The core idea
A percentage needs a base. Percentage change compares a difference with the starting value; percentage points subtract two percentages directly. Always name the population, time period and denominator.
1. Ask: a percentage of what?
Invented learning-club example: 30 of 50 registered learners attend a session. The attendance share is 30 ÷ 50 × 100 = 60%. Here 30 is the numerator, the number attending; 50 is the denominator, the group you compare it with. Writing “60% attended” without naming registered learners leaves the reader to guess who was counted.
If the club invited 100 people but only 50 registered, 30 attendees are 30% of invitees and 60% of registrants. Both calculations can be right. They answer different questions. Before arguing about a number, make the question and denominator explicit.
Sources: Office for National Statistics: Data visualisation principles ↗
2. Percentage change uses the starting value
Suppose registrations rise from 40 to 60. The absolute increase is 20 people. Relative to the original 40, that is 20 ÷ 40 × 100 = 50%. The formula is (new − old) ÷ old × 100. A fall from 60 back to 40 is −20 ÷ 60 × 100, about −33.3%, because the starting value is now 60.
This explains why equal percentage rises and falls do not cancel. A ₹500 item rising 20% becomes ₹600. A later 20% reduction subtracts ₹120, leaving ₹480. The second percentage acts on ₹600. Multiplying by 1.20 and then 0.80 gives 0.96 of the original amount.
Sources: Office for National Statistics: Percentages and percentage points ↗
3. Percentage points answer a different question
Keep the club size at 50. Attendance rises from 20 people to 30 people: the share moves from 40% to 60%. The difference is 20 percentage points. Relative growth in the share is (60 − 40) ÷ 40 × 100 = 50%. Saying “up 20%” would describe neither clearly; a 20% relative rise from 40% would produce 48%.
A good report states the endpoints first: “Attendance rose from 40% to 60%, an increase of 20 percentage points.” Add relative growth only if it helps the reader. The endpoints let someone check the calculation and understand the size of the original share.
One change, two measures
| Measure | Calculation | Result |
|---|---|---|
| Starting attendance | 20 ÷ 50 × 100 | 40% |
| Later attendance | 30 ÷ 50 × 100 | 60% |
| Percentage-point difference | 60 − 40 | 20 percentage points |
| Relative change | (60 − 40) ÷ 40 × 100 | 50% |
Sources: Office for National Statistics: Percentages and percentage points ↗
4. A rising count can accompany a falling share
In week one, 30 of 50 registered learners attend: 60%. In week two, 36 of 80 attend: 45%. Six more people came, a 20% rise in the count. Yet the share fell by 15 percentage points because registrations grew faster than attendance. “Attendance improved” is too broad: which measure improved?
Do not average group percentages blindly. If group A has 10 attendees out of 10 and group B has 10 out of 90, the combined share is 20 out of 100, or 20%. The simple average of 100% and about 11.1% is about 55.6%, which gives a tiny group the same weight as a large one. Add compatible counts first, then divide.
Sources: Office for National Statistics: Data visualisation principles ↗
5. Small bases and zero deserve care
A new club growing from 2 members to 6 has grown 200%, but added only 4 members. Another growing from 200 to 240 adds 40 members with 20% growth. Neither percentage alone tells you which club serves more people. Report the count and the percentage when a small base could exaggerate the practical impression.
The usual percentage-change formula cannot use an old value of zero: division by zero is undefined. If attendance goes from 0 to 12, say “12 more attendees” or “attendance began with 12”, not “infinite percentage growth”. Missing values are different from zero; an unrecorded session must not automatically become a zero-attendance session.
Sources: Office for National Statistics: Percentages and percentage points ↗
6. Turn a headline into an honest comparison
Write one sentence naming the measure, both values, period and base: “In this invented club, weekly attendance rose from 30 of 50 registrants to 36 of 80; the count rose while the share fell from 60% to 45%.” You can now discuss two distinct tasks: reaching more people and helping a larger share of registrants attend.
The arithmetic does not explain the cause. A change could involve session timing, transport, registration rules or different groups. Keep the numerical description separate from a proposed explanation. If definitions or periods changed, resolve that before claiming an improvement or decline.
PUT IT INTO PRACTICE
A short investigation with answers
- A share moves from 25% to 30%. Calculate both changes. Answer: 5 percentage points; (30 − 25) ÷ 25 × 100 = 20% relative growth.
- A price falls from ₹800 to ₹600. What rise restores it? Answer: ₹200 ÷ ₹600 × 100 ≈ 33.3%; not 25%, because the base changed.
- Combine 18 successful attempts out of 20 and 12 out of 30. Answer: 30 ÷ 50 = 60%; the unweighted average of 90% and 40% would wrongly give 65%.
- Rewrite “participation jumped 300%” when numbers rose from 1 to 4. Answer: “Participation rose from 1 to 4, adding 3 people; the relative rise was 300%.”
Check your understanding
What is a denominator?
The quantity used as the comparison base in a fraction or rate.
Is a rise from 10% to 15% a rise of 5%?
It is 5 percentage points, or 50% relative growth in the share.
Why do a 10% rise and 10% fall not cancel?
The fall uses the increased base; multiplying 1.10 by 0.90 leaves 0.99 of the original.
Can a count rise while its share falls?
Yes, if the denominator grows faster, as in 30/50 changing to 36/80.
How do you combine shares of unequal groups?
Sum compatible numerators and denominators, then divide; do not automatically average the percentages.
Can growth from zero use the usual formula?
No. State the absolute change and starting value instead.