Raw data
Values read or logged directly, with units, timestamps and measurement resolution.
IB Mathematics · Data analysis
Analyse indirectly measured quantities mathematically.
Separate raw readings, derived quantities, models and evaluation, and show transparently how measurements become a mathematical argument.
The mathematics behind the measurement
Raw readings become an investigation only through transformation, modelling and evaluation.
A transparent IA or EE separates direct readings, processed data, model parameters and model evaluation. This makes it clear which values were measured and which were derived mathematically.
Processed data
Means, resistances, powers, densities, rates, gradients or transformed coordinates.
Model
Equation, parameters, assumptions and the relationship between independent and dependent quantities.
Evaluation
Residuals, uncertainty, outliers, limitations, model comparison and improvements.
Five central transformations
Mathematical tools for indirect measurements.
Geometry, volume and density
Vcylinder = πr²h
Vsphere = 4πr³/3
ρ = m/V
Compare several geometric approximations and evaluate them through relative error against a reference measurement.
Rates of change from time series
dy/dt ≈ (yi+1 − yi)/(ti+1 − ti)
This gives an evaporation rate dm/dt, CO₂ rate dC/dt, cooling rate dT/dt or GPS speed ds/dt.
Areas under curves
E = ∫₀ᵀ P(t) dt
E ≈ Σ[(Pk+Pk+1)/2]·Δt
Power over time gives energy; flow rate over time gives volume; speed over time gives distance.
Calibration of DIY sensors
m = aS + b
m = aS² + bS + c
Compare linear and quadratic calibration using residuals, maximum deviation and physical plausibility, not R² alone.
Propagation of uncertainty
ΔR/R ≈ √[(ΔU/U)² + (ΔI/I)²]
ΔP/P ≈ √[(ΔU/U)² + (ΔI/I)²]
A derived quantity inherits uncertainty from several raw measurements. Keep resolution, calibration and systematic error distinct.
Battery: energy rather than voltage alone.
With a constant or documented load, record voltage and current as time series.
P(t) = U(t) · I(t)
E = ∫₀ᵀ U(t) · I(t) dt
The trapezoidal rule works with discrete readings. It links the discharge curve to the energy actually delivered.
Solar cell: angle, power and energy.
P(θ) = U(θ) · I(θ)
P(θ) = Pmax · cos(θ − θ₀)
Compare the cosine model with measured power, inspect residuals and determine the angle of the modelled maximum.
Data collection
Separate raw readings and derived quantities in the data table.
This structure makes repetitions, controls and uncertainty transparent.
| Run | Independent variable | Repeat | Raw value 1 | Raw value 2 | Temperature | Derived quantity | Uncertainty | Note |
|---|---|---|---|---|---|---|---|---|
| 1 | 0.10 | 1 | Uref | Usample | T | σ | Δσ | Stable setup |
| 2 | 0.10 | 2 | Uref | Usample | T | σ | Δσ | Stable setup |
| 3 | 0.20 | 1 | Uref | Usample | T | σ | Δσ | Electrodes rinsed |
Technical combinations
Hardware, data logging and Python as one transparent measurement chain.
Raspberry Pi + DS18B20 + fan
Cooling, thermal gradients or cooling performance per watt.
ESP32-CAM + image analysis
Dye spreading, motion analysis or visible changes in area.
USB GPS + Raspberry Pi
GPS spread, error ellipses and comparisons among environments.
HX711 load cell + camera
Force–deformation curves, spring constants and hysteresis.
Microphone + Python FFT
Resonance, frequency analysis, speed of sound and acoustic attenuation.
Raspberry Pi + several sensors
Parallel time series, correlations and time lags.
Quality check
What makes a strong experimental Mathematics IA.
- A precise mathematical research question rather than a purely scientific aim.
- A transparent derivation of the dependent quantity from raw data.
- A systematically varied independent variable and sufficient repetitions.
- Often about 8 to 15 levels of the independent variable, with repeats.
- A comparison of at least two plausible models.
- Analysis of residuals, outliers, resolution and systematic error.
- A mathematical interpretation of model parameters.
- Critical reflection on where the model fails and why.