate a correct computation from a stronger claim that the evidence cannot establish.
Why the denominator matters
The denominator in percent change is the starting value. When 60% becomes 75%, the increase of 15 is divided by 60, not by 75 and not by 100. However, a percentage-point difference simply subtracts the two reported percentages. Neither operation is universally better; the question decides which is informative.
Now reverse the example. A drop from 75% to 60% is a fall of 15 percentage points, but the relative decrease is 15 ÷ 75 × 100% = 20%. The rise and the fall have equal point differences yet different relative percentages because their starting values differ. This is one reason to write the direction and both endpoints.
The diagnostic: four questions before accepting a claim
- Are the two figures actually percentages with the same interpretation?
- Are we being asked for a direct gap in percentage points, or a relative change from an original level?
- Are the underlying groups, question counts and test difficulties comparable enough for the intended conclusion?
- Have we kept “percentage points,” “percent change,” and “number of pupils or correct answers” separate in our final sentence?
A student who subtracts 75 − 60 correctly but labels it “15% improvement” may understand the arithmetic and still miss the idea. The repair is not necessarily more subtraction drills. Ask them to explain what a point difference compares and what a relative increase divides by.
A second example: changing class sizes
Suppose 12 of 20 pupils complete a reading task independently in one observation and 30 of 40 do so in another. The observed shares are 60% and 75%; the gap is still 15 percentage points, with a 25% relative rise in the share. But it would be misleading to say that “18 more pupils learned the skill” as a measure of improvement: the class sizes are different and the two groups may not contain the same people.
When group sizes differ, report the numerator and denominator alongside the percentage. If the assessment or observation conditions differ, describe that limitation plainly. A trustworthy report should make it easy for another reader to reconstruct the calculation.
What changes when the starting value is zero?
If the original percentage is 0% and the new percentage is 10%, the increase is 10 percentage points. Ordinary relative percent change is undefined because it would divide by zero. Do not describe it as an infinite percent increase or invent a base. State the endpoints and the percentage-point rise; that is both accurate and useful.
For the separate calculation issue, eduKateSG explains how to report percentage change when the starting number is zero. The distinction here is choosing the right comparison when both results are already percentages.
A simple progression for students and parents
Begin by giving the learner pairs such as 20% to 30% and asking only for the percentage-point gap. Next, ask them to calculate the relative change with the original value written explicitly under the fraction. Finally, give a short school-style sentence with no label and ask them to rewrite it unambiguously. Independence means the student can pick the operation without a parent naming it first.
For a quick home check, use three cards: 40% to 50%, 50% to 40%, and 0% to 10%. Ask for a spoken explanation before a calculator. The accurate answers are +10 points and +25% relative; −10 points and −20% relative; and +10 points with undefined ordinary relative percent change. If any card is wrong, diagnose whether the error is subtraction, denominator choice or labelling.
Common mistakes and how to correct them
- Mixing labels: writing “15%” for 15 percentage points. Repair it by saying “15 points out of the percentage scale” rather than “15 percent of the starting value.”
- Dividing by the final number: the relative change should use the original number unless the task defines a different comparison.
- Ignoring unequal totals: keep raw counts beside percentages when groups are different.
- Making causal claims from score changes: improvement on two tasks is evidence about performance on those tasks, not proof of why it changed.
- Hiding awkward edge cases: if the starting level is zero, explain why an ordinary relative percentage is not available.
How teachers can observe genuine progress
Do not assess only whether the final numeral is correct. Give two short word problems that require different answers from the same pair of percentages. On the first, ask for percentage points; on the second, ask for relative percent change. A learner who labels both and explains the denominator has shown more understanding than someone who repeats a remembered calculation. Recheck a week later with new values and no prompts.
Parents can encourage this without turning home discussion into a test. Ask, “Compared with what?” and “Could another reader understand this without your working?” For students, the goal is to make everyday comparisons fairer, including their own progress reports.
Frequently asked questions
Is a ten-percentage-point increase the same as ten percent?
Usually not. A rise from 40% to 50% is ten percentage points but 25% relative to the original 40%. The two coincide numerically only in particular circumstances.
Should a child always report both measures?
No. Answer the question asked and label the unit clearly. When writing a research explanation or reporting an ambiguous change, both can help the reader; add the starting and final values for transparency.
Can test percentages prove that learning has improved?
They describe observed performance. Comparable questions completed independently over time provide better evidence, but interpretation should still account for difficulty, conditions and the learner’s reasoning.
A clearer number makes a clearer thinker
The deeper aim of this skill is not a clever formula. It is learning to name the comparison, check its basis and resist an attractive but imprecise headline. Explore the eduKateSG Education Hub and the Clementi Secondary 1 Mathematics teaching benchmark for diagnosis-first explanation and independently checked practice.