Probability and measurement uncertainty

• Summarize repeated measurements, separate random spread from bias, and use uncertainty to make a conservative threshold decision.

• units-estimation

Repeated measurements vary because the world and the instrument vary. A camera's range estimate may jitter with texture; a ruler reading changes with viewing angle. Probability gives language for variation, while uncertainty states what is not known about a reported value. Neither means “anything could happen.” They support bounded decisions.

For measurements x_i, the arithmetic mean is x̄ = (1/n)Σx_i. Deviations from the mean reveal spread; the sample standard deviation s = sqrt(Σ(x_i−x̄)²/(n−1)) is one common summary when repeated values are reasonably comparable. Range and median can also be useful, especially for small datasets or outliers. Precision is repeatability; accuracy is closeness to the relevant reference. Closely clustered readings can all share a calibration bias.

Suppose a desk robot must stop at least 100 mm from an obstacle. A sensor averaging 112 mm with several millimetres of noise should not use a 100 mm trigger merely because the mean exceeds it. A threshold can include observed spread, latency, and braking distance. Choosing 130 mm may trade workspace for robustness. The build decision should name that tradeoff and the evidence behind the margin.

A frequent misconception is that averaging removes every error. Averaging reduces independent random noise but preserves systematic offset. Another is interpreting a standard deviation as a guaranteed bound; it is a summary, not an absolute limit, and its probability meaning depends on a model. Record sample count, conditions, resolution, and actual observations. Do not invent confidence from five tidy readings. When stakes rise, improve the reference, collect data across operating conditions, and design for failure rather than polishing decimals.

Problem

A distance sensor returns 118, 121, 119, 122, and 120 mm for a fixed target. Find the mean and a simple half-range uncertainty summary.

Steps

1. Sum readings: 118 + 121 + 119 + 122 + 120 = 600 mm.
2. Divide by five: mean = 120 mm.
3. Range is 122 − 118 = 4 mm; half-range is 2 mm.
4. Report the descriptive result as 120 mm ± 2 mm half-range for these five trials, not as a universal guarantee.

Result

The sample mean is 120 mm, with observed values spanning 118–122 mm.

Sanity check

The mean lies inside the observed range and equals the visibly balanced center of these readings.

A miscalibrated sensor reads exactly 5 mm too high on every trial. Taking 100 readings and averaging mainly does what?

A. Reduces random scatter but leaves the 5 mm bias B. Eliminates the 5 mm bias C. Guarantees the true value lies within 1 mm D. Makes the sensor resolution finer

Your answer: ______________________________

Reasoning:

Goal

Measure repeatability and expose bias in a simple camera-based or ruler-based distance estimate.

Equipment

• ruler with millimetre marks
• fixed small object
• masking tape
• phone camera
• spreadsheet or calculator

Procedure

1. Tape the object at a ruler reference and take ten independent readings, repositioning your viewpoint each time.
2. Calculate mean, minimum, maximum, and range; keep the raw actual readings.
3. Compare the mean with the taped reference and discuss spread separately from offset.
4. Deliberately start the ruler at a worn or offset edge instead of the zero mark; diagnose the common shift that averaging fails to remove.
5. Choose a clearance margin for an imaginary moving camera carriage and justify it from observed spread plus a stated extra allowance.

What to observe

• Expected: viewpoint changes create some random spread; actual spread and any mean offset must be reported, even if zero to the ruler's resolution.
• Expected fault clue: the offset-edge readings may remain tightly grouped while being systematically wrong.

Transfer challenge

Design a short repeatability test for the printed width of five nominally identical parts and state how the result would affect a fit clearance.

Cautions

• Keep the phone supported away from table edges.
• This exercise characterizes a setup; it does not certify a safety-critical stopping system.

Learner: ____________________ Date: ____________________

Evidence artifact or file: ____________________________________________

Notes: