← Tutorials

Tutorial Series · Part 6 of 27

Measurement Noise: Why One Reading Is Not Enough

Introduce noise, bias, precision, accuracy and uncertainty ellipses.

By

  • uncertainty
  • statistics
  • sensors
Series Context

Navigation from First Principles

This article is part of an ordered course. Use the previous and next links to stay in sequence.

Progress Part 6 of 27
Previous Dead Reckoning: Navigating with Speed and Direction Next Combining Measurements: The Simplest Sensor Fusion Demo

A sensor reading is not a fact. It is evidence. The reading may contain random noise, bias, scale-factor error, environmental effects and modelling assumptions. GPS.gov notes that user GPS accuracy depends on satellite geometry, signal blockage, atmospheric conditions and receiver design, not just the satellites themselves.1 Analog Devices similarly lists null bias, scale-factor error and noise as major inertial-sensor error sources.2

The problem

Suppose a receiver reports a position fix. Should you trust it? A single dot on a map hides many questions:

  • Is the error random or biased?
  • Is the sensor precise but wrong?
  • Is it accurate on average but noisy instant to instant?
  • Has the environment changed?

The model

A simple scalar measurement model is:

z=x+b+ηz = x + b + \eta

where zz is measured value, xx is the true value, bb is bias and η\eta is random noise.

A common random-noise model is:

η∼N(0,σ2)\eta \sim \mathcal{N}(0,\sigma^2)

For a 2D position measurement:

z=x+b+η\mathbf{z}=\mathbf{x}+\mathbf{b}+\boldsymbol{\eta}

with covariance:

R=E[ηηT]\mathbf{R}=E[\boldsymbol{\eta}\boldsymbol{\eta}^T]

An uncertainty ellipse can be drawn from the eigenvalues and eigenvectors of R\mathbf{R}.

Interactive demo: measurement cloud playground

Use the playground to compare truth, noisy measurements and the statistics you estimate from the cloud.

Interactive Demo

Measurement Noise Playground

Drag the true position, then compare the raw measurement cloud with the sample mean, estimated bias, and covariance ellipse.

White marks truth, orange marks the sample mean, green shows the sample covariance ellipse, and red points are outliers.

Controls

Statistics

Mean error 0.0 m
Std deviation 0.0 m
RMS error 0.0 m
Bias estimate 0.0 N, 0.0 E

What to watch

  • More samples make the mean more stable but do not remove bias.
  • Increasing noise spreads the cloud and the estimated ellipse.
  • Increasing bias shifts the whole cloud away from the true point.
  • Outliers distort naive statistics much more than you might expect.

Try this

Set noise to 10 m10\,m and bias to zero. Increase sample count from 5 to 500. The average should move closer to truth. Then add a 20 m20\,m east bias. More samples now only reveal the bias more clearly; they do not remove it.

Where this breaks

Real GNSS and IMU errors are not always independent, Gaussian or stationary. Multipath, signal blockage, temperature drift and vibration can violate the simple model. But it is still the right first step because later filters need explicit measurement uncertainty.

Footnotes

  1. GPS.gov, “GPS Accuracy”. https://www.gps.gov/gps-accuracy - Official GPS accuracy FAQ covering satellite geometry, signal blockage, atmospheric conditions, receiver design, multipath, jamming and smartphone accuracy. Accessed 2026-06-11. ↩

  2. Analog Devices, “What are the major error sources for inertial sensors?”. https://www.analog.com/en/resources/faqs/faq_what_are_the_major_error_sources_for_inertial.html - Inertial sensor bias, scale factor and noise reference. Accessed 2026-06-11. ↩