PSLE Science Reality Lab Vol No.156 | “R² = 0.90” — Does That Mean 90% of the Predictions Were Correct?

PSLE-SCI-REALITY-0156

Wait, What? A Model Card Says R² = 0.90

A science fair team builds a model that uses the thickness of an insulating layer to estimate cooling time. Their report contains a scatter plot, a fitted curve and one large number:

R² = 0.90

A caption then says:

“The model is 90% correct.”

That is not what R² means.

For the fitted model and the data being analysed, R² is a summary of how much of the observed variation in the response is accounted for by the model compared with a simple baseline based on the response mean. It is not a score saying that 90 out of every 100 predictions are correct.

A model can have a high R² and still have important problems: curved residual patterns, a few influential observations, poor performance on new data, a limited tested range, or a relationship that should not be interpreted causally.

Reality Lab habit: A model-fit statistic describes one feature of a fitted relationship. Never turn it into a percentage-correct badge without checking what the statistic actually measures.

Quick Answer

R² is not the percentage of individual predictions that are correct.
R² summarises how much variation in the observed response is accounted for by the fitted model in the analysed data.
A high R² does not prove the model’s scientific explanation is true.
A high R² does not guarantee small prediction errors for every observation.
A high R² on the data used to fit a model does not guarantee equally strong performance on new data.
Residuals can reveal patterns that one R² number hides.
The tested range matters: a good fit inside one range does not guarantee safe extrapolation outside it.
Judge model usefulness using several pieces of evidence, not one impressive statistic.

The Exact Learner Job This Article Owns

This article owns one real-world evidence-transfer job: how a Primary 5/6 learner should read a scientific model graphic or report that gives an R² value without interpreting that number as “percent of predictions correct”.

It does not teach formal regression mathematics, sums of squares or statistical inference as standalone topics. It also does not replace the existing PSLE Science owners for model limits, extrapolation, causal reasoning or evidence strength. Reality Lab applies those owners to one common scientific communication object: a large R² printed beside a fitted model.

Original Reality Lab Case: The CoolBox Model

The following example is fictional and uses constructed data.

A student team tests insulated boxes with different foam thicknesses. For each box, they record how long it takes a fixed volume of warm water to cool by the same temperature difference.

Foam thickness / mm	Observed cooling time / min	Model prediction / min	Prediction error / min
5	18	20	−2
10	27	25	+2
15	31	30	+1
20	36	35	+1
25	38	40	−2
30	45	45	0
35	51	50	+1
40	53	55	−2

The software reports R² = 0.90 for the fitted relationship.

Can the team count the eight rows and say that 90% of them were correct?

No. R² is not created by marking each row “right” or “wrong”. A numerical prediction can be close by 1 minute, off by 2 minutes or much farther away. There is no universal boundary in this example that turns each prediction into a binary correct/incorrect score.

Observed, Calculated, Inferred and Overclaimed

Layer	CoolBox example
Observed	Cooling times measured for boxes within the tested thickness range
Calculated	A fitted model and R² = 0.90 for these analysed data
Reasonable interpretation	The fitted model accounts for a large fraction of the observed variation in cooling time relative to the baseline used by R²
Question still open	Are individual errors small enough for the intended use, and does the model work on new data?
Unsupported claim	90% of all predictions are correct

What Does “Variation” Mean Here?

The cooling times are not all identical. Some boxes cool faster and some slower. That spread of response values is variation.

A fitted model tries to account for some of that variation by relating the response to one or more measured predictors. R² summarises the fraction of the response variation accounted for by the fitted model in the analysed data, using a particular mathematical definition.

This is a statement about variation around a fitted relationship. It is not a statement about 100 imaginary checkboxes labelled correct or incorrect.

The “90% Correct” Trap

Suppose a model predicts 29.9 when the observed value is 30.0. Is that prediction “correct”?

What if it predicts 29? Or 27? Whether an error is acceptable depends on the scientific purpose and measurement context. R² does not supply one universal pass/fail boundary for each prediction.

That is why “90% correct” is the wrong translation. It silently changes a model-fit statistic into an accuracy percentage with a completely different meaning.

High R² Does Not Mean Every Point Is Close

A data set can show a strong overall relationship while still containing a few large prediction errors. Conversely, a narrow data range can produce small absolute errors even when R² is modest.

Therefore a careful model report should not stop at R². It should examine the actual sizes and patterns of errors as well.

