Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

How strongly is global temperature linked to ln(CO₂)?

AI SLAI HLAA SLAA HL Ambitious Also in: Statistics, Modelling

Research question

How well does a linear model of global temperature anomaly against the natural logarithm of CO₂ concentration fit the data since 1959, and what warming per doubling of CO₂ does the fitted gradient imply?

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Why it makes a good exploration

Physics predicts warming proportional to the logarithm of CO₂, so a regression of temperature on ln(CO₂) has a meaningful gradient. Estimating it from NASA and NOAA data — and discussing why it is not the full climate sensitivity — is a mature, rigorous IA.

The mathematics you'll need

Logarithmic transformation and why it is expected
Linear regression and correlation
Converting a gradient to warming per doubling (× ln 2)
Residuals and the role of natural variability

Modelling the data, step by step

This idea usually needs a straight-line model. See it worked step by step, with a criterion tip at every step: Straight line y = mx + c · Choosing and comparing models.

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for you.

Where the data comes from

Annual global temperature anomalies from NASA GISTEMP and annual mean CO₂ from NOAA GML.

NASA GISTEMP v4 — Global and hemispheric temperature anomalies from 1880, monthly and annual, CSV.
NOAA Global Monitoring Laboratory — Mauna Loa CO₂ — Monthly and annual mean atmospheric CO₂ since 1958 (the Keeling curve), CSV.

Real data for this idea

Primary source: NASA GISTEMP v4: GISTEMP v4 global means. Download the global-mean monthly, seasonal and annual means (CSV); the J-D column is the yearly anomaly. Licence: US government data; NASA asks you to cite the dataset.
Also useful: NOAA Global Monitoring Laboratory: Mauna Loa CO₂: Mauna Loa monthly and annual means. Download the monthly mean CSV (year, month, average). Licence: Free to use; cite the dataset, its version and the date you downloaded it.
Readable chart: Our World in Data: Temperature change relative to the pre-industrial period. Use it to see the pattern; download your numbers from the primary source or the chart's own download. Cite as: Our World in Data, ‘Temperature change relative to the pre-industrial period’, CC BY 4.0. Original source: Met Office Hadley Centre, HadCRUT5 (2026).
Readable chart: Our World in Data: Global atmospheric CO₂ concentration. Use it to see the pattern; download your numbers from the primary source or the chart's own download. Cite as: Our World in Data, ‘Global atmospheric CO₂ concentration’, CC BY 4.0. Original source: NOAA Global Monitoring Laboratory, Trends in Atmospheric Carbon Dioxide (2026).

The Our World in Data chart uses HadCRUT5, a different temperature series from GISTEMP: use one series for your model and name it. NOAA revises recent values, so write down the file's version date and the day you downloaded it. Links checked October 2026; we link to the data and never copy it. More free data sets.

Ready to use: CO₂ at Mauna Loa (yearly) · Global temperature (GISTEMP) in the IA data bank — cited, licensed for reuse, and one click to open in the grapher.

A possible outline

Explain why ln(CO₂) rather than CO₂.
Merge the two annual series.
Fit and interpret the regression.
Compute the implied warming per doubling.
Reflect: lag, other gases, aerosols, El Niño years, correlation vs causation.

Pitfalls that cost marks

Presenting the gradient as climate sensitivity without qualification.
Using mismatched years.
Ignoring autocorrelation in residuals (explain it simply).

Showing personal engagement

Compare your estimate with published ranges (cite them).
Identify outlier years and explain them.
Discuss how you would communicate the result responsibly.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Fits — ambitious at SL	Logarithmic transformation and why it is expected; Linear regression and correlation
AA HL	Good fit	Logarithmic transformation and why it is expected; Linear regression and correlation
AI SL	Fits — ambitious at SL	Logarithmic transformation and why it is expected; Linear regression and correlation
AI HL	Good fit	Logarithmic transformation and why it is expected; Linear regression and correlation

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: presenting the gradient as climate sensitivity without qualification; and using mismatched years. Say how each one affects your answer.

Use of mathematics (E)

SL: logarithmic transformation and why it is expected; linear regression and correlation; converting a gradient to warming per doubling (× ln 2); residuals and the role of natural variability, each explained in your own words and carried out correctly.

How environment explorations reach the top bands in general: Environment & climate IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Add a one-year lag or an El Niño indicator and compare the fit.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL environment exploration adds.

See a complete IA, marked

When can we launch? Modelling the tide at our sailing club's slipway (AA SL) is a complete environment exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AI SL, AI HL, AA SL and AA HL. We rate it Ambitious: it suits confident students, who should expect to learn some mathematics on their own.

What mathematics does this IA use?

Logarithmic transformation and why it is expected, linear regression and correlation, converting a gradient to warming per doubling (× ln 2) and residuals and the role of natural variability.

Where can I get data for this IA?

Annual global temperature anomalies from NASA GISTEMP and annual mean CO₂ from NOAA GML. Checked sources and tools: NASA GISTEMP v4, NOAA Global Monitoring Laboratory: Mauna Loa CO₂, Our World in Data and NOAA Global Monitoring Laboratory — Mauna Loa CO₂.

Turn this idea into your IA

The planner opens with “How strongly is global temperature linked to ln(CO₂)?” filled in: the question, the maths and the data sources.

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