If a ratio rises from 4% to 5.5%, the change is 1.5 percentage points, and the increase relative to the starting ratio is 37.5%. Both results are correct. The first expresses the difference between the two ratios, while the second expresses the magnitude of the increase relative to the previous ratio. In a report, you can present them together so that the scale of the change remains clear.
When comparing application rates, error rates or utilisation, first check that the same case counts as a success in both datasets and that cases were included in the denominator under identical conditions. The observation period should also be comparable: mark any still-open application period separately. With differing measurement rules, percentage calculations can still be performed, but you can only infer changes in performance from the result after clarifying the differences.
First, define the base
Let r be the numerical value of the old ratio expressed as a percentage, and u the new one. The change in percentage points is u − r. The relative percentage change is (u − r) / r × 100. For the latter, r cannot be zero. If 4% and 5.5% appear in your data, use 4 and 5.5 in these formulas.
The concept of percentage points is defined by the Eurostat definition; the change relative to the starting value is a separate concept of percentage change. The figures below are illustrative examples.
Imagine two groups of applicants, each with 2000 people. In the first, 80 people complete an optional form, and in the second, 110. The ratios are 80 / 2000 × 100 = 4%, and 110 / 2000 × 100 = 5.5%.
The difference is 5.5 − 4 = 1.5 percentage points. The relative increase is 1.5 / 4 × 100 = 37.5%. You can also check this using counts: 30 additional completions occurred, and 30 / 80 × 100 is also 37.5%. Because the base is the same 2000 people, the relative change in counts and ratios is identical here.
If you also need to state what change leads back to the original value, continue with the examples of increasing after a percentage decrease: the second percentage is already compared to the changed base.
Different group sizes yield different results
Let us modify only the size of the second group to 3000 people, keeping the number of completions at 110. The new ratio is 110 / 3000 × 100 = 3.6666…%, rounded to 3.67%. The change is now 3.6666… − 4 = −0.3333… percentage points, and −0.3333… / 4 × 100 = −8.3333…%.
The number of completions is still 37.5% higher than before, while the completion rate is approximately 8.33% lower. Therefore, both group sizes should be included in the report. The mere statement 'completions increased' does not reveal which metric you are discussing.
| Metric | First group | Second group |
|---|---|---|
| Group size | 2000 people | 3000 people |
| Completions | 80 items | 110 items |
| Completion rate | 4% | 3.67% |
The table shows rounded ratios, but the change was calculated from the values before rounding. If you continue calculating with 3.67, a small discrepancy will enter the relative result.
When every subgroup improves, the aggregate still decreases
In a new, separate example, we measure the success of two types of tasks. In the first period, 100 of each type are completed, and in the second, a total of 200 tasks are completed, but with a different composition.
| Task type | First period: successful / total | First ratio | Second period: successful / total | Second ratio |
|---|---|---|---|---|
| A | 90 / 100 | 90% | 19 / 20 | 95% |
| B | 40 / 100 | 40% | 90 / 180 | 50% |
The ratio for type A improved by 5 percentage points, and for type B by 10 percentage points. In aggregate, however, the first period is 130 / 200 = 65%, and the second is 109 / 200 = 54.5%. The change is −10.5 percentage points, relatively −10.5 / 65 × 100 ≈ −16.15%.
The discrepancy is explained by the distribution of tasks. The share of type B, which has lower success, increased from 50% to 90%. Therefore, the aggregate ratio alone does not imply that the effectiveness of any task type has deteriorated.
If you wish to compare the changes in ratios separately, you can also calculate with a fixed composition. Maintaining the original fifty-fifty distribution, the second period would correspond to 0.5 × 95% + 0.5 × 50% = 72.5%. This is a comparative value, not the success rate of the 200 tasks actually completed. Therefore, present the 72.5% only with the chosen weights noted.
From the data above, you can also decompose the total −10.5 percentage points. First, change the subgroup ratios while maintaining the original weights: 72.5 − 65 = +7.5 percentage points. Then switch to the new composition: 54.5 − 72.5 = −18 percentage points. The sum of the two is −10.5 percentage points.
You should also state the order of decomposition. If you swap the weights first, the weighted average of the old ratios with the new weights is 0.1 × 90% + 0.9 × 40% = 45%. Thus, −20 is attributed to the composition, and +9.5 percentage points to the improvement within the groups. The total discrepancy is the same, but the parts differ. Therefore, the chosen decomposition provides an arithmetic explanation; do not label the individual parts as proven causal effects without further investigation.
In your own report, retain the successful and total cases by group. If it is not even clear what counts as a case, first clarify the counting unit based on the article on the denominator of conversion rates (in Hungarian). The weighting of groups can then be performed consistently.
Zero and very small bases
When changing from zero to 2%, the difference is 2 percentage points. The formula for relative change would divide by zero, so do not write a finite percentage increase for this. Stating the previous zero and the new 2% is accurate in itself, provided you also include the case count and group size.
With very small ratios, large relative changes can also occur. An increase from 0.2% to 0.3% is a 50% relative increase, while the difference is 0.1 percentage points. This is an arithmetic fact; the two numbers alone do not determine the reliability or cause of the change.
In the percentage change calculator, select the percentage point and relative change mode. Enter the two ratios in the same percentage format. The final sentence should include the old ratio, the new ratio, and the compared period or group: these allow the calculation to be verified later.
Final knowledge check
Choose one answer for each question. Each correct answer earns 1 point; an incorrect answer, a skipped question or more than one selected answer earns 0 points. You can score 0–3 points in total. Read the solutions after answering.
1. A ratio increases from 4% to 5.5%. Which description is correct?
- A) +1.5% relative change and +37.5 percentage points.
- B) +5.5 percentage points.
- C) +1.5 percentage points and +37.5% relative change.
2. The success rate of task type A increases from 90% to 95%, and that of B from 40% to 50%. The weight of B changes from 50% to 90%. Can the overall success decrease?
- A) No, if both subgroups improved.
- B) Yes, but only due to a calculation error.
- C) Yes: in the example, the aggregate ratio decreases from 65% to 54.5%.
3. A ratio changes from 0% to 2%. How should you report the change?
- A) +2 percentage points; the usual relative percentage change is not interpretable from a zero base.
- B) +200% relative increase.
- C) +2% relative increase.
Solutions and scoring
1. C. The direct difference is 5.5 − 4 = 1.5 percentage points. Relative to the old ratio, 1.5/4 × 100 = 37.5%. For revision: First, define the base.
2. C. The first weighted ratio is 0.5 × 90% + 0.5 × 40% = 65%; the new one is 0.1 × 95% + 0.9 × 50% = 54.5%. The change in weights must be preserved separately. For revision: When every subgroup improves, the aggregate still decreases.
3. A. The difference is 2 percentage points, but the relative formula would divide by the old, zero value. Also show the old and new case counts and denominators. For revision: Zero and very small bases.
Each correct answer is worth 1 point; incorrect or skipped answers are worth 0 points. The total score can be 0–3.
- 0 points: work through the sections marked in the three solutions again, and write down the initial conditions of the examples.
- 1 point: derive the correct answer for the two missed questions; separately note which condition or calculation step was omitted.
- 2 points: return to the section linked to the single missed question, then verify the result again by changing one input.
- 3 points: you correctly applied the three relationships in this task set. Try one of the article's checks with your own data, noting the conditions and units.
Translated and adapted from the Hungarian article: Hungarian original.