Residuals: The Errors Have a Shape

A residual is the difference between an observed response and the value predicted by the fitted model. You do not need formal regression mathematics to use the idea.

Ask:

Are errors scattered on both sides of the model?
Do they become larger at one end of the range?
Do they form a curve or other pattern?
Is one observation carrying much of the fit?
Are certain groups consistently over- or under-predicted?

NIST’s model-validation guidance makes the point directly: R² is not enough. A high R² does not guarantee that a model fits the data well, and residual analysis can reveal problems that the one-number summary misses.

A High R² Can Hide a Curved Pattern

Imagine fitting a straight line to data that really bends upward. The line may still capture much of the overall change and produce a large R², especially across a wide range.

But the residuals may show a pattern: positive at low values, negative in the middle, positive again at high values. That pattern says the straight-line model is missing structure.

The lesson is not “R² is useless”. It is “R² answers one question, so ask other questions too”.

Fit Is Not Explanation

Suppose ice-cream sales and electricity use both rise on hot days. A fitted relationship between them might be strong. That does not mean buying ice cream causes household electricity use to rise.

A third condition—hot weather—can affect both.

R² therefore does not turn a pattern into a proven mechanism. The causal job belongs to experimental design, alternative explanations and domain science.

Fit on These Data Is Not Validation on New Data

A flexible model can sometimes fit the data used to build it very closely. The stronger question is whether it also performs well on new observations that did not determine the fit.

This is especially important when a report uses language such as “predictive”. Prediction is about future or otherwise unseen cases, not only about describing the data already supplied to the model.

For a Primary learner, the evidence habit is simple:

Ask whether the impressive statistic came from the same data used to build the model or from a genuinely separate check.

The Range Check: Where Was the Model Tested?

The CoolBox model was built using foam thicknesses from 5 mm to 40 mm.

Suppose someone uses it to predict cooling time for 200 mm of foam. Even if R² was high inside the original range, the relationship may not continue in the same way far beyond it. Physical limits, diminishing effects or new mechanisms can matter.

A fit statistic inside the tested range is not a passport to unlimited extrapolation.

The Scale Check: What Size of Error Matters?

Imagine Model A predicts crop height with errors of about 1 cm. Model B predicts a satellite’s position with errors of 1 km. The importance of those errors depends on the task.

R² alone does not tell a reader whether the prediction error is small enough for the intended use. A useful report should also communicate errors in meaningful units or other task-relevant measures.

Worked Case 1: High R², Large Error at One End

A model of plant height versus age reports R² = 0.94. Most predictions are close, but the oldest plants are consistently under-predicted by 8–10 cm.

The large R² does not erase the systematic error at the upper end. A learner should ask whether the model form or growth stage changes there.

Worked Case 2: R² Is High Because the Range Is Wide

Temperatures in a dataset range from 5°C to 45°C. A model has errors of several degrees but still tracks the overall wide trend strongly.

A high R² can coexist with errors that are too large for a precise control task. Judge the error in units that matter, not only the fraction of variation accounted for.

Worked Case 3: R² Drops on New Data

A model reports R² = 0.92 on the data used to fit it. A second independent dataset from the following month shows much poorer agreement and large errors.

The second dataset reveals that the original fit did not fully transfer. The correct response is to investigate changed conditions, overfitting or missing variables rather than advertise only the first number.

Worked Case 4: Same R², Different Error Patterns

Two models both report R² = 0.88. Model A’s residuals are scattered without an obvious pattern. Model B’s residuals form a clear curve.

The same R² does not make the models equally satisfactory. Model B’s patterned residuals indicate that its fitted form is missing structure.

Worked Case 5: One Extreme Point Inflates the Impression

Most observations occupy a narrow cluster, while one far-away observation lies near the fitted line. Removing or investigating that point changes the fitted relationship dramatically.

This does not mean the point should automatically be deleted. It means the reader should examine whether the conclusion depends heavily on one influential observation.

Worked Case 6: High R², Wrong Causal Story

A report finds that two environmental measurements move together and gives R² = 0.91. It claims variable X caused variable Y.

The fit statistic does not establish causation. Other variables, common drivers, selection effects or reverse relationships may explain the pattern. Causal claims require additional evidence.

Worked Case 7: Modest R² but Useful Scientific Signal

A biological system naturally varies because many factors influence it. One measured factor accounts for only part of that variation, yet the effect is repeatable and scientifically meaningful.

A modest R² does not automatically make the evidence worthless. Scientific usefulness depends on the question, design, effect, uncertainty and alternatives, not a single universal cutoff.

