Raw measurement
Voltage, time, mass, temperature, position, frequency or pixel values are recorded directly.
IB Mathematics · Indirect measurement
Indirect measurement methods for the Mathematics IA and Extended Essay.
Turn raw readings from sensors, video, GPS and simple instruments into transparent mathematical target quantities, models and comparisons.
From sensor reading to mathematical conclusion
A device provides raw data; mathematics provides the target quantity.
In an indirect measurement, the quantity of interest is not read directly. It is obtained from several raw measurements, a transparent model equation and an analysis. This chain turns an equipment-based idea into a Mathematics IA or EE question.
Model
A physical, geometric or statistical model connects the raw readings.
Target quantity
Resistance, power, energy, density, rate, gradient or an error ellipse is calculated.
Analysis
Models, residuals, parameters, uncertainties and limitations are compared.
Plan backwardsDo not begin with the device. Begin with the target quantity: what do you want to determine, which raw readings are required, which model connects them, and how will you test the model?
Combine equipment purposefully
Eight examples: what you read and what you can calculate from it.
The photographs show real equipment from the PreLearning measurement-tool directory. The mathematical work begins after the readings have been collected.
Density from geometry and mass
- Measured directly
- Diameter, length, depth and mass
- Derived indirectly
- Volume and density ρ = m/V
- Mathematical analysis
- Geometric models, relative error and uncertainty propagation
Cooling constant from a time series
- Measured directly
- Temperature T at fixed time intervals
- Derived indirectly
- Cooling rate and parameter k
- Mathematical analysis
- Exponential regression, residuals and parameter comparison
Temperature gradient and threshold area
- Measured directly
- Temperature values across a surface
- Derived indirectly
- Area mean, gradient and area above a threshold
- Mathematical analysis
- Image scaling, area analysis and spatial modelling
Air-exchange rate from CO₂ decay
- Measured directly
- CO₂ concentration C(t)
- Derived indirectly
- Decay constant and air-exchange rate
- Mathematical analysis
- Exponential model, sensitivity analysis and model comparison
Power law from field strength and distance
- Measured directly
- Relative field strength B and distance d
- Derived indirectly
- Exponent n in B(d) = a·d⁻ⁿ + c
- Mathematical analysis
- Power regression, log-linearisation and residuals
Force or mass from digital raw readings
- Measured directly
- Digital sensor readings S
- Derived indirectly
- Force or mass after calibration
- Mathematical analysis
- Linear and quadratic calibration, maximum deviation
Speed of sound from resonances
- Measured directly
- Resonance frequency f and tube length L
- Derived indirectly
- Speed of sound v
- Mathematical analysis
- Regression of f against 1/L, slope and error analysis
GPS error ellipse from coordinates
- Measured directly
- Many coordinate pairs at one fixed location
- Derived indirectly
- Spread, covariance and error ellipse
- Mathematical analysis
- Coordinate transformation, statistics and ellipse geometry
Worked measurement chain
Example: derive the conductivity of a salt solution from voltage measurements.
One possible series circuit uses a stable voltage source, a known reference resistor Rref and the solution between two electrodes. Measure Uref, Usolution, temperature, electrode spacing L and effective electrode area A.
CurrentI = Uref / Rref
Solution resistanceRsolution = Rref · Usolution / Uref
Approximate conductivityσ = L / (Rsolution · A)
To what extent does a linear, power or saturation model describe the relationship between sodium-chloride concentration and electrical conductivity derived from voltage measurements at constant temperature?
For a fair comparison, control electrode geometry, temperature, measurement duration, electrode material, applied voltage and mixing. The uncertainty in the derived conductivity depends on several raw measurements at once.
18 measurement chains
From a direct reading to a mathematically useful quantity.
The table shows how existing devices can be combined. On small screens, each row becomes a separate card.
| Device or combination | Measured directly | Derived indirectly | Possible research question | Mathematics |
|---|---|---|---|---|
| Calliper + scale | Diameter, length, mass | Density ρ = m/V | Which geometric model gives the most realistic density of an object? | Geometry, model comparison, relative error |
| Scale + timer | Mass over time | Evaporation rate dm/dt | Which model describes evaporation under different airflow conditions? | Difference quotients, regression |
| Scale + fan + temperature | Mass, temperature, time | Airflow-dependent evaporation | How do airflow and temperature change the evaporation constant? | Multiple regression, piecewise models |
| Measuring cylinder + timer | Volume, time | Flow rate Q = ΔV/Δt | How does the flow rate change during drainage? | Rates of change, regression |
| Temperature data logger | Temperature over time | Cooling constant | Which container insulation minimises the cooling constant? | Exponential regression, residuals |
| DS18B20 at several positions | Temperature, position, time | Temperature gradient ∂T/∂x | How does a temperature gradient develop along a metal rod? | Spatial regression, rates of change |
| Thermal imager | Temperature field | Area mean, gradient | How does the area above a temperature threshold change? | Image scaling, area analysis |
| CO₂ sensor | CO₂ concentration, time | Air-exchange rate | Which ventilation strategy maximises the air-exchange rate? | Exponential model, parameter comparison |
| EMF meter + tape measure | Field strength, distance | Exponent of a power law | Which exponent n describes B(d) = a·d⁻ⁿ + c? | Power regression, log-linearisation |
| HX711 load cell | Digital raw readings | Force or mass after calibration | Which calibration model minimises sensor error? | Calibration, residuals, uncertainty |
| Microphone + tape measure | Frequency, tube length | Speed of sound | How accurately can the speed of sound be obtained from resonances? | Regression of f against 1/L |
| Phone video + scale reference | Pixel position, frame number | Velocity, acceleration | Which model describes a projectile path? | Quadratic regression, derivatives |
| Phone gyroscope | Angular velocity | Angular position, damping | How does the damping rate change with friction? | Integration, exponential models |
| GPS + time | Position, time | Speed, path length | How do GPS and mapped path lengths differ? | Coordinates, numerical summation |
| GPS at a fixed point | Coordinate sequence | Spread, error ellipse | How does the built environment change GPS spread? | Covariance matrices, statistics |
| Compass + GPS/map | Bearing, position | Triangulated position | How does position uncertainty grow for small intersection angles? | Trigonometry, error propagation |
| ESP32-CAM + image analysis | Pixel values, image time | Area, spreading rate | How can dye spreading be modelled? | Scaling, power models, regression |
| Raspberry Pi + sensor logger | Several time series | Correlation, lag | How are temperature, CO₂ and ventilation related over time? | Time series, correlation, lag |
Next steps
Develop the measurement idea into a complete investigation.
Twelve complete project packages
Research question, raw data, derived quantity, mathematics, controls and suitable equipment.
Open project packagesAnalyse measurement data mathematically
Rates, integrals, calibration, uncertainty, residuals and data-table structure.
Plan the analysisView all measurement tools
Real product photographs, uses, data types, limitations and topic examples.
Open measurement tools