Uncertainty

This activity helps students understand the difference between signal and noise in a physical measurement. It develops critical thinking when faced with data and the statistical approach in experimental sciences.

Every measurement in science contains uncertainty. Even the most precise instruments in the world produce readings that fluctuate slightly around the true value due to electronic noise, environmental vibrations, and fundamental physical limits. Understanding, quantifying, and managing this uncertainty is not just a technical skill but a core principle of the scientific method: a result without an uncertainty estimate is essentially meaningless. The gyroscope in a smartphone provides an excellent case study. When the phone sits perfectly still, the gyroscope should read zero angular velocity. Instead, it produces a stream of small random values fluctuating around zero, the sensor's noise floor. When the phone actually rotates, the signal must be large enough to rise clearly above this noise. The boundary between noise and signal, characterized by the standard deviation of the noise distribution, defines the sensor's minimum detectable signal. This experiment teaches students to analyze this boundary using histograms and standard deviation, fundamental tools of experimental physics.

Learning objectives:

The student compares the data from the FizziQ gyroscope in two situations: smartphone stationary then rotating. By analyzing the histograms of the two measurements, the student observes the difference in distribution and standard deviation and deduces how to differentiate a real signal from the background noise inherent to any measuring instrument.

Level:

High school

FizziQ

Author:

Duration (minutes) :

25

What students will do :

- Record gyroscope data in two conditions: stationary (noise only) and rotating (signal + noise)
- Construct histograms of both datasets and compare their distributions
- Calculate the standard deviation of the noise distribution as a measure of sensor precision
- Determine the signal-to-noise ratio for the rotation measurement
- Understand the concepts of precision, accuracy, and minimum detectable signal

Scientific concepts:

- Statistical distribution of measurements
- Signal-to-noise ratio
- Standard deviation and variance
- MEMS gyroscope
- Accuracy vs Precision

Sensors:

- Gyroscope (angular velocity, Z-axis rotation)

What is required:

- Smartphone with the FizziQ application
- Clear space to turn safely
- Optional: turntable for additional comparison
- FizziQ experience notebook

Experimental procedure:

Open FizziQ and select the Gyroscope sensor (Z-axis angular velocity, in degrees per second or radians per second).
Place the smartphone on a flat, stable surface. Ensure it is completely stationary.
Record data for 20 seconds at the maximum sampling rate. This is your noise measurement.
Save this recording. Without looking at the data yet, predict: what should the average value be? What should the distribution look like?
Now pick up the phone and rotate it slowly and steadily around the Z-axis (yaw rotation) for 20 seconds. This is your signal measurement.
Save this recording.
Examine the noise data (stationary). Plot a histogram of the values. It should form a bell-shaped curve centered near zero.
Calculate the mean and standard deviation (σ) of the noise data.
Examine the signal data (rotating). Plot its histogram. It should be shifted away from zero and may be broader.
Calculate the mean and standard deviation of the signal data.
Calculate the signal-to-noise ratio (SNR): SNR = |mean_signal| / σ_noise.
Discuss: a signal is generally considered detectable when SNR > 2-3 (the signal exceeds 2-3 standard deviations of the noise). Based on your noise level, what is the minimum rotation speed the gyroscope can reliably detect?

Expected results:

The stationary gyroscope should produce values fluctuating around zero with a standard deviation of typically 0.01-0.10 °/s for a modern smartphone. The histogram should approximate a Gaussian (normal) distribution. During slow rotation, the mean shifts to a non-zero value (e.g., 5-50 °/s depending on rotation speed), and the distribution shifts accordingly. The SNR for a moderate rotation should be 10-100, well above the detection threshold. The minimum detectable rotation speed is approximately 2-3 times the noise standard deviation, typically 0.03-0.30 °/s. Students should clearly see the difference between the noise-only and signal+noise distributions.

Scientific questions:

- What is the physical origin of the noise in the gyroscope readings when the phone is stationary?
- Why is the standard deviation a good measure of sensor precision?
- What does a signal-to-noise ratio of 1 mean? Why is SNR > 3 typically required for reliable detection?
- How does averaging multiple measurements improve the effective precision of a sensor?
- What is the difference between precision (random error) and accuracy (systematic error)?
- How do these concepts apply to measurements in other fields of science?

Scientific explanations:

A smartphone's gyroscope is a MEMS (Micro Electro Mechanical Systems) sensor that measures the rotation speed around the three axes. Even when the device is perfectly still, the sensor records small, random fluctuations around zero, consisting mainly of electronic noise and environmental micro-vibrations.

These fluctuations generally follow a normal (Gaussian) distribution centered on zero, characterized by its standard deviation σ. This standard deviation defines the limit sensitivity of the sensor: any signal whose amplitude is less than 2-3σ risks being indistinguishable from the noise.

When the smartphone is rotated, a real signal is added to the background noise. The histogram then changes considerably: the distribution widens, shifts, and can even become bimodal or multimodal depending on the type of movement.

The standard deviation increases significantly, reflecting the wider dispersion of the measured values. In metrology, there are two fundamental concepts: precision (reproducibility of measurements, linked to the standard deviation) and accuracy (proximity to the real value).

A sensor can be precise but inaccurate if its measurements are consistent with each other but offset from reality (systematic bias). The experiment with a turntable allows us to deepen the analysis by comparing a regular movement (constant rotation) to a naturally more variable human movement.

Modern smartphone gyroscopes achieve a precision of around ±0.1-0.5°/s, sufficient for many applications such as games or augmented reality.

Extension activities:

Frequently asked questions:

Q: My stationary gyroscope shows a non-zero mean. Is the sensor broken?
R: A small systematic offset (bias) is normal for MEMS gyroscopes. This is a systematic error (accuracy issue), separate from the random fluctuations (precision issue). The standard deviation of the fluctuations is what matters for precision.

Q: The histogram does not look perfectly bell-shaped. Is this a problem?
R: With a limited number of data points, the histogram will show some irregularity. Collect more data (longer recording) for a smoother histogram. Small deviations from Gaussian shape are normal.

Q: What is the difference between signal-to-noise ratio and measurement uncertainty?
R: SNR compares the magnitude of the signal to the noise level. Measurement uncertainty includes both random noise and systematic errors. A high SNR means the signal is clearly detectable; low uncertainty means the measurement is both precise and accurate.

Q: Why is 2-3 standard deviations the threshold for detection?
R: In a Gaussian distribution, about 95% of random noise falls within ±2σ and 99.7% within ±3σ. A signal exceeding 3σ has less than a 0.3% probability of being caused by noise alone.