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Ten Points Up: Why Percent Change Depends on the Starting Score

A 10-point gain can be 25% or 12.5%. Worked examples show when points, ordinary percentages and log changes answer different questions.

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Sintesi

A change can be reported in score points or as a percentage of the starting value. Those summaries are not interchangeable. The percentage gives the starting score a second role: it is both an observation and the denominator.

This reproducible methods note uses constructed scores, exact arithmetic and a targeted primary-source review. A rise from 40 to 50 and a rise from 80 to 90 are both 10 points, but ordinary percent changes are 25% and 12.5%. A rise from 50 to 100 is +100%, while the return from 100 to 50 is -50%. The examples are teaching calculations, not observations of people, Altamens accounts or game telemetry.

Risultati chiave

  • Always report the starting value, ending value and score scale. A percentage without its denominator is incomplete.
  • Use ordinary percent change only when zero and ratios have a defensible meaning, the starting value is positive and the reader needs a multiplicative comparison.
  • For an arbitrary or bounded game score, raw point change plus the score range is often the more transparent description. Neither points nor percentages establish a training benefit or meaningful individual change.

1The denominator is an analytical decision

Point change is ending score minus starting score. Ordinary percent change divides that difference by the starting score and multiplies by 100. The division answers a new question: how large was the change relative to where the score began?

That can be useful when ratios are meaningful. It can also create a different story from the same point gain. Smaller starting values produce larger percentages for the same numerator, and a starting value of zero makes the ordinary percentage undefined.

2Worked example: the same 10 points tell two percentage stories

Consider two constructed score pairs on the same fixed scale. The first rises from 40 to 50; the second rises from 80 to 90. Both point changes are +10. Dividing by the starting score gives 10/40 = 25% for the first pair and 10/80 = 12.5% for the second.

Figura 1Equal point gains, unequal ordinary percentages
StartEnd

+10 points = +25%

40.0 points
50.0 points

+10 points = +12.5%

80.0 points
90.0 points

The numerator is identical. The percentage differs because the starting denominator differs.

Constructed scores. The chart demonstrates the denominator arithmetic, not learning, score quality or a meaningful-change threshold.

Download the worked examples CSV, standard-library calculator and methods and claim sheet. The files contain no participant or customer data.

3Ordinary percentage change is asymmetric

A rise from 50 to 100 is +100% because the gain equals the starting value. Returning from 100 to 50 is -50% because the same 50-point difference is now divided by 100. The two percentages do not cancel. This is ordinary percentage arithmetic, not an error.

Cole proposed 100 times the natural logarithm of end divided by start as a symmetric percentage difference for positive ratio-scale values. The 50-to-100 move is +69.31 log-percent and the reverse is -69.31. The measure is additive across successive positive ratios, but it does not make an arbitrary score origin meaningful.1

Figura 2Ordinary and log percentages answer different questions
Ordinary percent100 × log(end/start)

40 to 50

25.0%
22.3%

80 to 90

12.5%
11.8%

50 to 100

100.0%
69.3%

A symmetric formula solves direction asymmetry. It does not solve a bad scale, a noisy baseline or an unclear reader question.

Exact calculations rounded to one decimal place. Log change is defined only for positive values and needs a defensible multiplicative interpretation.

4Zero must mean zero before a ratio can mean much

For a ratio to carry the intended interpretation, the relationship between numerator and denominator needs appropriate scale properties. Curran-Everett notes that ratio normalization can misrepresent a relationship when its assumptions fail. A score whose zero is an arbitrary origin, offset or composite minimum does not automatically support statements such as twice as much performance.

