Evaluate the quality of a sensor

This activity helps students understand the concepts of measurement precision and uncertainty. It develops critical thinking regarding measuring instruments and introduces the statistical analysis of data.

Every scientific measurement comes with uncertainty. When a bathroom scale shows 70.0 kg, the true value might be anywhere from 69.5 to 70.5 kg. Understanding, quantifying, and minimizing measurement uncertainty is one of the foundational skills of experimental science. The accelerometer in a smartphone, despite its remarkable miniaturization, is subject to the same fundamental limitations as any measuring instrument: electronic noise, thermal drift, quantization error, and sensitivity to environmental vibrations. When the phone sits perfectly still on a table, the accelerometer should theoretically read a constant value equal to g. In reality, the readings fluctuate around this value, forming a distribution that reveals the sensor's precision. By recording these fluctuations and analyzing their statistical properties, students learn to characterize sensor quality using standard deviation, histograms, and normal distributions, tools that are essential in every branch of science and engineering.

Learning objectives:

The student measures the absolute acceleration of a stationary smartphone for 20 seconds then studies the dispersion of the values obtained using FizziQ statistical tools. By analyzing the histogram of the data and calculating the standard deviation, the student evaluates the accuracy of the accelerometer and compares its results with those obtained on other smartphone models.

Level:

High school

FizziQ

Author:

Duration (minutes) :

25

What students will do :

- Record a time series of accelerometer measurements with the smartphone at rest
- Analyze the distribution of measured values using histograms and statistical tools
- Calculate the mean, standard deviation, and relative uncertainty of the accelerometer
- Understand the difference between accuracy (systematic error) and precision (random error)
- Evaluate whether the measurement noise follows a normal (Gaussian) distribution

Scientific concepts:

- Normal distribution
- Measurement uncertainty
- Standard deviation
- Sensor accuracy
- Statistical analysis

Sensors:

- Accelerometer (absolute acceleration)

What is required:

- Smartphone with the FizziQ application
- Stable surface to place the smartphone
- Possibly several smartphones of different models for comparison
- FizziQ experience notebook
- Optional: spreadsheet software for additional analysis

Experimental procedure:

Open FizziQ and select the Accelerometer sensor. Choose absolute acceleration for a single-value measurement.
Place the smartphone on a flat, stable surface (a solid table, not a flexible desk or soft surface). Ensure nothing touches or vibrates near the phone.
Start recording and collect data for at least 20 seconds at the highest available sampling rate.
Stop recording and note the total number of data points collected.
Examine the raw acceleration-versus-time graph. The values should fluctuate slightly around a mean close to 9.81 m/s².
Use FizziQ's statistical tools to calculate the mean (average) and standard deviation of the recorded data.
Record these values. The standard deviation represents the precision of the sensor (random uncertainty).
If available, view the histogram of the data. It should approximate a bell curve (Gaussian distribution).
Calculate the relative uncertainty: (standard deviation / mean) × 100%. A good smartphone accelerometer should show relative uncertainty below 0.5%.
Compare the measured mean with the accepted value of g at your location. The difference between them represents the accuracy (systematic error) of the sensor.
If you have access to another smartphone, repeat the same measurement and compare the means and standard deviations of the two devices.
Discuss: if you average N measurements, the uncertainty of the mean decreases by a factor of √N. Calculate how many measurements would be needed to achieve a relative uncertainty of 0.01%.

Expected results:

Typical smartphone accelerometers show a standard deviation of 0.02-0.10 m/s² when stationary, corresponding to a relative precision of 0.2-1.0%. The mean value typically falls within 0.1-0.3 m/s² of the accepted value of g, reflecting the sensor's accuracy. The histogram should approximate a Gaussian distribution, though some sensors may show slight asymmetry or discrete quantization steps. Higher sampling rates may reveal high-frequency noise that is averaged out at lower rates. Different smartphone models can show significantly different precision levels. Environmental vibrations (nearby machinery, traffic, people walking) will increase the apparent standard deviation and may distort the Gaussian shape.

Scientific questions:

- What is the difference between the precision and accuracy of a measuring instrument?
- Why does the standard deviation decrease when you average more measurements?
- What factors contribute to the random fluctuations observed in the accelerometer readings?
- How could you improve the accuracy of the accelerometer (reduce systematic error)?
- Why is the Gaussian (normal) distribution so common in measurement science?
- How would you determine whether a sensor is suitable for a particular experiment?

Scientific explanations:

The accelerometer of a smartphone at rest should theoretically measure a constant value (around 9.81 m/s² for absolute acceleration, due to gravity). In reality, measurements fluctuate slightly around this value due to several factors: electronic noise from the sensor, minute vibrations from the environment, and intrinsic resolution limits of the MEMS sensor.

These fluctuations generally follow a normal (Gaussian) distribution, characterized by its mean μ and its standard deviation σ. The standard deviation quantifies the dispersion of measurements and is an excellent indicator of sensor accuracy: the lower σ, the more precise the sensor.

FizziQ allows direct visualization of this distribution via the histogram of the measurements. For an ideal sensor, the curve should be narrow and centered on the actual value.

In modern smartphones, accelerometers typically have an accuracy of ±0.01 to ±0.05 m/s², but this value varies between models and manufacturers. The quality of a sensor does not necessarily correlate with the price of the smartphone; some entry-level models may have excellent accelerometers.

Other statistical indicators such as skewness and kurtosis can provide additional information on the nature of sensor errors. This experiment illustrates the fundamental concepts of metrology, the science of measurement, and shows the importance of critical evaluation of measuring instruments in experimental science.

Extension activities:

Frequently asked questions:

Q: My standard deviation is much larger than expected. What could cause this?
R: Environmental vibrations are the most common cause. Ensure the phone is on a solid, isolated surface. Nearby construction, traffic, or even people walking can introduce measurable vibrations. Also check that no apps are causing the phone to vibrate (disable notifications).

Q: The mean value is not exactly 9.81 m/s². Does this mean the sensor is broken?
R: No. The actual value of g varies slightly with latitude and altitude (from about 9.78 to 9.83 m/s²), and sensor calibration introduces additional systematic offsets. Deviations of up to 0.2 m/s² from the nominal value are common and do not indicate a faulty sensor.

Q: Why does the histogram not look perfectly Gaussian?
R: With a limited number of data points (a few thousand), the histogram will show some irregularity. Additionally, sensor quantization (discrete output steps) can create peaks in the histogram. Environmental noise may also contribute non-Gaussian tails.

Q: How many measurements do I need for reliable statistics?
R: Generally, at least 100-200 data points are needed for a meaningful histogram and reliable standard deviation estimate. With 20 seconds of recording at 100 Hz, you get 2000 points, which is more than sufficient.