What Evidence Strengthens a Model Claim?

A clear description of what response and predictors were measured.
Data covering the range relevant to the claim.
Residual checks showing no important unexplained pattern.
Prediction errors reported in meaningful units.
Validation on data not used to fit the model where prediction is claimed.
Transparent handling of unusual or influential observations.
Alternative models or explanations considered when appropriate.
A conclusion that distinguishes pattern, prediction and cause.

What Weakens It?

Advertising only a large R² with no plot or error information.
Calling R² a percent-correct score.
Using the same data to fit and celebrate the model without a separate predictive check.
Ignoring patterned residuals.
Predicting far beyond the tested range.
Turning association into causation.
Hiding observations that make the fit look weaker.
Using “high R²” as if every scientific question has the same acceptable threshold.

Tempting Reasoning That Fails

“R² = 0.90 means 90 of 100 predictions are correct.” No. R² is not a count of correct cases.
“R² = 1 means the scientific explanation must be true.” A perfect fit does not prove a unique mechanism.
“High R² means every prediction error is tiny.” Individual errors still need inspection.
“Low R² means there is no real relationship.” Context, natural variation and the scientific question matter.
“The model worked from 5 to 40, so it must work at 200.” Extrapolation requires additional evidence.
“One high number is enough to validate a model.” Model checking is multi-part.

How Far Can the Conclusion Travel?

A careful statement can say:

For these analysed data and this fitted model, R² = 0.90 indicates that the model accounts for a large fraction of the observed variation in the response.

Without more evidence, it cannot automatically become:

90% of predictions are correct;
the model is 90% accurate;
the model will perform equally well on new data;
the model works outside the tested range;
the proposed mechanism is proven;
every residual is small or random.

PSLE-Style Transfer Case

A report models the relationship between light intensity and the measured growth rate of seedlings over a tested range. It states R² = 0.93 and claims, “93% of future growth-rate predictions will be correct.”

Explain why the claim is too strong.

Answer: R² does not count the percentage of future predictions that are correct. It summarises how much variation in the analysed response data is accounted for by the fitted model. To judge future prediction, the report should examine prediction errors and test the model on suitable new data within relevant conditions.

What additional graph would help?

A residual plot can show whether the errors are randomly scattered or whether a systematic pattern remains that the model failed to capture.

Explained Practice

Practice A: R² = 0.87. Can you say 87% of predictions are correct? No.

Practice B: A model has high R² but residuals form a curve. Is the high R² enough? No. The residual pattern suggests missing structure.

Practice C: A model is fitted on 5–40°C and used at 200°C. What is the main concern? Extrapolation beyond the tested range.

Practice D: The same dataset is used to fit the model and report its impressive fit. What additional evidence helps a predictive claim? Performance on suitable new data.

Practice E: Two variables have R² = 0.95. Is causation proven? No.

Delayed Independent Return: V-E-R-S

When a model report presents one impressive fit number, return later with four checks:

V — Variation: What variation is the statistic summarising?
E — Errors: How large are the residuals or prediction errors, and do they show a pattern?
R — Range: Is the claim staying inside the conditions actually tested?
S — Separate check: Was predictive performance tested on suitable new data?

This is a thinking aid, not an examiner-mandated template.

Parent and Tutor Teaching Guide

Begin without the symbol R². Draw a scatter of ten points around a rising line. Ask whether the line captures the overall change. Then circle one point that is far from the line and ask whether “good overall pattern” means “every point is predicted exactly”.

Next show two imaginary reports: one says “R² = 0.90”; the other says “90 out of 100 predictions were within ±1 unit”. Ask whether the two statements mean the same thing. They do not. The second defines a case-by-case accuracy rule; the first is a model-fit statistic.

Finally ask the learner to predict far beyond the plotted range. This exposes the difference between describing a fitted relationship and assuming it continues forever.

Authoritative Sources

NIST explicitly warns that R² alone is not enough for model validation and notes that a high R² does not guarantee a good model fit. SEAB’s 2026 PSLE Science objectives include interpreting and analysing information, evaluating observations, information and methods, and communicating explanations and reasoning. Those habits apply even when the statistic on the page belongs to more advanced mathematics.

The Quiet Return

R² = 0.90 can be useful information.

It becomes misleading only when we ask it to say something it was never designed to say.

Read the statistic for the job it actually performs. Then inspect the errors, the range and the evidence from new data before trusting the model farther.

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