A 0-to-10 change has a clear +10-point difference but no ordinary percent change because division by zero is undefined. Very small positive baselines can produce very large percentages. These are mathematical properties of the denominator, not evidence that the person changed more.6

5What the methods literature does and does not establish

Vickers used simulations of hypothetical controlled trials and found percentage change from baseline less statistically efficient than ANCOVA under the studied conditions. Curran-Everett and Williams likewise demonstrate that ratios can create or hide group differences, while Tu shows that testing percentage change against its own baseline can induce misleading associations.2,3,4

A 2026 simulation of neurite outgrowth assays compared endpoint, change, percentage-change, percent-of-control and ANCOVA methods across baseline imbalances and measurement-error conditions. Percentage change introduced denominator-related distortion in that model. The finding concerns a specific experimental estimand and assay simulation; it does not prove that every descriptive percentage is invalid or that ANCOVA is the right tool for a personal game dashboard.5

6A practical protocol for reporting a score change

  1. Start with the reader question: absolute progress on this score, proportional change or comparison with a recent level.
  2. Publish the start, end, direction and possible score range. Keep the raw values beside any transformed summary.
  3. Use point change for a fixed score scale when its units are the most interpretable and no defensible multiplicative meaning exists.
  4. Use ordinary percent change only when the baseline is positive, zero is meaningful and ratios answer the intended question.
  5. Consider a log change for positive ratio-scale measures when direction symmetry and additivity matter. Label the formula because it is not the ordinary percentage.
  6. Do not compare percentages across different games unless their score definitions, difficulty policies and denominators support that comparison.
  7. Show repeated results, accuracy, trial count and task changes. No single transformation removes measurement noise or strategy shifts.

For choosing the reference sessions, see One Session or Five? For denominator-aware accuracy reporting, see Same Accuracy, Different Certainty.

7Limitations, provenance and review status

All score pairs were constructed to expose arithmetic properties. The note does not estimate reliability, practice effects, measurement error, learning, ceiling effects or meaningful change for any Altamens game. The literature review was targeted rather than systematic, and evidence from controlled trials, physiological measurements and assay simulations was used only to bound general statistical claims.

No Altamens user records, accounts, analytics, devices or game telemetry were accessed. Literature search, drafting, calculations, source checks and publication preparation used an OpenAI automated assistant. The calculator and displayed outputs received automated consistency checks, but the paper has no independent human methods review and is not journal peer reviewed.

Riferimenti

  1. 1.Cole TJ Sympercents: symmetric percentage differences on the 100 log(e) scale simplify the presentation of log transformed data. Statistics in Medicine 19(22), 3109-3125, 2000. Proposes 100 × log(end/start) for positive ratio-scale values. DOI: 10.1002/1097-0258(20001130)19:22<3109::AID-SIM558>3.0.CO;2-F.. https://pubmed.ncbi.nlm.nih.gov/11113946/
  2. 2.Vickers AJ The use of percentage change from baseline as an outcome in a controlled trial is statistically inefficient: a simulation study. BMC Medical Research Methodology 1, 6, 2001. Simulation of controlled-trial analysis methods. DOI: 10.1186/1471-2288-1-6.. https://pubmed.ncbi.nlm.nih.gov/11459516/
  3. 3.Curran-Everett D, Williams CL Explorations in statistics: the analysis of change. Advances in Physiology Education 39(2), 49-54, 2015. Educational statistical methods article on absolute, percent and symmetrized change. DOI: 10.1152/advan.00018.2015.. https://pubmed.ncbi.nlm.nih.gov/26031718/
  4. 4.Tu YK Testing the relation between percentage change and baseline value. Scientific Reports 6, 23247, 2016. Methods paper with simulations of baseline-dependent percentage-change associations. DOI: 10.1038/srep23247.. https://pubmed.ncbi.nlm.nih.gov/26979481/
  5. 5.Hewitt K, Huang XF Baseline Adjustment and Normalisation in Neurite Outgrowth Assays: A Simulation-Based Comparison of Analytical Methods. Journal of Neuroscience Methods, 110940, 2026. Assay-specific simulation across baseline imbalance and measurement-error conditions. DOI: 10.1016/j.jneumeth.2026.110940.. https://pubmed.ncbi.nlm.nih.gov/42918554/
  6. 6.Curran-Everett D Explorations in statistics: the analysis of ratios and normalized data. Advances in Physiology Education 37(3), 213-219, 2013. Educational statistical methods article on ratio assumptions and regression alternatives. DOI: 10.1152/advan.00053.2013.. https://pubmed.ncbi.nlm.nih.gov/24022766/